Recent zbMATH articles in MSC 30https://www.zbmath.org/atom/cc/302022-05-16T20:40:13.078697ZWerkzeugA note on the zeros of approximations of the Ramanujan \(\Xi \)-functionhttps://www.zbmath.org/1483.111842022-05-16T20:40:13.078697Z"Chirre, Andrés"https://www.zbmath.org/authors/?q=ai:chirre.andres"Velásquez Castañón, Oswaldo"https://www.zbmath.org/authors/?q=ai:velasquez-castanon.oswaldoSummary: In this paper we review the study of the distribution of the zeros of certain approximations for the Ramanujan \(\Xi \)-function given by \textit{H. Ki} [Ramanujan J. 17, No. 1, 123--143 (2008; Zbl 1238.11080)], and we provide new proofs of his results. Our approach is motivated by the ideas of [the author, J. Anal. Math. 110, 67--127 (2010; Zbl 1203.11059)] in the study of the zeros of certain sums of entire functions with some condition of stability related to the Hermite-Biehler theorem.Uniqueness of \(L\) function with special class of meromorphic function under restricted sharing of setshttps://www.zbmath.org/1483.111902022-05-16T20:40:13.078697Z"Banerjee, Abhijit"https://www.zbmath.org/authors/?q=ai:banerjee.abhijit"Kundu, Arpita"https://www.zbmath.org/authors/?q=ai:kundu.arpitaSummary: The purpose of the paper is to rectify a series of errors occurred in [\textit{A. Banerjee} and \textit{A. Kundu}, Lith. Math. J. 61, No. 2, 161--179 (2021; Zbl 1469.11341); \textit{P. Sahoo} and \textit{A. Sarkar}, An. Științ. Univ. Al. I. Cuza Iași, Ser. Nouă, Mat. 66, No. 1, 81--92 (2020; Zbl 1474.11153); \textit{Q.-Q. Yuan} et al., Lith. Math. J. 58, No. 2, 249--262 (2018; Zbl 1439.11231)] for a particular situation. To get a fruitful solution and to overcome the issue, we introduce a new form of set sharing namely restricted set sharing, which is stronger than the usual one. We manipulate the newly introduced notion in this specific section of literature to resolve all the complications. Not only that we have subtly used the same sharing form to a well known unique range set [\textit{G. Frank} and \textit{M. Reinders}, Complex Variables, Theory Appl. 37, No. 1--4, 185--193 (1998; Zbl 1054.30519)] to settle a long time unsolved problem.The discrete case of the mixed joint universality for a class of certain partial zeta-functionshttps://www.zbmath.org/1483.111912022-05-16T20:40:13.078697Z"Kačinskaitė, Roma"https://www.zbmath.org/authors/?q=ai:kacinskaite.roma"Matsumoto, Kohji"https://www.zbmath.org/authors/?q=ai:matsumoto.kohjiAuthors' abstract: We give a new type of mixed discrete joint universality properties, which is satisfied by a wide class of zeta-functions. We study the universality for a certain modification of Matsumoto zeta-functions \(\varphi_h(s)\) and a collection of periodic Hurwitz zeta-functions \(\zeta (s;\alpha;\mathfrak B)\) under the condition that the common difference of arithmetical progression \(h > 0\) is such that \(\exp \{ \frac{2\pi}h\}\) is a rational number and parameter \(\alpha\) is a transcendental number.
Reviewer: Anatoly N. Kochubei (Kyïv)Value distribution of \(L\)-functions and a question of Chung-Chun Yanghttps://www.zbmath.org/1483.111922022-05-16T20:40:13.078697Z"Li, Xiao-Min"https://www.zbmath.org/authors/?q=ai:li.xiaomin"Yuan, Qian-Qian"https://www.zbmath.org/authors/?q=ai:yuan.qianqian"Yi, Hong-Xun"https://www.zbmath.org/authors/?q=ai:yi.hongxunSummary: We study the value distribution theory of \(L\)-functions and completely resolve a question from [\textit{L. Yang}, Value distribution theory. Berlin: Springer-Verlag; Beijing: Science Press (1993; Zbl 0790.30018)]. This question is related to \(L\)-functions sharing three finite values with meromorphic functions. The main result in this paper extends corresponding results from [\textit{B. Q. Li}, Proc. Am. Math. Soc. 138, No. 6, 2071--2077 (2010; Zbl 1195.30041)].Value distributions of \(L\)-functions concerning polynomial sharinghttps://www.zbmath.org/1483.111932022-05-16T20:40:13.078697Z"Mandal, Nintu"https://www.zbmath.org/authors/?q=ai:mandal.nintuSummary: We mainly study the value distributions of L-functions in the extended Selberg class. Concerning weighted sharing, we prove an uniqueness theorem when certain differential monomial of a meromorphic function share a polynomial with certain differential monomial of an L-function which improve and generalize some recent results due to \textit{F. Liu} et al. [Proc. Japan Acad., Ser. A 93, No. 5, 41--46 (2017; Zbl 1417.11144)], \textit{W.-J. Hao} and \textit{J.-F. Chen} [Open Math. 16, 1291--1299 (2018; Zbl 1451.11098)] and the author and \textit{N. K. Datta} [``Uniqueness of L-function and its certain differential monomial concerning small functions'', J. Math. Comput. Sci. 10, No. 5, 2155--2163 (2020; \url{doi:10.28919/jmcs/4836})].Special cases of the orbifold version of Zvonkine's \(r\)-ELSV formulahttps://www.zbmath.org/1483.140922022-05-16T20:40:13.078697Z"Borot, Gaëtan"https://www.zbmath.org/authors/?q=ai:borot.gaetan"Kramer, Reinier"https://www.zbmath.org/authors/?q=ai:kramer.reinier"Lewanski, Danilo"https://www.zbmath.org/authors/?q=ai:lewanski.danilo"Popolitov, Alexandr"https://www.zbmath.org/authors/?q=ai:popolitov.aleksandr"Shadrin, Sergey"https://www.zbmath.org/authors/?q=ai:shadrin.sergeySummary: We prove the orbifold version of Zvonkine's \(r\)-ELSV formula in two special cases: the case of \(r=2\) (completed 3-cycles) for any genus \(g\geq 0\) and the case of any \(r\geq 1\) for genus \(g=0\).Open and surjective mapping theorems for differentiable maps with critical pointshttps://www.zbmath.org/1483.260122022-05-16T20:40:13.078697Z"Li, Liangpan"https://www.zbmath.org/authors/?q=ai:li.liangpanSummary: Let \(\Omega\) be an open subset of \(\mathbb R^n\) \((n \ge 2)\), and let \(F : \Omega \rightarrow \mathbb R^n\) be a continuously differentiable map with countably many critical points. We show that \(F\) is an open map. Let \(G :\mathbb R^n \rightarrow \mathbb R^n\) \((n \ge 1)\) be a continuously differentiable map such that \(G(x) \rightarrow \infty\) as \(x \rightarrow \infty \). Then it is proved that \(G\) is surjective if and only if each connected component of the complement of the set of critical values of \(G\) contains at least one image of \(G\). Several applications of both theorems especially to complex analysis are presented.Nonlinear conditions for ultradifferentiabilityhttps://www.zbmath.org/1483.260232022-05-16T20:40:13.078697Z"Nenning, David Nicolas"https://www.zbmath.org/authors/?q=ai:nenning.david-nicolas"Rainer, Armin"https://www.zbmath.org/authors/?q=ai:rainer.armin"Schindl, Gerhard"https://www.zbmath.org/authors/?q=ai:schindl.gerhardSummary: A remarkable theorem of Joris states that a function \(f\) is \(C^{\infty}\) if two relatively prime powers of \(f\) are \(C^{\infty}\). Recently, Thilliez showed that an analogous theorem holds in Denjoy-Carleman classes of Roumieu type. We prove that a division property, equivalent to Joris's result, is valid in a wide variety of ultradifferentiable classes. Generally speaking, it holds in all dimensions for non-quasianalytic classes. In the quasianalytic case we have general validity in dimension one, but we also get validity in all dimensions for certain quasianalytic classes.Analog of the Hadamard theorem and related extremal problems on the class of analytic functionshttps://www.zbmath.org/1483.300012022-05-16T20:40:13.078697Z"Akopyan, R. R."https://www.zbmath.org/authors/?q=ai:akopyan.roman-razmikovichSummary: We study several related extremal problems for analytic functions in a finitely connected domain \(G\) with rectifiable Jordan boundary \(\Gamma \). A sharp inequality is established between values of a function analytic in \(G\) and weighted means of its boundary values on two measurable subsets \(\gamma_1\) and \(\gamma_0=\Gamma\setminus\gamma_1\) of the boundary:
\[ |f(z_0)|\leq\mathcal{C}\,\|f\|^{\alpha}_{L^q_{\varphi_1}(\gamma_1)}\, \|f\|^{\beta}_{L^p_{\varphi_0}(\gamma_0)},\quad z_0\in G,\quad 0<q,p\leq\infty.\]
The inequality is an analog of Hadamard's three-circle theorem and the Nevanlinna brothers' two-constant theorem. In the case of a doubly connected domain \(G\) and \(1\leq q,p\leq\infty \), we study the cases where the inequality provides the value of the modulus of continuity for a functional of analytic extension of a function from the part \(\gamma_1\) of the boundary to a given point of the domain. In these cases, the corresponding problem of optimal recovery of a function from its approximate boundary values on \(\gamma_1\) and the problem of the best approximation of a functional by bounded linear functionals are solved. The case of a simply connected domain \(G\) has been completely investigated previously.Some \(L^q\) inequalities for polynomialhttps://www.zbmath.org/1483.300022022-05-16T20:40:13.078697Z"Chanam, Barchand"https://www.zbmath.org/authors/?q=ai:chanam.barchand"Reingachan, N."https://www.zbmath.org/authors/?q=ai:reingachan.n"Devi, Khangembam Babina"https://www.zbmath.org/authors/?q=ai:devi.khangembam-babina"Devi, Maisnam Triveni"https://www.zbmath.org/authors/?q=ai:devi.maisnam-triveni"Krishnadas, Kshetrimayum"https://www.zbmath.org/authors/?q=ai:krishnadas.kshetrimayumSummary: Let \(p(z)\) be a polynomial of degree \(n\). Then Bernstein's inequality is
\[
{\max\limits_{|z|=1} |p'(z)| \leq n \max\limits_{|z|=1} |(z)|}.
\]
For \(q>0\), we denote
\[
\|p\|_q = \left\{\frac{1}{2\pi}\int_0^{2\pi} |p(e^{i\theta})|^q d\theta\right\}^{\frac{1}{q}},
\]
and a well-known fact from analysis gives
\[
\lim_{q\to\infty}\left\{\frac{1}{2\pi}\int_0^{2\pi}\big\vert p(e^{i\theta})\big\vert^q d\theta\right\}^{\frac{1}{q}}=\max_{\vert z\vert=1} \vert p(z)\vert.
\]
Above Bernstein's inequality was extended by \textit{A. Zygmund} [Proc. Lond. Math. Soc., II. Ser. 34, 392--400 (1932; Zbl 0005.35301)] into \(L^q\) norm by proving
\[
\|p'\|_q \leq n\|p\|_q, \quad q \geq 1.
\]
Let \(p(z) = a_0 + \sum_{\nu=\mu}^n a_\nu z^\nu,\) \(1 \leq \mu n,\) be a polynomial of degree n having no zero in \(|z| < k, k \geq 1.\) Then for \(0 < r \leq R \leq k\), \textit{A. Aziz} and \textit{B. A. Zargar} [Math. Inequal. Appl. 1, No. 4, 543--550 (1998; Zbl 0914.30002)] proved
\[
{\max\limits_{|z|=R} |p'(z)| \leq \frac{nR^{\mu-1}(R^\mu + k^\mu)^{\frac{n}{\mu}-1}}{(r^\mu + k^\mu)^{\frac{n}{\mu}}} \max\limits_{|z|=r} |p'(z)|}.
\]
In this paper, we obtain the \(L^q\) version of the above inequality for \(q > 0\). Further, we extend a result of \textit{A. Aziz} and \textit{W. M. Shah} [Math. Inequal. Appl. 7, No. 3, 379--391 (2004; Zbl 1061.30001)] into \(L^q\) analogue for \(q > 0\). Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.A generalization of the Polia-Szego and Makai inequalities for torsional rigidityhttps://www.zbmath.org/1483.300032022-05-16T20:40:13.078697Z"Gafiyatullina, L. I."https://www.zbmath.org/authors/?q=ai:gafiyatullina.l-i"Salakhudinov, R. G."https://www.zbmath.org/authors/?q=ai:salakhudinov.rustem-gumerovichSummary: We establish some generalizations of the classical inequalities by Polya-Szego and Makai about torsional rigidity of convex domains. The main idea of the proof is in using an exact isoperimetric inequality for Euclidean moments of domains. This inequality has a wide class of extremal regions and is of independent interest.Bohr-type inequalities with one parameter for bounded analytic functions of Schwarz functionshttps://www.zbmath.org/1483.300042022-05-16T20:40:13.078697Z"Hu, Xiaojun"https://www.zbmath.org/authors/?q=ai:hu.xiaojun"Wang, Qihan"https://www.zbmath.org/authors/?q=ai:wang.qihan"Long, Boyong"https://www.zbmath.org/authors/?q=ai:long.boyongSummary: In this article, some Bohr-type inequalities with one parameter or involving convex combination for bounded analytic functions of Schwarz functions are established. Some previous inequalities are generalized. All the results are sharp.The Bohr inequality for the generalized Cesáro averaging operatorshttps://www.zbmath.org/1483.300052022-05-16T20:40:13.078697Z"Kayumov, Ilgiz R."https://www.zbmath.org/authors/?q=ai:kayumov.ilgiz-rifatovich"Khammatova, Diana M."https://www.zbmath.org/authors/?q=ai:khammatova.diana-m"Ponnusamy, Saminathan"https://www.zbmath.org/authors/?q=ai:ponnusamy.saminathanSummary: The main aim of this paper is to prove a generalization of the classical Bohr theorem and as an application, we obtain a counterpart of Bohr theorem for the generalized Cesáro operator.Some Bohr-type inequalities with one parameter for bounded analytic functionshttps://www.zbmath.org/1483.300062022-05-16T20:40:13.078697Z"Wu, Le"https://www.zbmath.org/authors/?q=ai:wu.le"Wang, Qihan"https://www.zbmath.org/authors/?q=ai:wang.qihan"Long, Boyong"https://www.zbmath.org/authors/?q=ai:long.boyongSummary: In this paper, for bounded analytic function, some Bohr-type inequalities with one parameter or involving convex combination are established. Most of the results are sharp. Some previous results are generalized.A bilogarithmic criterion for the existence of a regular minorant that does not satisfy the bang conditionhttps://www.zbmath.org/1483.300072022-05-16T20:40:13.078697Z"Gaisin, R. A."https://www.zbmath.org/authors/?q=ai:gaisin.rashit-akhtyarovichSummary: Problems of constructing regular majorants for sequences \(\mu=\{\mu_n\}_{n=0}^{\infty}\) of numbers \(\mu_n\ge0\) that are the Taylor coefficients of integer transcendental functions of minimal exponential type are investigated. A new criterion for the existence of regular minorants of associated sequences of the extended half-line \((0,+\infty]\) in terms of the Levinson bilogarithmic condition \(M=\{\mu_n^{-1}\}_{n=0}^{\infty}\) is obtained. The result provides a necessary and sufficient condition for the nontriviality of the important subclass defined by J. A. Siddiqi. The proofs of the main statements are based on properties of the Legendre transform.Real roots of random polynomials with coefficients of polynomial growth: a comparison principle and applicationshttps://www.zbmath.org/1483.300082022-05-16T20:40:13.078697Z"Do, Yen Q."https://www.zbmath.org/authors/?q=ai:do.yen-qSummary: This paper seeks to further explore the distribution of the real roots of random polynomials with non-centered coefficients. We focus on polynomials where the typical values of the coefficients have power growth and count the average number of real zeros. Almost all previous results require coefficients with zero mean, and it is non-trivial to extend these results to the general case. Our approach is based on a novel comparison principle that reduces the general situation to the mean-zero setting. As applications, we obtain new results for the Kac polynomials, hyperbolic random polynomials, their derivatives, and generalizations of these polynomials. The proof features new logarithmic integrability estimates for random polynomials (both local and global) and fairly sharp estimates for the local number of real zeros.Riesz projection and bounded mean oscillation for Dirichlet serieshttps://www.zbmath.org/1483.300092022-05-16T20:40:13.078697Z"Konyagin, Sergei"https://www.zbmath.org/authors/?q=ai:konyagin.sergey-v"Queffélec, Hervé"https://www.zbmath.org/authors/?q=ai:queffelec.herve"Saksman, Eero"https://www.zbmath.org/authors/?q=ai:saksman.eero"Seip, Kristian"https://www.zbmath.org/authors/?q=ai:seip.kristianSummary: We prove that the norm of the Riesz projection from \(L^\infty (\mathbb{T}^n)\) to \(L^p(\mathbb{T}^n)\) is \(1\) for all \(n\ge 1\) only if \(p\le 2\), thus solving a problem posed by \textit{J. Marzo} and the fourth author [Bull. Sci. Math. 135, No. 3, 324--331 (2011; Zbl 1221.42013)]. This shows that \(H^p(\mathbb{T}^{\infty})\) does not contain the dual space of \(H^1(\mathbb{T}^{\infty})\) for any \(p > 2\). We then note that the dual of \(H^1(\mathbb{T}^{\infty})\) contains, via the Bohr lift, the space of Dirichlet series in BMOA of the right half-plane. We give several conditions showing how this BMOA space relates to other spaces of Dirichlet series. Finally, relating the partial sum operator for Dirichlet series to Riesz projection on \(\mathbb{T}\), we compute its \(L^p\) norm when \(1 < p < \infty\), and we use this result to show that the \(L^\infty\) norm of the \(N\) th partial sum of a bounded Dirichlet series over \(d\)-smooth numbers is of order \(\log\log N\).Bernstein-Walsh type inequalities for derivatives of algebraic polynomialshttps://www.zbmath.org/1483.300102022-05-16T20:40:13.078697Z"Abdullayev, Fahreddin G."https://www.zbmath.org/authors/?q=ai:abdullayev.fahreddin-g"Gün, Cevahir D."https://www.zbmath.org/authors/?q=ai:gun.cevahir-dSummary: In this work, we study Bernstein-Walsh-type estimations for the derivative of an arbitrary algebraic polynomial in regions with piecewise smooth boundary without cusps of the complex plane. Also, estimates are given on the whole complex plane.Inequalities for the derivative of a polynomial with restricted zeroshttps://www.zbmath.org/1483.300112022-05-16T20:40:13.078697Z"Ahanger, Uzma Mubeen"https://www.zbmath.org/authors/?q=ai:ahanger.uzma-mubeen"Shah, W. M."https://www.zbmath.org/authors/?q=ai:shah.wali-mohammadSummary: For a polynomial \(p(z)\) of degree \(n\), it is known that
\[
\begin{aligned}\max_{|z|=1}|p'(z)|\leq \frac{n}{1+k}\max_{|z|=1}|p(z)|,\end{aligned}
\]
if \(p(z)\neq 0\) in \(|z|<k,k \geq 1\) and
\[
\begin{aligned}\max_{|z|=1}|p'(z)|\geq \frac{n}{1+k}\max_{|z|=1}|p(z)|,\end{aligned}
\]
if \(p(z)\neq 0\) for \(|z|>k\), \(k \leq 1\). In this paper, we assume that there is a zero of multiplicity \(s\), \(s <n\) at a point inside \(|z|=1\) and prove some generalizations and improvements of these inequalities.Inequalities for complex rational functionshttps://www.zbmath.org/1483.300122022-05-16T20:40:13.078697Z"Bidkham, M."https://www.zbmath.org/authors/?q=ai:bidkham.mahmood"Khojastehnezhad, E."https://www.zbmath.org/authors/?q=ai:khojastehnezhad.elaheSummary: For a rational function \(r(z) = p(z)/H(z)\) all zeros of which are in \(|z| \leq 1\), it is known that
\[ \left|r'(z)\right|\ge \frac{1}{2}\left|B'(z)\right|\left|r(z)\right| \text{ for }\left|z\right|=1,\]
where \(H(z)={\prod}_{j=1}^n\left(z-{c}_j\right)\), \(\left|{c}_j\right|>1\), \(n\) is a positive integer, \(B(z) = H^\ast (z)/H(z)\), and \({H}^{\ast }(z)={z}^n\overline{H\left(1/\overline{z}\right)}\). We improve the above-mentioned inequality for the rational function \(r(z)\) with all zeros in \(|z| \leq 1\) and a zero of order \(s\) at the origin. Our main results refine and generalize some known rational inequalities.On an inequality of S. Bernsteinhttps://www.zbmath.org/1483.300132022-05-16T20:40:13.078697Z"Chanam, Barchand"https://www.zbmath.org/authors/?q=ai:chanam.barchand"Devi, Khangembam Babina"https://www.zbmath.org/authors/?q=ai:devi.khangembam-babina"Krishnadas, Kshetrimayum"https://www.zbmath.org/authors/?q=ai:krishnadas.kshetrimayum"Devi, Maisnam Triveni"https://www.zbmath.org/authors/?q=ai:devi.maisnam-triveni"Ngamchui, Reingachan"https://www.zbmath.org/authors/?q=ai:ngamchui.reingachan"Singh, Thangjam Birkramjit"https://www.zbmath.org/authors/?q=ai:singh.thangjam-birkramjitSummary: If \(p(z) = \sum_{\nu=0}^n a_\nu z^\nu\) is a polynomial of degree \(n\) having all its zeros on \(|z|=k\), \(k\leq 1,\) then \textit{N. K. Govil} [J. Math. Phys. Sci. 14, 183--187 (1980; Zbl 0444.30007)] proved that
\[
\max\limits_{|z|=1}|p'(z)| \leq \frac{n}{k^n+k^{n-1}}\max\limits_{|z|=1}|p(z)|.
\]
In this paper, by involving certain coefficients of \(p(z)\), we not only improve the above inequality but also improve a result proved by \textit{K. K. Dewan} and \textit{A. Mir} [Southeast Asian Bull. Math. 31, No. 4, 691--695 (2007; Zbl 1150.30001)].Matrix orthogonality in the plane versus scalar orthogonality in a Riemann surfacehttps://www.zbmath.org/1483.300142022-05-16T20:40:13.078697Z"Charlier, Christophe"https://www.zbmath.org/authors/?q=ai:charlier.christopheSummary: We consider a non-Hermitian matrix orthogonality on a contour in the complex plane. Given a diagonalizable and rational matrix valued weight, we show that the Christoffel-Darboux (CD) kernel, which is built in terms of matrix orthogonal polynomials, is equivalent to a scalar valued reproducing kernel of meromorphic functions in a Riemann surface. If this Riemann surface has genus \(0\), then the matrix valued CD kernel is equivalent to a scalar reproducing kernel of polynomials in the plane. Interestingly, this scalar reproducing kernel is not necessarily a scalar CD kernel. As an application of our result, we show that the correlation kernel of certain doubly periodic lozenge tiling models admits a double contour integral representation involving only a scalar CD kernel. This simplifies a formula of Duits and Kuijlaars.Homeomorphisms of \(S^1\) and factorizationhttps://www.zbmath.org/1483.300152022-05-16T20:40:13.078697Z"Dalthorp, Mark"https://www.zbmath.org/authors/?q=ai:dalthorp.mark"Pickrell, Doug"https://www.zbmath.org/authors/?q=ai:pickrell.dougSummary: For each \(n>0\) there is a one complex parameter family of homeomorphisms of the circle consisting of linear fractional transformations ``conjugated by \(z\to z^n\)''. We show that these families are free of relations, which determines the structure of ``the group of homeomorphisms of finite type''. We next consider factorization for more robust groups of homeomorphisms. We refer to this as root subgroup factorization (because the factors correspond to root subgroups). We are especially interested in how root subgroup factorization is related to triangular factorization (i.e., conformal welding) and correspondences between smoothness properties of the homeomorphisms and decay properties of the root subgroup parameters. This leads to interesting comparisons with Fourier series and the theory of Verblunsky coefficients.\(L^r\) inequalities for the derivative of a polynomialhttps://www.zbmath.org/1483.300162022-05-16T20:40:13.078697Z"Devi, Khangembam Babina"https://www.zbmath.org/authors/?q=ai:devi.khangembam-babina"Krishnadas, Kshetrimayum"https://www.zbmath.org/authors/?q=ai:krishnadas.kshetrimayum"Chanam, Barchand"https://www.zbmath.org/authors/?q=ai:chanam.barchandSummary: Let \(p(z)\) be a polynomial of degree \(n\) having no zero in \(|z|< k\), \(k\leq 1\), then \textit{N. K. Govil} [Proc. Natl. Acad. Sci. India, Sect. A 50, 50--52 (1980; Zbl 0493.30003)] proved
\[
\max\limits_{|z|=1}|p'(z)|\leq \frac{n}{1+k^n}\max\limits_{|z|=1}|p(z)|,
\]
provided \(|p'(z)|\) and \(|q'(z)|\) attain their maxima at the same point on the circle \(|z|=1\), where
\[
q(z)=z^n\overline{p\left(\frac{1}{\overline{z}}\right)}.
\]
In this paper, we not only obtain an integral mean inequality for the above inequality but also extend an improved version of it into \(L^r\) norm.Bounds for the derivative of a certain class of rational functionshttps://www.zbmath.org/1483.300172022-05-16T20:40:13.078697Z"Gupta, Preeti"https://www.zbmath.org/authors/?q=ai:gupta.preeti"Hans, Sunil"https://www.zbmath.org/authors/?q=ai:hans.sunil"Mir, Abdullah"https://www.zbmath.org/authors/?q=ai:mir.abdullahSummary: In this paper, we shall obtain the bounds for the derivative of a rational function in the supremum norm on the unit circle in both the directions by involving the moduli of all its zeros. The obtained results strengthen some recently proved results.On the location of zeros of polynomialshttps://www.zbmath.org/1483.300182022-05-16T20:40:13.078697Z"Kumar, Prasanna"https://www.zbmath.org/authors/?q=ai:kumar.prasanna-v-k"Dhankhar, Ritu"https://www.zbmath.org/authors/?q=ai:dhankhar.rituSummary: In this paper, we discuss the necessary and sufficient conditions for a polynomial \(P(z)\) to have all its zeros inside the open unit disc. These results involve two associated polynomials namely, the derivative of the reciprocal polynomial of \(P(z)\) and the reciprocal of the derivative of \(P(z)\). We also derive some generalizations of the classical Theorem of Laguerre.Extremal problems of Bernstein-type and an operator preserving inequalities between polynomialshttps://www.zbmath.org/1483.300192022-05-16T20:40:13.078697Z"Milovanović, G. V."https://www.zbmath.org/authors/?q=ai:milovanovic.gradimir-v"Mir, A."https://www.zbmath.org/authors/?q=ai:mir.abdullah"Hussain, A."https://www.zbmath.org/authors/?q=ai:hussain.adilSummary: Under consideration are the well-known extremal problems of Bernstein-type which relate the uniform norm between polynomials on the unit disk in the plane. We establish a few new inequalities in both directions for the generalized \({\mathcal{B}}_n \)-operator while accounting for the placement of the zeros of the underlying polynomials. Also, we obtain various estimates for the maximum modulus of a polynomial as well as some inequalities of Erdös-Lax type.A Turán-type inequality for polynomialshttps://www.zbmath.org/1483.300202022-05-16T20:40:13.078697Z"Mir, Abdullah"https://www.zbmath.org/authors/?q=ai:mir.abdullahSummary: In this paper, we consider the class of polynomials \(P(z):=\sum \limits_{j=0}^nc_jz^j\) having all zeros in a closed disk \(|z|\le k,\text{where}~ k\ge 1\) and obtain a result that improves and generalizes the results of Govil, Jain and others by using certain coefficients of \(P(z)\).Some integral inequalities for a polynomial with zeros outside the unit diskhttps://www.zbmath.org/1483.300212022-05-16T20:40:13.078697Z"Mir, Abdullah"https://www.zbmath.org/authors/?q=ai:mir.abdullahSummary: The goal of this paper is to generalize and refine some previous inequalities between the \(L^P\)- norms of the \(s^{\mathrm{th}}\) derivative and of the polynomial itself, in the case when the zeros are outside of the open unit disk.A note on a recent result: on the location of the zeros of polynomials (Lacunary type)https://www.zbmath.org/1483.300222022-05-16T20:40:13.078697Z"Mogbademu, Adesanmi Alao"https://www.zbmath.org/authors/?q=ai:mogbademu.adesanmi-alaoSummary: In this paper, we give some corrections and comments about a result which is contained in a published paper in [\textit{D. Tripathi} et al., Nonlinear Funct. Anal. Appl. 24, No. 3, 555--564 (2019; Zbl 1427.30005)].Weighted Chebyshev polynomials on compact subsets of the complex planehttps://www.zbmath.org/1483.300232022-05-16T20:40:13.078697Z"Novello, Galen"https://www.zbmath.org/authors/?q=ai:novello.galen"Schiefermayr, Klaus"https://www.zbmath.org/authors/?q=ai:schiefermayr.klaus"Zinchenko, Maxim"https://www.zbmath.org/authors/?q=ai:zinchenko.maximSummary: We study weighted Chebyshev polynomials on compact subsets of the complex plane with respect to a bounded weight function. We establish existence and uniqueness of weighted Chebyshev polynomials and derive weighted analogs of Kolmogorov's criterion, the alternation theorem, and a characterization due to Rivlin and Shapiro. We derive invariance of the Widom factors of weighted Chebyshev polynomials under polynomial pre-images and a comparison result for the norms of Chebyshev polynomials corresponding to different weights. Finally, we obtain a lower bound for the Widom factors in terms of the Szegő integral of the weight function and discuss its sharpness.
For the entire collection see [Zbl 1479.47003].Number of zeros of polar derivatives of polynomialshttps://www.zbmath.org/1483.300242022-05-16T20:40:13.078697Z"Ramulu, P."https://www.zbmath.org/authors/?q=ai:ramulu.p"Reddy, G. L."https://www.zbmath.org/authors/?q=ai:reddy.g-lakshma(no abstract)Generalizations and sharpenings of certain Bernstein and Turán types of inequalities for the polar derivative of a polynomialhttps://www.zbmath.org/1483.300252022-05-16T20:40:13.078697Z"Singh, Thangjam Birkramjit"https://www.zbmath.org/authors/?q=ai:singh.thangjam-birkramjit"Chanam, Barchand"https://www.zbmath.org/authors/?q=ai:chanam.barchandSummary: Let \(p(z)\) be a polynomial of degree \(n\). The polar derivative of \(p(z)\) with respect to a complex number \(\alpha\) is defined by
\[
D_\alpha p(z)=np(z)+(\alpha-z)p'(z).
\]
If \(p(z)\) has all its zeros in \(|z|\leq k\), \(k\geq 1\), then for \(|\alpha|\geq k\), \textit{A. Aziz} and \textit{N. A. Rather} [Math. Inequal. Appl. 1, No. 2, 231--238 (1998; Zbl 0911.30002)] proved
\[
\max\limits_{|z|=1}|D_\alpha p(z)|\geq n\left(\frac{|\alpha|-k}{1+k^n}\right)\max\limits_{|z|=1}|p(z)|.
\]
In this paper, we first improve as well as generalize the above inequality. Besides, we are able to prove an improvement of a result due to \textit{N. K. Govil} and \textit{G. N. McTume} [Acta Math. Hung. 104, No. 1--2, 115--126 (2004; Zbl 1060.30004)] and also prove an inequality for a subclass of polynomials having all its zeros in \(|z|\geq k\), \(k\leq 1\).Bernstien type inequalities for polynomials with restricted zeroshttps://www.zbmath.org/1483.300262022-05-16T20:40:13.078697Z"Wali, S. L."https://www.zbmath.org/authors/?q=ai:wali.shah-lubna"Shah, W. M."https://www.zbmath.org/authors/?q=ai:shah.wali-mohammadSummary: In this paper we prove results by using a simple but elegant techniques to improve and strengthen known generalisations and refinements of some widely known polynomial inequalities and thereby deduce useful corollaries from these results.The fundamental theorem of algebra and Liouville's theorem geometrically revisitedhttps://www.zbmath.org/1483.300272022-05-16T20:40:13.078697Z"Almira, Jose Maria"https://www.zbmath.org/authors/?q=ai:almira.jose-maria"Romero, Alfonso"https://www.zbmath.org/authors/?q=ai:romero.alfonsoSummary: If \(f(z)\) is either a polynomial with no zeroes or a bounded entire function, then a Riemannian metric \(g_f\) is constructed on the complex plane \(\mathbb{C}\). This metric \(g_f\) is shown to be flat and geodesically complete. Therefore, the Riemannian manifold \((\mathbb{C}, g_f)\) must be isometric to \((\mathbb{C}, |dz|^2)\), which implies that \(f(z)\) is a constant.Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomialshttps://www.zbmath.org/1483.300282022-05-16T20:40:13.078697Z"Beliaev, D."https://www.zbmath.org/authors/?q=ai:beliaev.dmitri-b"Muirhead, S."https://www.zbmath.org/authors/?q=ai:muirhead.stephen"Wigman, I."https://www.zbmath.org/authors/?q=ai:wigman.igorSummary: Beginning with the predictions of Bogomolny-Schmit for the random plane wave, in recent years the deep connections between the level sets of smooth Gaussian random fields and percolation have become apparent. In classical percolation theory a key input into the analysis of global connectivity are scale-independent bounds on crossing probabilities in the critical regime, known as Russo-Seymour-Welsh (RSW) estimates. Similarly, establishing RSW-type estimates for the nodal sets of Gaussian random fields is a major step towards a rigorous understanding of these relations.
The Kostlan ensemble is an important model of Gaussian homogeneous random polynomials. The nodal set of this ensemble is a natural model for a `typical' real projective hypersurface, whose understanding can be considered as a statistical version of Hilbert's 16th problem. In this paper we establish RSW-type estimates for the nodal sets of the Kostlan ensemble in dimension two, providing a rigorous relation between random algebraic curves and percolation. The estimates are uniform with respect to the degree of the polynomials, and are valid on all relevant scales; this, in particular, resolves an open question raised recently by Beffara-Gayet. More generally, our arguments yield RSW estimates for a wide class of Gaussian ensembles of smooth random functions on the sphere or the flat torus.Conformal welding for critical Liouville quantum gravityhttps://www.zbmath.org/1483.300292022-05-16T20:40:13.078697Z"Holden, Nina"https://www.zbmath.org/authors/?q=ai:holden.nina"Powell, Ellen"https://www.zbmath.org/authors/?q=ai:powell.ellenSummary: Consider two critical Liouville quantum gravity surfaces (i.e., \(\gamma\)-LQG for \(\gamma =2)\), each with the topology of \(\mathbb{H}\) and with infinite boundary length. We prove that there a.s. exists a conformal welding of the two surfaces, when the boundaries are identified according to quantum boundary length. This results in a critical LQG surface decorated by an independent \(\mathrm{SLE}_4\). Combined with the proof of uniqueness for such a welding, recently established by \textit{O. McEnteggart, J. Miller} and \textit{W. Qian} [``Uniqueness of the welding problem for SLE and Liouville quantum gravity'', Preprint, \url{arXiv:1809.02092}], this shows that the welding operation is well-defined. Our result is a critical analogue of \textit{S. Sheffield}'s quantum gravity zipper theorem [Ann. Probab. 44, No. 5, 3474--3545 (2016; Zbl 1388.60144)], which shows that a similar conformal welding for subcritical LQG (i.e., \(\gamma\)-LQG for \(\gamma\in (0,2))\) is well-defined.Time-reversal of multiple-force-point \(\mathrm{SLE}_{\kappa}(\underline{\rho})\) with all force points lying on the same sidehttps://www.zbmath.org/1483.300302022-05-16T20:40:13.078697Z"Zhan, Dapeng"https://www.zbmath.org/authors/?q=ai:zhan.dapeng.1|zhan.dapengSummary: We define intermediate \(\mathrm{SLE}_{\kappa}(\underline{\rho})\) and reversed intermediate \(\mathrm{SLE}_{\kappa}(\underline{\rho})\) processes using Appell-Lauricella multiple hypergeometric functions, and use them to describe the time-reversal of multiple-force-point chordal \(\mathrm{SLE}_{\kappa}(\underline{\rho})\) curves in the case that all force points are on the boundary and lie on the same side of the initial point, and \(\kappa\) and \(\underline{\rho}=(\rho_1,\dots,\rho_m)\) satisfy that either \(\kappa\in (0,4]\) and \(\sum_{{j=1}^k}{\rho_j}> -2\) for all \(1\le k\le m\), or \(\kappa\in (4,8)\) and \(\sum_{{j=1}^k}\rho_j\ge \frac{\kappa}{2}-2\) for all \(1\le k\le m\).Approximate properties of the \(p\)-Bieberbach polynomials in regions with simultaneously exterior and interior zero angleshttps://www.zbmath.org/1483.300312022-05-16T20:40:13.078697Z"Abdullayev, F. G."https://www.zbmath.org/authors/?q=ai:abdullayev.fahreddin-g"Imashkyzy, M."https://www.zbmath.org/authors/?q=ai:imashkyzy.meerim"Özkartepe, P."https://www.zbmath.org/authors/?q=ai:ozkartepe.naciye-pelinSummary: In this paper, we study the uniform convergence of \(p\)-Bieberbach polynomials in regions with a finite number of both interior and exterior zero angles at the boundary.Hankel, Toeplitz, and Hermitian-Toeplitz determinants for certain close-to-convex functionshttps://www.zbmath.org/1483.300322022-05-16T20:40:13.078697Z"Allu, Vasudevarao"https://www.zbmath.org/authors/?q=ai:allu.vasudevarao"Lecko, Adam"https://www.zbmath.org/authors/?q=ai:lecko.adam"Thomas, Derek K."https://www.zbmath.org/authors/?q=ai:thomas.derek-keithSummary: Let \(f\) be analytic in \(\mathbb{D}=\{z\in \mathbb{C} : |z|<1\}\), and be given by \(f(z) = z+\sum_{n=2}^\infty a_nz^n\). We give sharp bounds for the second Hankel determinant, some Toeplitz, and some Hermitian-Toeplitz determinants of functions in the class of Ozaki close-to-convex functions, together with a sharp bound for the Zalcman functional \(J_{2, 3}(f)\).Certain estimates of normalized analytic functionshttps://www.zbmath.org/1483.300332022-05-16T20:40:13.078697Z"Anand, Swati"https://www.zbmath.org/authors/?q=ai:anand.swati"Jain, Naveen Kumar"https://www.zbmath.org/authors/?q=ai:jain.naveen-kumar"Kumar, Sushil"https://www.zbmath.org/authors/?q=ai:kumar.sushilSummary: Let \(\phi\) be a normalized convex function defined on open unit disk \(\mathbb{D}\). For a unified class of normalized analytic functions which satisfy the second order differential subordination \(f'(z)+\alpha zf''(z)\prec\varphi(z)\) for all \(z\in\mathbb{D}\), we investigate the distortion theorem and growth theorem. Further, the bounds on initial logarithmic coefficients, inverse coefficient and the second Hankel determinant involving the inverse coefficients are examined.Paatero's \(V(k)\) space and a claim by Pinchukhttps://www.zbmath.org/1483.300342022-05-16T20:40:13.078697Z"Andreev, Valentin V."https://www.zbmath.org/authors/?q=ai:andreev.valentin-v"Bekker, Miron B."https://www.zbmath.org/authors/?q=ai:bekker.miron-b"Cima, Joseph A."https://www.zbmath.org/authors/?q=ai:cima.joseph-aSummary: In this article we obtain a factorization theorem for the functions in Paatero's \(V(k)\) space. We bring attention to a significant result of Pinchuk which unfortunately is false. This result relates measures associated to functions in \(V(k)\) and an integral representation theorem for such functions. We prove necessary and sufficient conditions for a wide class of functions (in particular, the polynomials) to belong to the Paatero class based on the geometry of their critical points, and obtain explicit representation of the measures associated to a wide class of such polynomials that includes the Suffridge polynomials.Coefficient estimates for Libera type bi-close-to-convex functionshttps://www.zbmath.org/1483.300352022-05-16T20:40:13.078697Z"Bulut, Serap"https://www.zbmath.org/authors/?q=ai:bulut.serapSummary: In a very recent paper, \textit{Z. Wang} and the author [C. R., Math., Acad. Sci. Paris 355, No. 8, 876--880 (2017; Zbl 1376.30015)]
determined the estimates for the general Taylor-Maclaurin coefficients of functions belonging to the bi-close-to-convex function class. In this study, we introduce the class of Libera type bi-close-to-convex functions and obtain the upper bounds for the coefficients of functions belonging to this class. Our results generalize the results in the above mentioned paper.Convolution conditions for two subclasses of analytic functions defined by Jackson \(q\)-difference operatorhttps://www.zbmath.org/1483.300362022-05-16T20:40:13.078697Z"El-Emam, Fatma Z."https://www.zbmath.org/authors/?q=ai:el-emam.fatma-zSummary: By using Jackson \(q\)-derivative, some characterizations in terms of convolutions for two classes of analytic functions in the open unit disc are given. Also, coefficient conditions and inclusion properties for functions in these classes are found.A proof of Hall's conjecture on length of ray images under starlike mappings of order \(\alpha\)https://www.zbmath.org/1483.300372022-05-16T20:40:13.078697Z"Hästö, Peter"https://www.zbmath.org/authors/?q=ai:hasto.peter-a"Ponnusamy, Saminathan"https://www.zbmath.org/authors/?q=ai:ponnusamy.saminathanSummary: Assume that \(f\) lies in the class of starlike functions of order \(\alpha\in[0,1)\), that is, which are regular and univalent for \(|z|< 1\) and such that
\[
\mathrm{Re}\left(\frac{zf'(z)}{f(z)}\right)> \alpha\quad\text{for } |z|<1.
\]
In this paper we show that for each \(\alpha\in[0,1)\), the following sharp inequality holds:
\[
|f(re^{i\theta})|^{-1}\int_0^r|f'(ue^{i\theta})|\,du\leq\frac{\Gamma(\frac{1}{2})\Gamma(2-\alpha)}{\Gamma(\frac{3}{2}-\alpha)} \quad\text{for every } r< 1 \text{ and } \theta.
\]
This settles the conjecture of \textit{R. R. Hall} [Bull. Lond. Math. Soc. 12, 119--126 (1980; Zbl 0442.30007)] positively.Littlewood-Paley conjecture associated with certain classes of analytic functionshttps://www.zbmath.org/1483.300382022-05-16T20:40:13.078697Z"Kumar, Virendra"https://www.zbmath.org/authors/?q=ai:kumar.virendra"Srivastava, Rekha"https://www.zbmath.org/authors/?q=ai:srivastava.rekha"Cho, Nak Eun"https://www.zbmath.org/authors/?q=ai:cho.nakeun|cho.nak-eunSummary: The Littlewood-Paley conjecture hardly holds for any subclass of univalent functions except the class of starlike functions as verified, in general, by the researchers until now. Therefore, it is interesting to consider the classes where the Littlewood-Paley conjecture holds completely or partially. For such investigation, the classes of normalized strongly \(\alpha\)-close-to-convex functions and \(\alpha\)-quasiconvex functions of order \(\beta\) are considered in this paper. In the main, bounds on the initial coefficients and related Fekete-Szegö inequalities are derived in this paper. Furthermore, it is seen that the Littlewood-Paley conjecture holds for all values of the parameter \(\gamma >0\) in case of the first coefficient. However for the second coefficient, it holds for large positive values of \(\gamma\). Relevant connections of our results with the existing results are also pointed out.Certain subclasses of univalent functions involving Pascal distribution serieshttps://www.zbmath.org/1483.300392022-05-16T20:40:13.078697Z"Lashin, Abdel Moneim Y."https://www.zbmath.org/authors/?q=ai:lashin.abdel-moneim-yousof"Badghaish, Abeer O."https://www.zbmath.org/authors/?q=ai:badghaish.abeer-o"Bajamal, Amani Z."https://www.zbmath.org/authors/?q=ai:bajamal.amani-zSummary: In this paper, our aim is to find the necessary and sufficient conditions for univalent functions involving Pascal distribution to be in some subclasses of analytic functions.On a subclass of analytic functions that are starlike with respect to a boundary point involving exponential functionhttps://www.zbmath.org/1483.300402022-05-16T20:40:13.078697Z"Lecko, Adam"https://www.zbmath.org/authors/?q=ai:lecko.adam"Murugusundaramoorthy, Gangadharan"https://www.zbmath.org/authors/?q=ai:murugusundaramoorthy.gangadharan"Sivasubramanian, Srikandan"https://www.zbmath.org/authors/?q=ai:sivasubramanian.srikandanSummary: In the present exploration, the authors define and inspect a new class of functions that are regular in the unit disc \(\mathfrak{D}:=\{\varsigma \in \mathbb{C} : |\varsigma| < 1\}\), by using an adapted version of the interesting analytic formula offered by Robertson (unexploited) for starlike functions with respect to a boundary point by subordinating to an exponential function. Examples of some new subclasses are presented. Initial coefficient estimates are specified, and the familiar Fekete-Szegö inequality is obtained. Differential subordinations concerning these newly demarcated subclasses are also established.Fourth Hankel determinant for a subclass of starlike functions based on modified sigmoidhttps://www.zbmath.org/1483.300412022-05-16T20:40:13.078697Z"Mashwani, Wali Khan"https://www.zbmath.org/authors/?q=ai:mashwani.wali-khan"Ahmad, Bakhtiar"https://www.zbmath.org/authors/?q=ai:ahmad.bakhtiar"Khan, Nazar"https://www.zbmath.org/authors/?q=ai:khan.nazar"Khan, Muhammad Ghaffar"https://www.zbmath.org/authors/?q=ai:khan.muhammad-ghaffar"Arjika, Sama"https://www.zbmath.org/authors/?q=ai:arjika.sama"Khan, Bilal"https://www.zbmath.org/authors/?q=ai:khan.bilal"Chinram, Ronnason"https://www.zbmath.org/authors/?q=ai:chinram.ronnasonSummary: In our present investigation, we obtain the improved third-order Hankel determinant for a class of starlike functions connected with modified sigmoid functions. Further, we investigate the fourth-order Hankel determinant, Zalcman conjecture, and also evaluate the fourth-order Hankel determinants for 2-fold, 3-fold, and 4-fold symmetric starlike functions.A family of holomorphic functions defined by differential inequalityhttps://www.zbmath.org/1483.300422022-05-16T20:40:13.078697Z"Mohammed, Nafya Hameed"https://www.zbmath.org/authors/?q=ai:hameed-mohammed.nafya"Adegani, Ebrahim Analouei"https://www.zbmath.org/authors/?q=ai:adegani.ebrahim-analouei"Bulboacă, Teodor"https://www.zbmath.org/authors/?q=ai:bulboaca.teodor"Cho, Nak Eun"https://www.zbmath.org/authors/?q=ai:cho.nak-eunSummary: The aim of the present paper is to introduce and study a subfamily of holomorphic and normalized functions defined by a differential inequality. Some geometric properties of this family of holomorphic functions and different problems of a family of such functions are presented.Fekete-Szegö inequality for certain subclasses of analytic functions related with crescent-shaped domain and application of poison distribution serieshttps://www.zbmath.org/1483.300432022-05-16T20:40:13.078697Z"Murugusundaramoorthy, Gangadharan"https://www.zbmath.org/authors/?q=ai:murugusundaramoorthy.gangadharanSummary: The purpose of this paper is to define a new class of analytic, normalized functions in the open unit disk \(\mathbb{D}=\{ z:z\in \mathbb{C}\text{ and } \left\vert z\right\vert <1\}\) subordinating with crescent shaped regions, and to derive certain coefficient estimates \(a_2, a_3\) and Fekete-Szegö inequality for \(f\in\mathcal{M}_q(\alpha,\beta,\lambda)\). A similar result have been done for the function \(f^{-1}\). Further application of our results to certain functions defined by convolution products with a normalized analytic function is given, in particular we obtain Fekete-Szegö inequalities for certain subclasses of functions defined through Poisson distribution series.On an extension of Nunokawa's lemmahttps://www.zbmath.org/1483.300442022-05-16T20:40:13.078697Z"Nunokawa, Mamoru"https://www.zbmath.org/authors/?q=ai:nunokawa.mamoru"Sokół, Janusz"https://www.zbmath.org/authors/?q=ai:sokol.januszSummary: Jack's Lemma says that if \(f(z)\) is regular in the disc \(|z|\le r\), \(f(0)=0\), and \(|f(z)|\) assumes its maximum at \(z_0\) on the circle \(|z|=r\), then \(z_0f'(z)_0/f(z_0)\ge 1\). This Lemma was generalized in several directions. In this paper we consider an improvement of some first author's results of this type.Coefficients of the inverse of functions for the subclass of the class \(\mathcal{U} (\lambda)\)https://www.zbmath.org/1483.300452022-05-16T20:40:13.078697Z"Obradović, Milutin"https://www.zbmath.org/authors/?q=ai:obradovic.milutin"Tuneski, Nikola"https://www.zbmath.org/authors/?q=ai:tuneski.nikolaSummary: Let \(\mathcal{A}\) be the class of functions \(f\) that are analytic in the unit disk \(\mathbb{D}\) and normalized such that \(f(z)=z+a_2z^2+a_3z^3+\cdots\). Let \(0<\lambda \leq 1\) and
\[
\begin{aligned} \mathcal{U} (\lambda) = \left\{f\in \mathcal{A} : \left| \left(\frac{z}{f(z)} \right)^2f'(z)-1\right| < \lambda, z\in \mathbb{D}\right\}. \end{aligned}
\]
In this paper sharp upper bounds of the first three coefficients of the inverse function \(f^{-1}\) are given in the case when
\[
\begin{aligned} \frac{f(z)}{z}\prec \frac{1}{(1-z)(1-\lambda z)}. \end{aligned}
\]Properties of functions with symmetric points involving subordinationhttps://www.zbmath.org/1483.300462022-05-16T20:40:13.078697Z"Raza, Malik Ali"https://www.zbmath.org/authors/?q=ai:raza.malik-ali"Bukhari, Syed Zakar Hussain"https://www.zbmath.org/authors/?q=ai:bukhari.syed-zakar-hussain"Ahmed, Imtiaz"https://www.zbmath.org/authors/?q=ai:ahmed.imtiaz"Ashfaq, Muhammad"https://www.zbmath.org/authors/?q=ai:ashfaq.muhammad"Nazir, Maryam"https://www.zbmath.org/authors/?q=ai:nazir.maryamSummary: We study a new subclass of functions with symmetric points and derive an equivalent formulation of these functions in term of subordination. Moreover, we find coefficient estimates and discuss characterizations for functions belonging to this new class. We also obtain distortion and growth results. We relate our results with the existing literature of the subject.On sufficient conditions for strongly starlikeness of order a and type \(\beta\)https://www.zbmath.org/1483.300472022-05-16T20:40:13.078697Z"Sharma, Vidyadhar"https://www.zbmath.org/authors/?q=ai:sharma.vidyadhar"Mathur, Nisha"https://www.zbmath.org/authors/?q=ai:mathur.nisha"Soni, Amit"https://www.zbmath.org/authors/?q=ai:soni.amitSummary: By making use of differential subordination technique, we derive certain conditions for \(p\)-valent strongly starlike functions of order \(\alpha\) and type \(\beta\). The results presented here are sharp.Properties of analytic solutions of three similar differential equations of the second orderhttps://www.zbmath.org/1483.300482022-05-16T20:40:13.078697Z"Sheremeta, M. M."https://www.zbmath.org/authors/?q=ai:sheremeta.myroslav-m"Trukhan, Yu. S."https://www.zbmath.org/authors/?q=ai:trukhan.yu-sSummary: An analytic univalent in \(\mathbb{D}=\{z:\;|z|<1\}\) function \(f(z)\) is said to be convex if \(f(\mathbb{D})\) is a convex domain. It is well known that the condition \(\operatorname{Re}\{1+zf''(z)/f'(z)\}>0\), \(z\in\mathbb{D} \), is necessary and sufficient for the convexity of \(f\). The function \(f\) is said to be close-to-convex in \(\mathbb{D}\) if there exists a convex in \(\mathbb{D}\) function \(\Phi\) such that \(\operatorname{Re}(f'(z)/\Phi'(z))>0\), \(z\in\mathbb{D} \).
S.M. Shah indicated conditions on real parameters \(\beta_0\), \(\beta_1\), \(\gamma_0\), \(\gamma_1\), \(\gamma_2\) of the differential equation \(z^2w''+(\beta_0 z^2+\beta_1 z)w'+(\gamma_0z^2+\gamma_1 z+\gamma_2) w=0\), under which there exists an entire transcendental solution \(f\) such that \(f\) and all its derivatives are close-to-convex in \(\mathbb{D} \).
Let \(0<R\le+\infty\), \(\mathbb{D}_R=\{z:\;|z|<R\}\) and \(l\) be a positive continuous function on \([0,R)\), which satisfies \((R-r)l(r)>C\), \(C=\operatorname{const}>1\). An analytic in \(\mathbb{D}_R\) function \(f\) is said to be of bounded \(l\)-index if there exists \(N\in\mathbb{Z}_+\) such that for all \(n\in\mathbb{Z}_+\) and \(z\in\mathbb{D}_R\)
\[\frac{|f^{(n)}(z)|}{n!l^n(|z|)}\le \max\left\{\frac{|f^{(k)}(z)|}{k!l^k(|z|)}:\;0\le k\le N\right\}.\]
Here we investigate close-to-convexity and the boundedness of the \(l\)-index for analytic in \(\mathbb{D}\) solutions of three analogues of Shah differential equation: \(z(z-1) w''+\beta z w'+\gamma w=0\), \((z-1)^2 w''+\beta z w'+\gamma w=0\) and \((1-z)^3 w''+\beta(1- z) w'+\gamma w=0\). Despite the similarity of these equations, their solutions have different properties.A note on spirallike functionshttps://www.zbmath.org/1483.300492022-05-16T20:40:13.078697Z"Sim, Y. J."https://www.zbmath.org/authors/?q=ai:sim.young-jong"Thomas, D. K."https://www.zbmath.org/authors/?q=ai:thomas.derek-keithSummary: Let \(f\) be analytic in the unit disk \(\mathbb{D}=\{z\in \mathbb{C}:|z|<1 \}\) and let \(\mathcal{S}\) be the subclass of normalised univalent functions with \(f(0)=0\) and \(f'(0)=1\), given by \(f(z)=z+\sum_{n=2}^{\infty }a_n z^n\). Let \(F\) be the inverse function of \(f\), given by \(F(\omega )=\omega +\sum_{n=2}^{\infty }A_n \omega^n\) for \(|\omega |\le r_0(f)\). Denote by \(\mathcal{S}_p^{*}(\alpha )\) the subset of \(\mathcal{S}\) consisting of the spirallike functions of order \(\alpha\) in \(\mathbb{D} \), that is, functions satisfying
\[ \operatorname{Re} \left\{e^{-i\gamma}\dfrac{zf'(z)}{f(z)}\right\}>\alpha\cos \gamma, \]
for \(z\in \mathbb{D}\), \(0\le \alpha <1\) and \(\gamma \in (-\pi /2,\pi /2)\). We give sharp upper and lower bounds for both \(|a_3|-|a_2|\) and \(|A_3|-|A_2|\) when \(f\in \mathcal{S}_p^{* }(\alpha )\), thus solving an open problem and presenting some new inequalities for coefficient differences.A certain subclass of analytic functions with negative coefficients defined by Gegenbauer polynomialshttps://www.zbmath.org/1483.300502022-05-16T20:40:13.078697Z"Venkateswarlu, Bolineni"https://www.zbmath.org/authors/?q=ai:venkateswarlu.bollineni"Reddy, Pinninti Thirupathi"https://www.zbmath.org/authors/?q=ai:reddy.pinninti-thirupathi"Sridevi, Settipalli"https://www.zbmath.org/authors/?q=ai:sridevi.settipalli"Sujatha, Vaishnavy"https://www.zbmath.org/authors/?q=ai:sujatha.vaishnavySummary: In this paper, we introduce a new subclass of analytic functions with negative coefficients defined by Gegenbauer polynomials. We obtain coefficient bounds, growth and distortion properties, extreme points and radii of starlikeness, convexity and close-to-convexity for functions belonging to the class \(TS_\lambda^m(\gamma,\varrho, k, \vartheta)\). Furthermore, we obtained the Fekete-Szego problem for this class.A conjecture on Marx-Strohhäcker type inclusion relation between \(q\)-convex and \(q\)-starlike functionshttps://www.zbmath.org/1483.300512022-05-16T20:40:13.078697Z"Verma, Sarika"https://www.zbmath.org/authors/?q=ai:verma.sarika"Kumar, Raj"https://www.zbmath.org/authors/?q=ai:kumar.raj"Sokół, Janusz"https://www.zbmath.org/authors/?q=ai:sokol.januszSummary: We prove that the class \(\mathcal{K}\) of normalized univalent convex functions defined in the unit disk \(\mathbb{E}\) is contained in \(\mathcal{K}_q\left(\frac{1-q}{1+q^2}\right)\) (\(0<q<1\)), the class of normalized univalent \(q\)-convex functions of order \((1-q)/(1+q^2)\). We provide examples that exhibit a Marx-Strohhäcker type inclusion relation, i.e. \(\mathcal{K}_q\left(\frac{1-q}{1+q^2}\right)\subset\mathcal{S}_q^\ast\left(\frac{1}{1+q}\right)\), where \(\mathcal{S}_q^\ast\left(\frac{1}{1+q}\right)\) is the class of \(q\)-starlike functions of order \(1/(1+q)\). Note that for \(q\to 1^-\) this relation coincides with the well-known result, \(\mathcal{K}\subset\mathcal{S}^\ast\left(\frac{1}{2}\right)\), of \textit{A. Marx} [Math. Ann. 107, 40--67 (1932; JFM 58.0363.01)]
and \textit{E. Strohhäcker} [M. Z. 37, 356--380 (1933; JFM 59.0353.02)].On the third and fourth Hankel determinants for a subclass of analytic functionshttps://www.zbmath.org/1483.300522022-05-16T20:40:13.078697Z"Wang, Zhi-Gang"https://www.zbmath.org/authors/?q=ai:wang.zhigang"Raza, Mohsan"https://www.zbmath.org/authors/?q=ai:raza.mohsan"Arif, Muhammad"https://www.zbmath.org/authors/?q=ai:arif.muhammad"Ahmad, Khurshid"https://www.zbmath.org/authors/?q=ai:ahmad.khurshidSummary: The objective of this paper is to investigate the third and fourth Hankel determinants for the class of functions with bounded turning associated with Bernoulli's lemniscate. The fourth Hankel determinants for 2-fold symmetric and 3-fold symmetric functions are also studied.Quasiconformal Whitney partitionhttps://www.zbmath.org/1483.300532022-05-16T20:40:13.078697Z"Gol'dshtein, Vladimir"https://www.zbmath.org/authors/?q=ai:goldshtein.vladimir"Zobin, N."https://www.zbmath.org/authors/?q=ai:zobin.nahum|zobin.naum-mSummary: Whitney partition is a very important concept in modern geometric analysis. We discuss here a quasiconformal version of Whitney
partition that can be useful for Sobolev spaces.Beurling-Ahlfors extension by heat kernel, \(\mathrm{A}_{\infty}\)-weights for VMO, and vanishing Carleson measureshttps://www.zbmath.org/1483.300542022-05-16T20:40:13.078697Z"Wei, Huaying"https://www.zbmath.org/authors/?q=ai:wei.huaying"Matsuzaki, Katsuhiko"https://www.zbmath.org/authors/?q=ai:matsuzaki.katsuhiko.1|matsuzaki.katsuhikoAn increasing homeomorphism \(h\) of the real line \(\mathbb R\) onto itself is said to be quasisymmetric if there exists some \(M>0\) such that \[\frac 1M\le\frac{h(x+t)-h(x)}{h(x)-h(x-t))}\le M\] for all \(x\in\mathbb R\) and \(t>0\). In an important paper [Acta Math. 96, 125--142 (1956; Zbl 0072.29602)] \textit{A. Beurling} and \textit{L. V. Ahlfors} constructed a quasiconformal homeomorphism \(f\) of the upper half plane \(\mathbb U=\{z=x+iy: y>0\}\) onto itself which has boundary values \(h\) when \(h\) is a quasisymmetric homeomorphism. A variant of the Beurling-Ahlfors extension was investigated by \textit{R. A. Fefferman} et al. [Ann. Math. (2) 134, No. 1, 65--124 (1991; Zbl 0770.35014)] when the quasisymmetric homeomorphism \(h\) is induced by an \(A^{\infty}\) weight. Precisely, when \(h\) is locally absolutely continuous on the real line such that \(h'\) is an \(A^{\infty}\) weight, Fefferman et al. [loc. cit.] showed that \(h\) can be extended to a quasiconformal mapping \(F\) of the upper half plane, which is a variant of the Beurling-Ahlfors extension by the heat kernel, such that the complex dilatation \(\mu\) of \(F\) induces a Carleson measure \(|\mu(x+iy)|^2/y\). In this paper, the authors give a rather detailed exposition of this result, and show that the the complex dilatation \(\mu\) of \(F\) induces a vanishing Carleson measure \(|\mu(x+iy)|^2/y\) under the additional assumption that \(\log h'\) is a VMO function. This answers a question raised recently by the reviewer [Ann. Fenn. Math. 47, No. 1, 57--82 (2022; Zbl 1482.30057)].
Reviewer: Yuliang Shen (Suzhou)Weighted Hardy spaces of quasiconformal mappingshttps://www.zbmath.org/1483.300552022-05-16T20:40:13.078697Z"Benedict, Sita"https://www.zbmath.org/authors/?q=ai:benedict.sita"Koskela, Pekka"https://www.zbmath.org/authors/?q=ai:koskela.pekka"Li, Xining"https://www.zbmath.org/authors/?q=ai:li.xiningSummary: We study the integral characterizations of weighted Hardy spaces of quasiconformal mappings on the \(n\)-dimensional unit ball using the weight \((1-r)^{n-2 + \alpha}\). We extend the known results for univalent functions on the unit disk. Some of our results are new even in the unweighted setting for quasiconformal mappings.Sphericalization and flattening preserve uniform domains in nonlocally compact metric spaceshttps://www.zbmath.org/1483.300562022-05-16T20:40:13.078697Z"Li, Yaxiang"https://www.zbmath.org/authors/?q=ai:li.yaxiang"Ponnusamy, Saminathan"https://www.zbmath.org/authors/?q=ai:ponnusamy.saminathan"Zhou, Qingshan"https://www.zbmath.org/authors/?q=ai:zhou.qingshanSummary: The main aim of this paper is to investigate the invariant properties of uniform domains under flattening and sphericalization in nonlocally compact complete metric spaces. Moreover, we show that quasi-Möbius maps preserve uniform domains in nonlocally compact spaces as well.Open-door lemma for functions with fixed second coefficientshttps://www.zbmath.org/1483.300572022-05-16T20:40:13.078697Z"Amani, M."https://www.zbmath.org/authors/?q=ai:amani.mostafa"Aghalary, R."https://www.zbmath.org/authors/?q=ai:aghalary.rasoul"Ebadian, A."https://www.zbmath.org/authors/?q=ai:ebadian.aliSummary: We extend the well-known open-door lemma by using the theory of differential subordination for functions with fixed initial coefficient and apply it to study some integral operators. Further, using an extension of Nunokawa lemma, we determine some sufficient conditions for the radii and order of starlike functions with fixed second coefficient. Our results improve and generalize some previously known results.The use of the isometry of function spaces with different numbers of variables in the theory of approximation of functionshttps://www.zbmath.org/1483.300582022-05-16T20:40:13.078697Z"Bushev, D. M."https://www.zbmath.org/authors/?q=ai:bushev.d-m"Abdullayev, F. G."https://www.zbmath.org/authors/?q=ai:abdullayev.fahreddin-g"Kal'chuk, I. V."https://www.zbmath.org/authors/?q=ai:kalchuk.inna-v"Imashkyzy, M."https://www.zbmath.org/authors/?q=ai:imashkyzy.meerimSummary: In the work, we found integral representations for function spaces that are isometric to spaces of entire functions of exponential type, which are necessary for giving examples of equality of approximation characteristics in function spaces isometric to spaces of non-periodic functions. Sufficient conditions are obtained for the nonnegativity of the delta-like kernels used to construct isometric function spaces with various numbers of variables.Rate of growth of distributionally chaotic functionshttps://www.zbmath.org/1483.300592022-05-16T20:40:13.078697Z"Gilmore, Clifford"https://www.zbmath.org/authors/?q=ai:gilmore.clifford"Martínez-Giménez, Félix"https://www.zbmath.org/authors/?q=ai:martinez-gimenez.felix"Peris, Alfred"https://www.zbmath.org/authors/?q=ai:peris.alfredoSummary: We investigate the permissible growth rates of functions that are distributionally chaotic with respect to differentiation operators. We improve on the known growth estimates for \(D\)-distributionally chaotic entire functions, where growth is in terms of average \(L^p\)-norms on spheres of radius \(r>0\) as \(r\rightarrow\infty\), for \(1\leq p\leq\infty\). We compute growth estimates of \(\partial/\partial x_k\)-distributionally chaotic harmonic functions in terms of the average \(L^2\)-norm on spheres of radius \(r>0\) as \(r\rightarrow\infty\). We also calculate sup-norm growth estimates of distributionally chaotic harmonic functions in the case of the partial differentiation operators \(D^\alpha\).Global boundedness of functions of finite order that are bounded outside small setshttps://www.zbmath.org/1483.300602022-05-16T20:40:13.078697Z"Khabibullin, Bulat N."https://www.zbmath.org/authors/?q=ai:khabibullin.b-nOn the number of real zeros of real entire functions with a non-decreasing sequence of the second quotients of Taylor coefficientshttps://www.zbmath.org/1483.300612022-05-16T20:40:13.078697Z"Nguyen, Thu Hien"https://www.zbmath.org/authors/?q=ai:nguyen.thu-hien-thi"Vishnyakova, Anna"https://www.zbmath.org/authors/?q=ai:vishnyakova.anna-mSummary: For an entire function \(f(z)=\sum^\infty_{k=0}a_kz^k\), \(a_k>0\), we define the sequence of the second quotients of Taylor coefficients \(Q:=\bigg(\frac{a^2_k}{a_{k-1}a_{k+1}}\bigg)^\infty_{k=1}\). We find new necessary conditions for a function with a non-decreasing sequence \(Q\) to belong to the Laguerre-Pólya class of type I. We also estimate the possible number of non-real zeros for a function with a non-decreasing sequence \(Q\).Uniqueness of differential \(q\)-shift difference polynomials of entire functionshttps://www.zbmath.org/1483.300622022-05-16T20:40:13.078697Z"Mathai, Madhura M."https://www.zbmath.org/authors/?q=ai:mathai.madhura-m"Manjalapur, Vinayak V."https://www.zbmath.org/authors/?q=ai:manjalapur.vinayak-vSummary: In this paper, we prove the uniqueness theorems of differential \(q\)-shift difference polynomials of transcendental entire functions.On meromorphic solutions of nonlinear delay-differential equationshttps://www.zbmath.org/1483.300632022-05-16T20:40:13.078697Z"Mao, Zhiqiang"https://www.zbmath.org/authors/?q=ai:mao.zhiqiang"Liu, Huifang"https://www.zbmath.org/authors/?q=ai:liu.huifangSummary: Using Cartan's second main theorem and Nevanlinna's theorem concerning a group of meromorphic functions, we obtain the growth and zero distribution of meromorphic solutions of the nonlinear delay-differential equation \(f^n(z) + P(z) f^{( k )}(z + \eta) = H_0(z) + H_1(z) e^{\omega_1 z^q} + \cdots + H_m(z) e^{\omega_m z^q}\), where \(n, k, q, m\) are positive integers, \( \eta, \omega_1, \cdots, \omega_m\) are complex numbers with \(\omega_1 \cdots \omega_m \neq 0\), and \(P, H_0, H_1, \cdots, H_m\) are entire functions of order less than \(q\) with \(P H_1 \cdots H_m \not\equiv 0\). Especially for \(\eta = 0\), some sufficient conditions are given to guarantee the above equation has no meromorphic solutions of few poles.Paired Hayman conjecture and uniqueness of complex delay-differential polynomialshttps://www.zbmath.org/1483.300642022-05-16T20:40:13.078697Z"Gao, Yingchun"https://www.zbmath.org/authors/?q=ai:gao.yingchun"Liu, Kai"https://www.zbmath.org/authors/?q=ai:liu.kai.4|liu.kai.1|liu.kai.2|liu.kai|liu.kai.3|liu.kai.5Summary: In this paper, the paired Hayman conjecture of different types are considered, namely, the zeros distribution of \(f(z)^nL(g)-a(z)\) and \(g(z)^nL(f)-a(z)\), where \(L(h)\) takes the derivatives \(h^{(k)}(z)\) or the shift \(h(z+c)\) or the difference \(h(z+c)-h(z)\) or the delay-differential \(h^{(k)}(z+c)\), where \(k\) is a positive integer, \(c\) is a non-zero constant and \(a(z)\) is a non-zero small function with respect to \(f(z)\) and \(g(z)\). The related uniqueness problems of complex delay-differential polynomials are also considered.Entire solutions of differential-difference equations of Fermat typehttps://www.zbmath.org/1483.300652022-05-16T20:40:13.078697Z"Hu, Peichu"https://www.zbmath.org/authors/?q=ai:hu.peichu"Wang, Wenbo"https://www.zbmath.org/authors/?q=ai:wang.wenbo"Wu, Linlin"https://www.zbmath.org/authors/?q=ai:wu.linlinSummary: In this paper, we extend some previous works by Liu et al. on the existence of transcendental entire solutions of differential-difference equations of Fermat type. In addition, we also present a precise description of the associated entire solutions.Some results on uniqueness of meromorphic functions concerning differential polynomialshttps://www.zbmath.org/1483.300662022-05-16T20:40:13.078697Z"Husna, V."https://www.zbmath.org/authors/?q=ai:husna.vSummary: In this paper, we study the uniqueness problem of certain differential polynomials generated by two meromorphic functions. The results of the paper extend some recent results due to \textit{C. Meng} and \textit{X. Li} [J. Anal. 28, No. 3, 879--894 (2020; Zbl 1455.30028)].One-way sharing of sets with derivatives and normal functionshttps://www.zbmath.org/1483.300672022-05-16T20:40:13.078697Z"Singh, Virender"https://www.zbmath.org/authors/?q=ai:singh.virender-pal"Lal, Banarsi"https://www.zbmath.org/authors/?q=ai:lal.banarsiSummary: The aim of this paper is to study normal functions under the weaker condition of one-way sharing of sets and further to improve and generalize the earlier work of \textit{Q. Chen} and \textit{D. Tong} [Bol. Soc. Mat. Mex., III. Ser. 25, No. 3, 589--596 (2019; Zbl 1430.30016)], \textit{Y. Xu} and \textit{H. Qiu} [Filomat 30, No. 2, 287--292 (2016; Zbl 1474.30243)].Approximation of functions and all derivatives on compact setshttps://www.zbmath.org/1483.300682022-05-16T20:40:13.078697Z"Armeniakos, Sotiris"https://www.zbmath.org/authors/?q=ai:armeniakos.sotiris"Kotsovolis, Giorgos"https://www.zbmath.org/authors/?q=ai:kotsovolis.giorgos"Nestoridis, Vassili"https://www.zbmath.org/authors/?q=ai:nestoridis.vassiliSummary: In Mergelyan type approximation we uniformly approximate functions on compact sets \(K\) by polynomials or rational functions or holomorphic functions on varying open sets containing \(K\). In the present paper we consider analogous approximation, where uniform convergence on \(K\) is replaced by uniform approximation on \(K\) of all order derivatives.Induced fields in isolated elliptical inhomogeneities due to imposed polynomial fields at infinityhttps://www.zbmath.org/1483.300692022-05-16T20:40:13.078697Z"Calvo-Jurado, Carmen"https://www.zbmath.org/authors/?q=ai:calvo-jurado.carmen"Parnell, William J."https://www.zbmath.org/authors/?q=ai:parnell.william-jSummary: The Eshelby inhomogeneity problem plays a crucial role in the micromechanical analysis of the effective mechanical behaviour of inhomogeneous media since it provides a mechanism to predict interior fields associated with ellipsoidal inhomogeneities. In the context of linear elasticity, Eshelby showed that given an isolated elliptical (two dimensions) or ellipsoidal (three dimensions) inhomogeneity embedded in a homogeneous material of infinite extent, then for any uniform strain or traction imposed in the far field, the induced strain inside the inhomogeneity is also uniform. In the case of non-uniform far-field conditions, Eshelby showed that if the loading is a polynomial of order \(n\), the associated interior field is characterized by a polynomial of the same order. This is often called `Eshelby's polynomial conservation theorem'. Since then, the problem has been studied by many, but in most cases for the uniform loading scenario, i.e. when strains or tractions in the far field are uniform. However, in many applications, e.g. permittivity, conductivity, elasticity, etc., the case of non-uniform conditions is also of interest and furthermore, methods to deal with non-elliptical and non-ellipsoidal inhomogeneities are required. In this work, for prescribed non-uniform polynomial far-field conditions, we introduce a method to approximate interior fields for isolated inhomogeneities of elliptical shape. This subproblem is relevant for approximating effective properties of numerous composites since constituent inhomogeneities are often of this form, or limiting forms, e.g. layered and fibre reinforced composites. We verify that the obtained results agree with the polynomial conservation property and with results determined using conformal mappings or the classical circle inclusion theorem. We close with a discussion of how the method can be straightforwardly extended to the case of non-elliptical inhomogeneities.Regularization of a class of summary equationshttps://www.zbmath.org/1483.300702022-05-16T20:40:13.078697Z"Garif'yanov, F. N."https://www.zbmath.org/authors/?q=ai:garifyanov.farkhat-nurgayazovich"Strezhneva, E. V."https://www.zbmath.org/authors/?q=ai:strezhneva.elena-vasilevnaSummary: Let \(D\) be an arbitrary quadrangle with boundary \(\Gamma \). We consider a four-element linear summary equation. The solution is sought in the class of functions which are holomorphic outside \(D\) and vanish at infinity. The boundary values satisfy the Hölder condition on any compact set which does not contain the vertices. At the vertices, singularities at most of logarithmic order are allowed. The coefficients of the equation are holomorphic in \(D\) and their boundary values satisfy the Hölder condition on \(\Gamma \). The free term satisfies the same conditions. The solution is sought in the form of the Cauchy type integral over \(\Gamma\) with unknown density. To regularize the obtained functional equation, we use the Carleman problem. Previously, a Carleman shift is introduced on \(\Gamma \); it transfers each side to itself and reverses orientation; the midpoints of the sides are fixed under the shift. We indicate some applications of this summary equation to the problem of moments for entire functions of exponential type.The Dirichlet-Neumann boundary value problem for the inhomogeneous Bitsadze equation in a ring domainhttps://www.zbmath.org/1483.300712022-05-16T20:40:13.078697Z"Gençtürk, İlker"https://www.zbmath.org/authors/?q=ai:gencturk.ilkerSummary: In this study, by using some integral representations formulas, we study solvability conditions and explicit solution of the Dirichlet-Neumann problem, an example for a combined boundary value problem, for the Bitsadze equation in a ring domain.Polyanalytic boundary value problems for planar domains with harmonic Green functionhttps://www.zbmath.org/1483.300722022-05-16T20:40:13.078697Z"Begehr, Heinrich"https://www.zbmath.org/authors/?q=ai:begehr.heinrich"Shupeyeva, Bibinur"https://www.zbmath.org/authors/?q=ai:shupeyeva.bibinurThe authors characterize the solvability of three boundary value problems for the inhomogeneous polyanalytic equation in planar domains (having a harmonic Green function), namely the well-posed iterated Schwarz problem, and two over-determined iterated problems of Dirichlet and Neumann type. Solutions formulas are also obtained and, in particular, it is concluded that the polyanalytic Cauchy-Pompeiu representation formula provides the solution to the Dirichlet problem (for any degree \(n\), and in the cases for which the solution exists).
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)Asymptotics of the Riemann-Hilbert problem for the Somov model of magnetic reconnection of long shock waveshttps://www.zbmath.org/1483.300732022-05-16T20:40:13.078697Z"Bezrodnykh, S. I."https://www.zbmath.org/authors/?q=ai:bezrodnykh.sergei-i"Vlasov, V. I."https://www.zbmath.org/authors/?q=ai:vlasov.vladimir-ivanovichSummary: We consider the Riemann-Hilbert problem in a domain of complicated shape (the exterior of a system of cuts), with the condition of growth of the solution at infinity. Such a problem arises in the Somov model of the effect of magnetic reconnection in the physics of plasma, and its solution has the physical meaning of a magnetic field. The asymptotics of the solution is obtained for the case of infinite extension of four cuts from the given system, which have the meaning of shock waves, so that the original domain splits into four disconnected components in the limit. It is shown that if the coefficient in the condition of growth of the magnetic field at infinity consistently decreases in this case, then this field basically coincides in the limit with the field arising in the Petschek model of the effect of magnetic reconnection.Noether property and approximate solution of the Riemann boundary value problem on closed curveshttps://www.zbmath.org/1483.300742022-05-16T20:40:13.078697Z"Bory-Reyes, Juan"https://www.zbmath.org/authors/?q=ai:bory-reyes.juan|moreno-garcia.tania"Katz, David"https://www.zbmath.org/authors/?q=ai:katz.david-f|katz.david-bBased on classic methods, the authors analyse the Noether property of a Riemann boundary value problem in the Banach algebra of continuous functions over closed curves, as well as consequent approximate solutions (by using quasi-Fredholm operators).
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)On the splitting type of holomorphic vector bundles induced from regular systems of differential equationhttps://www.zbmath.org/1483.300752022-05-16T20:40:13.078697Z"Giorgadze, Grigori"https://www.zbmath.org/authors/?q=ai:giorgadze.grigory"Gulagashvili, Gega"https://www.zbmath.org/authors/?q=ai:gulagashvili.gegaSummary: We calculate the splitting type of holomorphic vector bundles on the Riemann sphere induced by a Fuchsian system of differential equations. Using this technique, we indicate the relationship between Hölder continuous matrix functions and a moduli space of vector bundles on the Riemann sphere. For second order systems with three singular points we give a complete characterization of the corresponding vector bundles by the invariants of Fuchsian system.On the number of linearly independent solutions of the Riemann boundary value problem on the Riemann surface of an algebraic functionhttps://www.zbmath.org/1483.300762022-05-16T20:40:13.078697Z"Kruglov, V. E."https://www.zbmath.org/authors/?q=ai:kruglov.vladislav-e|kruglov.viktor-eSummary: We suggest a modified solution to the Riemann boundary value problem on a Riemann surface of an algebraic function of genus \(\rho \). This allows us to to reduce the problem of finding the number \(l\) of linearly independent algebraic functions (LIAF), that are multiples of a fractional divisor \(Q\), to finding the number of LIAF that are multiples of an effective divisor \(J \) (\(\operatorname{ord}J = \rho\)); this provides a solution to the Jacobi inversion problem given in this paper. We study the case, where the exponents of the normal basis elements coincide, and solve the problem of finding the number of LIAF, multiples of an effective divisor. The definitions of conjugate points of Riemann surface and hyperorder of an effective divisor are introduced. Depending on the structure of divisor \(J\), exact formulas are obtained for number \(l\); they are expressed in terms of the order of divisor \(Q\), the hyperorder of divisor \(J\), and numbers \(\rho\) and \(n\), where \(n\) is the number of sheets of the algebraic Riemann surface.Free boundary problems in the spirit of Sakai's theoremhttps://www.zbmath.org/1483.300772022-05-16T20:40:13.078697Z"Vardakis, Dimitris"https://www.zbmath.org/authors/?q=ai:vardakis.dimitris"Volberg, Alexander"https://www.zbmath.org/authors/?q=ai:volberg.alexander-lSummary: A Schwarz function on an open domain \(\Omega\) is a holomorphic function satisfying \(S(\zeta)=\bar{\zeta}\) on \(\Gamma\), which is part of the boundary of \(\Omega\). \textit{M. Sakai} [Acta Math. 166, No. 3-4, 263--297 (1991; Zbl 0728.30007)] gave a complete characterization of the boundary of a domain admitting a Schwarz function. In fact, if \(\Omega\) is simply connected and \(\Gamma =\partial\Omega\cap D(\zeta_0,r)\), then \(\Gamma\) has to be regular real analytic (with possible cusps). Sakai's result has natural applications to 1) quadrature domains, 2) free boundary problem for \(\Delta u=1\) equation. In our scenarios \(\Gamma\) can be, respectively, from real-analytic to just \(C^\infty\), regular except for a harmonic-measure-zero set, or regular except finitely many points.Uniformization of branched surfaces and Higgs bundleshttps://www.zbmath.org/1483.300782022-05-16T20:40:13.078697Z"Biswas, Indranil"https://www.zbmath.org/authors/?q=ai:biswas.indranil"Bradlow, Steven"https://www.zbmath.org/authors/?q=ai:bradlow.steven-b"Dumitrescu, Sorin"https://www.zbmath.org/authors/?q=ai:dumitrescu.sorin"Heller, Sebastian"https://www.zbmath.org/authors/?q=ai:heller.sebastian-gregorOn \(s\)-extremal Riemann surfaces of even genushttps://www.zbmath.org/1483.300792022-05-16T20:40:13.078697Z"Kozłowska-Walania, Ewa"https://www.zbmath.org/authors/?q=ai:kozlowska-walania.ewaSummary: We consider Riemann surfaces of even genus \(g\) with the action of the group \(\mathcal{D}_n\times \mathbb{Z}_2\), with \(n\) even. These surfaces have the maximal number of 4 non-conjugate symmetries and shall be called \textit{\(s\)-extremal}. We show various results for such surfaces, concerning the total number of ovals, topological types of symmetries, hyperellipticity degree and the minimal genus problem. If in addition an \(s\)-extremal Riemann surface has the maximal total number of ovals, then it shall simply be called \textit{extremal}. In the main result of the paper we find all the families of extremal Riemann surfaces of even genera, depending on if one of the symmetries is fixed-point free or not.The dual volume of quasi-Fuchsian manifolds and the Weil-Petersson distancehttps://www.zbmath.org/1483.300802022-05-16T20:40:13.078697Z"Mazzoli, Filippo"https://www.zbmath.org/authors/?q=ai:mazzoli.filippoSummary: Making use of the dual Bonahon-Schläfli formula, we prove that the dual volume of the convex core of a quasi-Fuchsian manifold \(M\) is bounded by an explicit constant, depending only on the topology of \(M\), times the Weil-Petersson distance between the hyperbolic structures on the upper and lower boundary components of the convex core of \(M\).Quadratic differentials and signed measureshttps://www.zbmath.org/1483.300812022-05-16T20:40:13.078697Z"Baryshnikov, Yuliy"https://www.zbmath.org/authors/?q=ai:baryshnikov.yuliy-m"Shapiro, Boris"https://www.zbmath.org/authors/?q=ai:shapiro.boris-zalmanovichSummary: In this paper, motivated by the classical notion of a Strebel quadratic differential on a compact Riemann surface without boundary, we introduce several more general classes of quadratic differentials (called non-chaotic, gradient, and positive gradient) which possess certain properties of Strebel differentials and often appear in applications. We discuss the relation between gradient differentials and special signed measures supported on their set of critical trajectories. We provide a characterization of gradient differentials for which there exists a positive measure in the latter class.On existence of quasi-Strebel structures for meromorphic \(k\)-differentialshttps://www.zbmath.org/1483.300822022-05-16T20:40:13.078697Z"Shapiro, Boris"https://www.zbmath.org/authors/?q=ai:shapiro.boris-zalmanovich"Tahar, Guillaume"https://www.zbmath.org/authors/?q=ai:tahar.guillaumeA meromorphic differential \(\Psi\) of order \(k\geq 2\) on a compact orientable Riemann surface \(Y\) without boundary is a meromorphic section of the \(k\)-th tensor power \((T^{*}_{\mathbb{C}}Y)^{\otimes k}\) of the holomorphic cotangent bundle of \(Y\). Zeros and poles of \(\Psi\) constitute the set of critical points.
For a differential of order \(k\) given locally by \(f(z)dz^{k}\) in a neighborhood of a non-critical point, there are \(k\) locally distinct directions, called \textit{horizontal}, which are given by the conditions that \(f(z)dz^{k}\) is real and positive.
Recall that a quadratic differential is called \textit{Strebel} if almost all horizontal trajectories are closed. Such a phenomenon can never happen for a \(k\)-differential of order \(k\geq 3\) unless it is a power of a \(1\)-form or a quadratic differential. The authors introduce thus the notion of quasi-Strebel structure and give sufficient conditions for a meromorphic \(k\)-differential to be quasi-Strebel. The main result is the following:
\textbf{Theorem 1.} Let \(\Psi\) be a meromorphic \(k\)-differential such that
\begin{itemize}
\item no poles have of order smaller than \(-k\);
\item at a pole of order \(-k\), the residue belongs to \(i^{k}\mathbb{R}\).
\end{itemize}
Then,
\begin{itemize}
\item[1.] if \(k>2\) is even, then \(\Psi\) has a quasi-Strebel structure;
\item[2.] if \(k>2\) is odd, and, up to a common factor, the period of \(\sqrt[k]{\Psi}\) along every path connecting any two singularities of \(\Psi\) belongs to \(\mathbb{Q}[e^{\frac{2i\pi}{k}}]\), then \(\Psi\) has a quasi-Strebel structure.
\end{itemize}
Reviewer: Andrea Tamburelli (Houston)On the discreteness of states accessible via right-angled paths in hyperbolic spacehttps://www.zbmath.org/1483.300832022-05-16T20:40:13.078697Z"Lessa, Pablo"https://www.zbmath.org/authors/?q=ai:lessa.pablo"Garcia, Ernesto"https://www.zbmath.org/authors/?q=ai:garcia.ernestoSummary: We consider the control problem where, given an orthonormal tangent frame in the hyperbolic plane or three dimensional hyperbolic space, one is allowed to transport the frame a fixed distance \(r>0\) along the geodesic in direction of the first vector, or rotate it in place a right angle. We characterize the values of \(r>0\) for which the set of orthonormal frames accessible using these transformations is discrete.
In the hyperbolic plane this is equivalent to solving the discreteness problem (see [\textit{J.Gilman}, Geom. Dedicata 201, 139--154 (2019; Zbl 1421.30056)] and the references therein) for a particular one parameter family of two-generator subgroups of \(\mathrm{PSL}_2(\mathbb{R})\). In the three dimensional case we solve this problem for a particular one parameter family of subgroups of the isometry group which have four generators.The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti-de Sitter geometryhttps://www.zbmath.org/1483.300842022-05-16T20:40:13.078697Z"Bonsante, Francesco"https://www.zbmath.org/authors/?q=ai:bonsante.francesco"Danciger, Jeffrey"https://www.zbmath.org/authors/?q=ai:danciger.jeffrey"Maloni, Sara"https://www.zbmath.org/authors/?q=ai:maloni.sara"Schlenker, Jean-Marc"https://www.zbmath.org/authors/?q=ai:schlenker.jean-marcA theorem by Alexandrov and Pogorelov says that any smooth Riemannian metric on the 2-sphere with curvature \(K>-1\) coincides with the induced metric on the boundary of some compact convex subset of hyperbolic 3-space with smooth boundary and, furthermore, that this compact convex subset is unique up to a global isometry of hyperbolic 3-space. In the paper under review, the authors study a generalization of this result to unbounded convex subsets of hyperbolic 3-space, more especially to convex subsets bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, they supplement the notion of induced metric on the boundary of the convex set so that it includes a gluing map at infinity which records how the asymptotic geometries of the two surfaces fit together near the limiting Jordan curve. They restrict their study to the case where the induced metrics on the two bounding surfaces have constant curvature \(K \in [-1, 0)\) and were the Jordan curve at infinity is a quasicircle. In this case the gluing map becomes a quasisymmetric homeomorphism of the circle and the authors prove that for \(K\) in the given interval, any quasisymmetric map can be obtained as the gluing map at infinity along some quasicircle. They also obtain Lorentzian analogous of these results, in which hyperbolic 3-space is replaced by the 3-dimensional anti-de Sitter space \(\mathbb{A}d\mathbb{S}^3\), whose natural boundary is the Einstein space \(\mathrm{Ein}^{1,1}\), a conformal Lorentzian analogue of the Riemannian sphere. The authors say that their results may be viewed as universal versions of a conjecture of Thurston about the realization of metrics on boundaries of convex cores of quasifuchsian hyperbolic manifolds and of an analogue of this conjecture, due to Mess, in the setting of globally hyperbolic anti-de Sitter spacetimes.
Reviewer: Athanase Papadopoulos (Strasbourg)From hierarchical to relative hyperbolicityhttps://www.zbmath.org/1483.300852022-05-16T20:40:13.078697Z"Russell, Jacob"https://www.zbmath.org/authors/?q=ai:russell.jacobSummary: We provide a simple, combinatorial criteria for a hierarchically hyperbolic space to be relatively hyperbolic by proving a new formulation of relative hyperbolicity in terms of hierarchy structures. In the case of clean hierarchically hyperbolic groups, this criteria characterizes relative hyperbolicity. We apply our criteria to graphs associated to surfaces and prove that the separating curve graph of a surface is relatively hyperbolic when the surface has zero or two punctures. We also recover a celebrated theorem of Brock and Masur on the relative hyperbolicity of the Weil-Petersson metric on Teichmüller space for surfaces with complexity three.Intersection pairings for higher laminationshttps://www.zbmath.org/1483.300862022-05-16T20:40:13.078697Z"Le, Ian"https://www.zbmath.org/authors/?q=ai:le.ianSummary: One can realize higher laminations as positive configurations of points in the affine building [the author, Geom. Topol. 20, No. 3, 1673--1735 (2016; Zbl 1348.30023)]. The duality pairings of \textit{V. Fock} and \textit{A. Goncharov} [Publ. Math., Inst. Hautes Étud. Sci. 103, 1--211 (2006; Zbl 1099.14025)] give pairings between higher laminations for two Langlands dual groups \(G\) and \(G^{\vee}\). These pairings are a generalization of the intersection pairing between measured laminations on a topological surface.
We give a geometric interpretation of these intersection pairings in a wide variety of cases. In particular, we show that they can be computed as the minimal weighted length of a network in the building. Thus we relate the intersection pairings to the metric structure of the affine building. This proves several of the conjectures from [the author and \textit{E. O'Dorney}, Doc. Math. 22, 1519--1538 (2017; Zbl 1383.51009)]. We also suggest the next steps toward giving geometric interpretations of intersection pairings in general.
The key tools are linearized versions of well-known classical results from combinatorics, like Hall's marriage lemma, König's theorem, and the Kuhn-Munkres algorithm, which are interesting in themselves.On an Enneper-Weierstrass-type representation of constant Gaussian curvature surfaces in 3-dimensional hyperbolic spacehttps://www.zbmath.org/1483.300872022-05-16T20:40:13.078697Z"Smith, Graham"https://www.zbmath.org/authors/?q=ai:smith.graham-a|smith.graham-mSummary: For all \(k\in ]0,1[\), we construct a canonical bijection between the space of ramified coverings of the sphere of hyperbolic type and the space of complete immersed surfaces in 3-dimensional hyperbolic space of finite area and of constant extrinsic curvature equal to \(k\). We show, furthermore, that this bijection restricts to a homeomorphism over each stratum of the space of ramified coverings of the sphere.
For the entire collection see [Zbl 1473.53006].On the extended class of SUPM and their generating URSM over non-Archimedean fieldhttps://www.zbmath.org/1483.300882022-05-16T20:40:13.078697Z"Banerjee, Abhijit"https://www.zbmath.org/authors/?q=ai:banerjee.abhijit"Maity, Sayantan"https://www.zbmath.org/authors/?q=ai:maity.sayantanSummary: In this article, we investigate an extended class of strong uniqueness polynomial over non-Archimedean field than that was recently studied by \textit{H.H. Khoai} and \textit{V. H. An} [\(p\)-Adic Numbers Ultrametric Anal. Appl. 12, No. 4, 276--284 (2020; Zbl 1456.30082)]. We also find the unique range set of weight 2 corresponding to the SUPM which improve and generalize significantly the results of the paper [loc. cit.] and an earlier one due to \textit{P.-C. Hu} and \textit{C.-C. Yang} [Acta Math. Vietnam. 24, No. 1, 95--108 (1999; Zbl 0986.30025)].Bounded extremal problems in Bergman and Bergman-Vekua spaceshttps://www.zbmath.org/1483.300892022-05-16T20:40:13.078697Z"Delgado, Briceyda B."https://www.zbmath.org/authors/?q=ai:delgado.briceyda-b"Leblond, Juliette"https://www.zbmath.org/authors/?q=ai:leblond.julietteSummary: We analyze Bergman spaces \(A_f^p(\mathbb{D})\) of generalized analytic functions of solutions to the Vekua equation \(\bar{\partial}w = (\bar{\partial}f/f)\bar{w}\) in the unit disc of the complex plane, for Lipschitz-smooth non-vanishing real valued functions \(f\) and \(1<p<\infty\). We consider a family of bounded extremal problems (best constrained approximation) in the Bergman space \(A^p(\mathbb{D})\) and in its generalized version \(A^p_f(\mathbb{D})\), that consists in approximating a function in subsets of \(\mathbb{D}\) by the restriction of a function belonging to \(A^p(\mathbb{D})\) or \(A^p_f(\mathbb{D})\) subject to a norm constraint. Preliminary constructive results are provided for \(p = 2\).On the investigation of isotropic thick-walled shellshttps://www.zbmath.org/1483.300902022-05-16T20:40:13.078697Z"Khvoles, A."https://www.zbmath.org/authors/?q=ai:khvoles.a-r|khvoles.alexander|khvoles.a-a"Zgenti, V."https://www.zbmath.org/authors/?q=ai:zgenti.v"Vashakmadze, T."https://www.zbmath.org/authors/?q=ai:vashakmadze.tamaz-s|vashakmadze.tamara-sSummary: We consider the problems of creating 2-dim models for thin-walled structures and satisfaction of boundary conditions when the generalized stress vector is given on the surfaces for elastic plates and shells. This problem was open also both for refined theories in the wide sense and hierarchical type models.Generalized growth of special monogenic functions having finite convergence radiushttps://www.zbmath.org/1483.300912022-05-16T20:40:13.078697Z"Kumar, Susheel"https://www.zbmath.org/authors/?q=ai:kumar.susheelSummary: In the present paper, we study the growth of special monogenic functions having finite convergence radius. The characterizations of generalized order and generalized type of special monogenic functions having finite convergence radius have been obtained in terms of their Taylor's series coefficients.\(k\)-CF functions and \(\Box_b\) on the quaternionic Heisenberg grouphttps://www.zbmath.org/1483.300922022-05-16T20:40:13.078697Z"Shi, Yun"https://www.zbmath.org/authors/?q=ai:shi.yun"Wang, Wei"https://www.zbmath.org/authors/?q=ai:wang.wei.18Summary: The tangential \(k\)-Cauchy-Fueter operator and \(k\)-CF functions on the quaternionic Heisenberg group are quaternionic counterparts of the tangential CR operator \(\overline{\partial}_b\) and CR functions on the Heisenberg group in the theory of several complex variables. We analyze the operator \(\Box_b\) associated the tangential 2-Cauchy-Fueter operator and give its fundamental solution when the coefficients of the group satisfy the condition \(\sum_{l=0}^{n-1}a_l\ne\pm\sum_{l=0}^{n-1}|a_l|\). As an application, we prove that the \(L^p\)-integrable 2-CF function space is trivial in this case. We also discuss the results for general \(k\).Short-time special affine Fourier transform for quaternion-valued functionshttps://www.zbmath.org/1483.300932022-05-16T20:40:13.078697Z"Srivastava, H. M."https://www.zbmath.org/authors/?q=ai:srivastava.hari-mohan"Shah, Firdous A."https://www.zbmath.org/authors/?q=ai:shah.firdous-ahmad"Teali, Aajaz A."https://www.zbmath.org/authors/?q=ai:teali.aajaz-aSummary: The special affine Fourier transform is a promising tool for analyzing transient signals with more degrees of freedom via a chirp-like basis. In this article, our goal is to introduce a novel quaternion-valued short-time special affine Fourier transform in the context of two-dimensional quaternion-valued signals. In addition to studying all fundamental properties of the proposed transform, we also formulate some notable uncertainty inequalities including the Heisenberg-Weyl inequality, logarithmic inequality and local-type inequalities by employing the machinery of quaternionic Fourier transforms. Nevertheless, an illustrative example is presented to endorse the obtained results.Variability regions for the second derivative of bounded analytic functionshttps://www.zbmath.org/1483.300942022-05-16T20:40:13.078697Z"Chen, Gangqiang"https://www.zbmath.org/authors/?q=ai:chen.gangqiang"Yanagihara, Hiroshi"https://www.zbmath.org/authors/?q=ai:yanagihara.hiroshiSummary: Let \(z_0\) and \(w_0\) be given points in the open unit disk \({\mathbb{D}}\) with \(|w_0| < |z_0|\). Let \({\mathcal{H}}_0\) be the class of all analytic self-maps \(f\) of \({\mathbb{D}}\) normalized by \(f(0)=0\), and \({\mathcal{H}}_0 (z_0,w_0) = \{ f \in{\mathcal{H}}_0 : f(z_0) =w_0\} \). In this paper, we explicitly determine the variability region of \(f''(z_0)\) when \(f\) ranges over \({\mathcal{H}}_0 (z_0,w_0)\). Moreover, we approximate this region numerically in some special cases, to illustrate our main result.Corona theorem for the Dirichlet-type spacehttps://www.zbmath.org/1483.300952022-05-16T20:40:13.078697Z"Luo, Shuaibing"https://www.zbmath.org/authors/?q=ai:luo.shuaibingSummary: This paper utilizes Cauchy's transform and duality for the Dirichlet-type space \(D(\mu)\) with positive superharmonic weight \(U_{\mu}\) on the unit disk \(\mathbb{D}\) to establish the corona theorem for the Dirichlet-type multiplier algebra \(M\big( D(\mu)\big)\) that: if
\[
\{ f_1,\ldots ,f_n\} \subseteq M\big( D(\mu )\big) \quad \text{and}\quad \inf_{z\in \mathbb{D}}\sum_{j=1}^n |f_j (z)|>0
\]
then
\[
\exists \,\{ g_1,\ldots,g_n\}\subseteq M\big( D(\mu)\big) \quad \text{such that}\quad \sum_{j=1}^n f_j g_j =1,
\]
thereby generalizing \textit{L. Carleson}'s corona theorem for \(M(H^2)=H^{\infty}\) in [Ann. Math. (2) 76, 547--559 (1962; Zbl 0112.29702)] and \textit{J.Xiao}'s corona theorem for \(M(\mathscr{D})\subset H^{\infty}\) in [Manuscr. Math. 97, No. 2, 217--232 (1998; Zbl 1049.30025)] thanks to
\[
D(\mu )=
\begin{cases}
\text{Hardy space } H^2 \quad & \text{as}\quad \text{d}\mu (z)=(1-|z|^2)\,\text{d}A(z)\quad \forall z\in \mathbb{D}; \\
\text{Dirichlet space}\; \mathscr{D} & \text{as}\quad \text{d}\mu (z)=|\text{d}z|\quad \forall z\in \mathbb{T} =\partial\mathbb{D}. \end{cases}
\]Composition of analytic paraproductshttps://www.zbmath.org/1483.300962022-05-16T20:40:13.078697Z"Aleman, Alexandru"https://www.zbmath.org/authors/?q=ai:aleman.alexandru"Cascante, Carme"https://www.zbmath.org/authors/?q=ai:cascante.carme"Fàbrega, Joan"https://www.zbmath.org/authors/?q=ai:fabrega.joan"Pascuas, Daniel"https://www.zbmath.org/authors/?q=ai:pascuas.daniel"Peláez, José Ángel"https://www.zbmath.org/authors/?q=ai:pelaez.jose-angelSummary: For a fixed analytic function \(g\) on the unit disc \(\mathbb{D}\), we consider the analytic paraproducts induced by \(g\), which are defined by \(T_g f(z)=\int_0^z f(\zeta)g^\prime(\zeta)d\zeta\), \(S_g f(z)=\int_0^z f^\prime(\zeta) g(\zeta)d\zeta\), and \(M_g f(z)=f(z) g(z)\). The boundedness of these operators on various spaces of analytic functions on \(\mathbb{D}\) is well understood. The original motivation for this work is to understand the boundedness of compositions of two of these operators, for example \(T_g^2\), \(T_gS_g\), \(M_g T_g\), etc. Our methods yield a characterization of the boundedness of a large class of operators contained in the algebra generated by these analytic paraproducts acting on the classical weighted Bergman and Hardy spaces in terms of the symbol \(g\). In some cases it turns out that this property is not affected by cancellation, while in others it requires stronger and more subtle restrictions on the oscillation of the symbol \(g\) than the case of a single paraproduct.A note on the phase retrieval of holomorphic functionshttps://www.zbmath.org/1483.300972022-05-16T20:40:13.078697Z"Perez, Rolando III"https://www.zbmath.org/authors/?q=ai:perez.rolando-iiiSummary: We prove that if \(f\) and \(g\) are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then \(f=g\) up to the multiplication of a unimodular constant, provided the segments make an angle that is an irrational multiple of \(\pi\). We also prove that if \(f\) and \(g\) are functions in the Nevanlinna class, and if \(|f|=|g|\) on the unit circle and on a circle inside the unit disc, then \(f=g\) up to the multiplication of a unimodular constant.Norms of inclusions between some spaces of analytic functionshttps://www.zbmath.org/1483.300982022-05-16T20:40:13.078697Z"Llinares, Adrián"https://www.zbmath.org/authors/?q=ai:llinares.adrianSummary: The inclusions between the Besov spaces \(B^q\), the Bloch space \(\mathcal{B}\) and the standard weighted Bergman spaces \(A^p_{\alpha}\) are completely understood, but the norms of the corresponding inclusion operators are in general unknown. In this work, we compute or estimate asymptotically the norms of these inclusions.Complete interpolating sequences for small Fock spaceshttps://www.zbmath.org/1483.300992022-05-16T20:40:13.078697Z"Omari, Youssef"https://www.zbmath.org/authors/?q=ai:omari.youssefSummary: We give a characterization of complete interpolating sequences for the Fock spaces \(\mathcal{F}_\varphi^p\), \(1\leq p< \infty\), where \(\varphi(z)=\alpha(\log^+|z|)^2\), \(\alpha > 0\). Our results are analogous to the classical Kadets-Ingham's 1/4-Theorem on perturbation of Riesz bases of complex exponentials, and they answer a question asked by \textit{A. Baranov} et al. [J. Math. Pures Appl. (9) 103, No. 6, 1358--1389 (2015; Zbl 1315.30009)].On Bloch seminorm of finite Blaschke products in the unit diskhttps://www.zbmath.org/1483.301002022-05-16T20:40:13.078697Z"Baranov, Anton D."https://www.zbmath.org/authors/?q=ai:baranov.anton-d"Kayumov, Ilgiz R."https://www.zbmath.org/authors/?q=ai:kayumov.ilgiz-rifatovich"Nasyrov, Semen R."https://www.zbmath.org/authors/?q=ai:nasyrov.semen-rSummary: We prove that, for any finite Blaschke product \(w = B(z)\) in the unit disk, the corresponding Riemann surface over the \(w\)-plane contains a one-sheeted disk of the radius 0.5. Moreover, it contains a unit one-sheeted disk with a radial slit. We apply this result to obtain a universal lower estimate of the Bloch seminorm for finite Blaschke products.Weak quasicircles have Lipschitz dimension 1https://www.zbmath.org/1483.301012022-05-16T20:40:13.078697Z"Freeman, David M."https://www.zbmath.org/authors/?q=ai:freeman.david-mandellSummary: We prove that the Lipschitz dimension of any bounded turning Jordan circle or arc is equal to 1. Equivalently, the Lipschitz dimension of any weak quasicircle or arc is equal to 1.Quasiconformal Jordan domainshttps://www.zbmath.org/1483.301022022-05-16T20:40:13.078697Z"Ikonen, Toni"https://www.zbmath.org/authors/?q=ai:ikonen.toniSummary: We extend the classical Carathéodory extension theorem to quasiconformal Jordan domains \((Y, d_Y)\). We say that a metric space \((Y, d_Y)\) is a \textit{quasiconformal Jordan domain} if the completion \(\overline{Y}\) of \((Y, d_Y)\) has finite Hausdorff 2-measure, the \textit{boundary} \(\partial Y = \overline{Y}\setminus Y\) is homeomorphic to \(\mathbb{S}^1\), and there exists a homeomorphism \(\phi : \mathbb{D} \rightarrow (Y, d_Y)\) that is quasiconformal in the geometric sense.
We show that \(\phi\) has a continuous, monotone, and surjective extension \(\Phi : \overline{\mathbb{D}} \rightarrow \overline{Y}\). This result is best possible in this generality. In addition, we find a necessary and sufficient condition for \(\Phi\) to be a quasiconformal homeomorphism. We provide sufficient conditions for the restriction of \(\Phi\) to \(\mathbb{S}^1\) being a quasisymmetry and to \(\partial Y\) being bi-Lipschitz equivalent to a quasicircle in the plane.Uniformization of metric surfaces using isothermal coordinateshttps://www.zbmath.org/1483.301032022-05-16T20:40:13.078697Z"Ikonen, Toni"https://www.zbmath.org/authors/?q=ai:ikonen.toniSummary: We establish a uniformization result for metric surfaces -- metric spaces that are topological surfaces with locally finite Hausdorff \(2\)-measure. Using the geometric definition of quasiconformality, we show that a metric surface that can be covered by quasiconformal images of Euclidean domains is quasiconformally equivalent to a Riemannian surface. To prove this, we construct an atlas of suitable isothermal coordinates.Quasisymmetrically minimal Moran sets on packing dimensionhttps://www.zbmath.org/1483.301042022-05-16T20:40:13.078697Z"Li, Yanzhe"https://www.zbmath.org/authors/?q=ai:li.yanzhe"Fu, Xiaohui"https://www.zbmath.org/authors/?q=ai:fu.xiaohui"Yang, Jiaojiao"https://www.zbmath.org/authors/?q=ai:yang.jiaojiaoQuasisymmetric Koebe uniformization with weak metric doubling measureshttps://www.zbmath.org/1483.301052022-05-16T20:40:13.078697Z"Rajala, Kai"https://www.zbmath.org/authors/?q=ai:rajala.kai"Rasimus, Martti"https://www.zbmath.org/authors/?q=ai:rasimus.marttiSummary: We give a characterization of metric spaces quasisymmetrically equivalent to a finitely connected circle domain. This result generalizes the uniformization of Ahlfors 2-regular spaces by \textit{S. Merenkov} and \textit{K. Wildrick} [Rev. Mat. Iberoam. 29, No. 3, 859--910 (2013; Zbl 1294.30043)].Semiconcavity and sensitivity analysis in mean-field optimal control and applicationshttps://www.zbmath.org/1483.301062022-05-16T20:40:13.078697Z"Bonnet, Benoît"https://www.zbmath.org/authors/?q=ai:bonnet.benoit"Frankowska, Hélène"https://www.zbmath.org/authors/?q=ai:frankowska.heleneSummary: In this article, we investigate some of the fine properties of the value function associated with an optimal control problem in the Wasserstein space of probability measures. Building on new interpolation and linearisation formulas for non-local flows, we prove semiconcavity estimates for the value function, and establish several variants of the so-called sensitivity relations which provide connections between its superdifferential and the adjoint curves stemming from the maximum principle. We subsequently make use of these results to study the propagation of regularity for the value function along optimal trajectories, as well as to investigate sufficient optimality conditions and optimal feedbacks for mean-field optimal control problems.On the coefficients of certain subclasses of harmonic univalent mappings with nonzero polehttps://www.zbmath.org/1483.310012022-05-16T20:40:13.078697Z"Bhowmik, Bappaditya"https://www.zbmath.org/authors/?q=ai:bhowmik.bappaditya"Majee, Santana"https://www.zbmath.org/authors/?q=ai:majee.santanaSummary: Let \(Co(p), p\in (0,1]\) be the class of all meromorphic univalent functions \(\varphi\) defined in the open unit disc \(\mathbb{D}\) with normalizations \(\varphi (0)=0=\varphi^{\prime} (0)-1\) and having simple pole at \(z=p\in (0,1]\) such that the complement of \(\varphi (\mathbb{D})\) is a convex domain. The class \(Co(p)\) is called the class of concave univalent functions. Let \(S_H^0 (p)\) be the class of all sense preserving univalent harmonic mappings \(f\) defined on \(\mathbb{D}\) having simple pole at \(z=p\in (0,1)\) with the normalizations \(f(0)=f_z (0)-1=0\) and \(f_{\bar{z}}(0)=0\). We first derive the exact regions of variability for the second Taylor coefficients of \(h\) where \(f=h+\overline{g}\in S_H^0 (p)\) with \(h-g\in Co(p)\). Next we consider the class \(S_H^0 (1)\) of all sense preserving univalent harmonic mappings \(f\) in \(\mathbb{D}\) having simple pole at \(z=1\) with the same normalizations as above. We derive exact regions of variability for the coefficients of \(h\) where \(f=h+\overline{g}\in S_H^0 (1)\) satisfying \(h-e^{2i\theta}g\in Co(1)\) with dilatation \(g^{\prime} (z)/h^{\prime} (z)=e^{-2i\theta}z\), for some \(\theta, 0\leq \theta <\pi\).Extreme points and support points of families of harmonic Bloch mappingshttps://www.zbmath.org/1483.310022022-05-16T20:40:13.078697Z"Deng, Hua"https://www.zbmath.org/authors/?q=ai:deng.hua"Ponnusamy, Saminathan"https://www.zbmath.org/authors/?q=ai:ponnusamy.saminathan"Qiao, Jinjing"https://www.zbmath.org/authors/?q=ai:qiao.jinjingSummary: In this paper, the main aim is to discuss the existence of the extreme points and support points of families of harmonic Bloch mappings and little harmonic Bloch mappings. First, in terms of the Bloch unit-valued set, we prove a necessary condition for a harmonic Bloch mapping (resp. a little harmonic Bloch mapping) to be an extreme point of the unit ball of the normalized harmonic Bloch spaces (resp. the normalized little harmonic Bloch spaces) in the unit disk \(\mathbb{D}\). Then we show that a harmonic Bloch mapping \(f\) is a support point of the unit ball of the normalized harmonic Bloch spaces in \(\mathbb{D}\) if and only if the Bloch unit-valued set of \(f\) is not empty. We also give a characterization for the support points of the unit ball of the harmonic Bloch spaces in \(\mathbb{D}\).Harmonic measure of the outer boundary of colander setshttps://www.zbmath.org/1483.310032022-05-16T20:40:13.078697Z"Glücksam, Adi"https://www.zbmath.org/authors/?q=ai:glucksam.adiSummary: We present two companion results: Phragmén-Lindelöf type tight bounds on the minimal possible growth of subharmonic functions with a recurrent zero set, and tight bounds on the maximal possible decay of the harmonic measure of the outer boundary of colander sets.The existence of minimizers of energy for diffeomorphisms between two-dimensional annuli in \(\mathbb{R}^2\) and \(\mathbb{R}^3\)https://www.zbmath.org/1483.310042022-05-16T20:40:13.078697Z"Kalaj, David"https://www.zbmath.org/authors/?q=ai:kalaj.david"Zhu, Jian-Feng"https://www.zbmath.org/authors/?q=ai:zhu.jianfengSummary: In this paper, we consider the minimization for the Dirichlet energy of Sobolev homeomorphisms between two-dimensional annuli in \(\mathbb{R}^2\) and \(\mathbb{R}^3\), respectively. It should be noticed that in this case a Nitsche phenomenon occurs. The main result of this paper partly extends the corresponding result in [\textit{K. Astala} et al., Arch. Ration. Mech. Anal. 195, No. 3, 899--921 (2010; Zbl 1219.30011)].Completely monotone sequences and harmonic mappingshttps://www.zbmath.org/1483.310052022-05-16T20:40:13.078697Z"Long, Bo-Yong"https://www.zbmath.org/authors/?q=ai:long.boyong"Sugawa, Toshiyuki"https://www.zbmath.org/authors/?q=ai:sugawa.toshiyuki"Wang, Qi-Han"https://www.zbmath.org/authors/?q=ai:wang.qihanSummary: In the present paper, we will study geometric properties of harmonic mappings whose analytic and co-analytic parts are (shifted) generated functions of completely monotone sequences.Some properties of certain close-to-convex harmonic mappingshttps://www.zbmath.org/1483.310072022-05-16T20:40:13.078697Z"Wang, Xiao-Yuan"https://www.zbmath.org/authors/?q=ai:wang.xiaoyuan"Wang, Zhi-Gang"https://www.zbmath.org/authors/?q=ai:wang.zhigang|wang.zhigang.1"Fan, Jin-Hua"https://www.zbmath.org/authors/?q=ai:fan.jinhua"Hu, Zhen-Yong"https://www.zbmath.org/authors/?q=ai:hu.zhenyongSummary: In this paper, we determine the sharp estimates for Toeplitz determinants of a subclass of close-to-convex harmonic mappings. Moreover, we obtain an improved version of Bohr's inequalities for a subclass of close-to-convex harmonic mappings, whose analytic parts are Ma-Minda convex functions.How to keep a spot cool?https://www.zbmath.org/1483.310112022-05-16T20:40:13.078697Z"Solynin, Alexander Yu."https://www.zbmath.org/authors/?q=ai:solynin.alexander-yuSummary: Let \(D\) be a planar domain, let \(a\) be a \textit{reference} point fixed in \(D\), and let \(b_k, k=1,\dots,n\), be \(n\) \textit{controlling} points fixed in \(D\setminus\{a\}\). Suppose further that each \(b_k\) is connected to the boundary \(\partial D\) by an arc \(l_k\). In this paper, we propose the problem of finding a shape of arcs \(l_k\), \(k=1,\dots,n\), which provides the minimum to the harmonic measure \(\omega(a,\bigcup_{k=1}^n l_k,D\setminus \bigcup_{k=1}^n l_k)\). This problem can also be interpreted as a problem on the minimal temperature at \(a\), in the steady-state regime, when the arcs \(l_k\) are kept at constant temperature \(T_1\) while the boundary \(\partial D\) is kept at constant temperature \(T_0< T_1\).
In this paper, we mainly discuss the first non-trivial case of this problem when \(D\) is the unit disk \(\mathbf{D}=\{z:|z|< 1\}\) with the reference point \(a=0\) and two controlling points \(b_1=ir\), \(b_2=-ir\), \(0< r< 1\). It appears, that even in this case our minimization problem is highly nontrivial and the arcs \(l_1\) and \(l_2\) providing minimum for the harmonic measure are not the straight line segments as it could be expected from symmetry properties of the configuration of points under consideration.\(q\)-analyticity in the sense of Ahern \& Brunahttps://www.zbmath.org/1483.320032022-05-16T20:40:13.078697Z"Daghighi, Abtin"https://www.zbmath.org/authors/?q=ai:daghighi.abtinSummary: We consider an alternative notion of polyanalyticity in several complex variables based upon a previous work of \textit{P. Ahern} and \textit{J. Bruna} [Rev. Mat. Iberoam. 4, No. 1, 123--153 (1988; Zbl 0685.42008)]. We also generalize this notion to the case of generic embedded
submanifolds of \(\mathbb{C}^n\) and give characterizations of the notions involved.Necessary density conditions for \(d\)-approximate interpolation sequences in the Bargmann-Fock spacehttps://www.zbmath.org/1483.320042022-05-16T20:40:13.078697Z"Li, Haodong"https://www.zbmath.org/authors/?q=ai:li.haodong"Mitkovski, Mishko"https://www.zbmath.org/authors/?q=ai:mitkovski.mishkoSummary: Inspired by \textit{A. Olevskiĭ} and \textit{A. Ulanovskiĭ} [St. Petersbg. Math. J. 21, No. 6, 1015--1025 (2010); translation from Algebra Anal. 21, No. 6, 227--240 (2009; Zbl 1207.30038)], we introduce the concept of \(d\)-approximate interpolation in weighted Bargmann-Fock spaces as a natural extension of the classical concept of interpolation. We then show that d-approximate interpolation sets satisfy a density condition, similar to the one that classical interpolation sets satisfy. More precisely, we show that the upper Beurling density of any \(d\)-approximate interpolation set must be bounded from above by \(1/(1-d^2)\).Degeneracy second main theorems for meromorphic mappings into projective varieties with hypersurfaceshttps://www.zbmath.org/1483.320082022-05-16T20:40:13.078697Z"Quang, Si Duc"https://www.zbmath.org/authors/?q=ai:si-duc-quang.Summary: The purpose of this paper is twofold. The first purpose is to establish a second main theorem with truncated counting functions for algebraically nondegenerate meromorphic mappings into an arbitrary projective variety intersecting a family of hypersurfaces in subgeneral position. In our result, the truncation level of the counting functions is estimated explicitly. Our result is an extension of the classical second main theorem of H. Cartan and is also a generalization of the recent second main theorem of M. Ru and improves some recent results. The second purpose of this paper is to give another proof for the second main theorem for the special case where the projective variety is a projective space, by which the truncation level of the counting functions is estimated better than that of the general case.Corona theorem for strictly pseudoconvex domainshttps://www.zbmath.org/1483.320162022-05-16T20:40:13.078697Z"Gwizdek, Sebastian"https://www.zbmath.org/authors/?q=ai:gwizdek.sebastianSummary: Nearly 60 years have passed since Lennart Carleson gave his proof of Corona Theorem for unit disc in the complex plane. It was only recently that \textit{M. Kosiek} and \textit{K. Rudol} [``Corona theorem'', Preprint, \url{arXiv:2106.15683}] obtained the first positive result for Corona Theorem in multidimensional case. Using duality methods for uniform algebras the authors proved ``abstract'' Corona Theorem which allowed to solve Corona Problem for a wide class of regular domains. In this paper we expand Corona Theorem to strictly pseudoconvex domains with smooth boundaries.Holomorphic approximation: the legacy of Weierstrass, Runge, Oka-Weil, and Mergelyanhttps://www.zbmath.org/1483.320202022-05-16T20:40:13.078697Z"Fornæss, John Erik"https://www.zbmath.org/authors/?q=ai:fornass.john-erik"Forstnerič, Franc"https://www.zbmath.org/authors/?q=ai:forstneric.franc"Wold, Erlend F."https://www.zbmath.org/authors/?q=ai:wold.erlend-fornaessHolomorphic approximation -- i.e., approximation of, for instance, continuous functions by holomorphic functions -- plays a fundamental role in analysis, as an aim as well as a tool. The authors present the development of holomorphic approximation over a period of almost two centuries, starting with Weierstrass' and Runge's theorems and ending up with recent results on approximation of and by manifold-valued maps due to, in particular, F. Forstnerič. Several of these latter results appear here for the first time. Along the way the authors present and discuss important theorems and methods due to Mergelyan, Oka, Weil, Grauert, and others. Proofs are partly given -- completely, of course, for the new results -- and the ideas behind the proofs are thoroughly discussed. The authors have worked through a tremendous amount of literature (180 items); the paper is valuable for whoever wants to pursue questions of approximation in analysis of one or several complex variables.
For the entire collection see [Zbl 1443.30001].
Reviewer: Ingo Lieb (Bonn)Growth of solutions of non-homogeneous linear differential equations and its applicationshttps://www.zbmath.org/1483.341202022-05-16T20:40:13.078697Z"Pramanik, Dilip Chandra"https://www.zbmath.org/authors/?q=ai:pramanik.dilip-chandra"Biswas, Manab"https://www.zbmath.org/authors/?q=ai:biswas.manabLet \(H\subset \mathbb{C}\) be set with positive upper density, let \(a_0(z),\ldots,a_k(z)\), \(b(z)\) and \(c(z)\) be entire functions and let \(0\leq q < p\). In the paper under review it is shown that if there exists a constant \(\eta >0\) such that \[ |a_j(z)|\leq e^{q|z|^\eta}, \qquad |b(z)| \geq e^{p|z|^\eta}, \qquad |c(z)| \leq e^{q|z|^\eta}, \] for all \(z\in H\), then all meromorphic solutions \(f\not\equiv 0\) of \[ a_k(z) f^{(k)} + \cdots + a_1(z)f' + a_0(z) f = b(z) f + c(z) \] are of infinite order. The paper is concluded by two results on sharing value problems related to the equation above.
Reviewer: Risto Korhonen (Joensuu)The translating soliton equationhttps://www.zbmath.org/1483.351782022-05-16T20:40:13.078697Z"López, Rafael"https://www.zbmath.org/authors/?q=ai:lopez.rafael-beltran|lopez-camino.rafaelSummary: We give an analytic approach to the translating soliton equation with a special emphasis in the study of the Dirichlet problem in convex domains of the plane.
For the entire collection see [Zbl 1473.53006].Escaping Fatou components of transcendental self-maps of the punctured planehttps://www.zbmath.org/1483.370542022-05-16T20:40:13.078697Z"Martí-Pete, David"https://www.zbmath.org/authors/?q=ai:marti-pete.davidHolomorphic self-maps of the Riemann sphere are the most well-studied families of systems in complex dynamics (rational dynamics), followed by self-maps of the once punctured plane (transcendental dynamics). This paper concerns the study of holomorphic self-maps of the twice punctured Riemann sphere, a subject still in its early days.
In transcendental dynamics, an important role is played by the escaping set, which consists of those points which iterate to the essential singularity of the function. In the present setting, however, there are two essential singularities. The main result is that any possible way of escaping is possible for a wandering Fatou component. The author's main technique is approximation theory.
Reviewer: Kirill Lazebnik (New Haven)Exponential polynomials with Fatou and non-escaping sets of finite Lebesgue measurehttps://www.zbmath.org/1483.370552022-05-16T20:40:13.078697Z"Wolff, Mareike"https://www.zbmath.org/authors/?q=ai:wolff.mareikeSummary: We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial \(f\) both have finite Lebesgue measure. Essentially, these conditions are designed such that \(|f(z)|\geq \exp (|z|^\alpha)\) for some \(\alpha>0\) and all \(z\) outside a set of finite Lebesgue measure.Asymptotic upper bound for tangential speed of parabolic semigroups of holomorphic self-maps in the unit dischttps://www.zbmath.org/1483.370572022-05-16T20:40:13.078697Z"Cordella, Davide"https://www.zbmath.org/authors/?q=ai:cordella.davideThe author studies continuous semigroups of holomorphic maps in the unit disc \((\Phi_t)_{t\ge 0}\). For a non-elliptic semigroup, \textit{F. Bracci} [Ann. Univ. Mariae Curie-Skłodowska, Sect. A 73, No. 2, 21--43 (2019; Zbl 1436.30007)] introduced and studied three kinds of speeds: the total speed, the orthogonal speed, and the tangential speed. The tangential speed \(v^T(t)\) is related to the slope of convergence of orbits to the Denjoy-Wolff point of the semigroup. In the present paper, the author proves a conjecture in [loc. cit.] claiming that \(\limsup_{t\to\infty}\left(v^T(t)-\frac 12\log t\right)<\infty\) holds for parabolic semigroups.
Reviewer: Barbara Drinovec Drnovsek (Ljubljana)Complex best \(r\)-term approximations almost always exist in finite dimensionshttps://www.zbmath.org/1483.410102022-05-16T20:40:13.078697Z"Qi, Yang"https://www.zbmath.org/authors/?q=ai:qi.yang"Michałek, Mateusz"https://www.zbmath.org/authors/?q=ai:michalek.mateusz"Lim, Lek-Heng"https://www.zbmath.org/authors/?q=ai:lim.lek-hengSummary: We show that in finite-dimensional nonlinear approximations, the best \(r\)-term approximant of a function \(f\) almost always exists over \(\mathbb{C}\) but that the same is not true over \(\mathbb{R}\), i.e., the infimum \(\inf_{f_1, \dots, f_r \in D} \| f - f_1 - \dots - f_r \|\) is almost always attainable by complex-valued functions \(f_1, \ldots, f_r\) in \(D\), a set (dictionary) of functions (atoms) with some desired structures. Our result extends to functions that possess properties like symmetry or skew-symmetry under permutations of arguments. When \(D\) is the set of separable functions, this is the best rank-\(r\) tensor approximation problem. We show that over \(\mathbb{C}\), any tensor almost always has a unique best rank-\(r\) approximation. This extends to other notions of ranks such as symmetric and alternating ranks, to best \(r\)-block-terms approximations, and to best approximations by tensor networks. Applied to sparse-plus-low-rank approximations, we obtain that for any given \(r\) and \(k\), a general tensor has a unique best approximation by a sum of a rank-\(r\) tensor and a \(k\)-sparse tensor with a fixed sparsity pattern; a problem arising in covariance estimation of Gaussian model with \(k\) observed variables conditionally independent given \(r\) hidden variables. The existential (but not uniqueness) part of our result also applies to best approximations by a sum of a rank-\(r\) tensor and a \(k\)-sparse tensor with no fixed sparsity pattern, and to tensor completion problems.Recent progress in bilinear decompositionshttps://www.zbmath.org/1483.420132022-05-16T20:40:13.078697Z"Fu, Xing"https://www.zbmath.org/authors/?q=ai:fu.xing"Chang, Der-Chen"https://www.zbmath.org/authors/?q=ai:chang.der-chen-e"Yang, Dachun"https://www.zbmath.org/authors/?q=ai:yang.dachunSummary: The targets of this article are twofold. The first one is to give a survey on bilinear decompositions for products of functions in Hardy spaces and their dual spaces, as well as their variants associated with the Schrödinger operator on Euclidean spaces. The second one is to give a new proof of the bilinear decomposition for products of functions in the Hardy space \(H^1\) and BMO on metric measure spaces of homogeneous type. Some applications to div-curl lemmas and commutators are also presented.Some biorthogonal polynomials arising in numerical analysis and approximation theoryhttps://www.zbmath.org/1483.420162022-05-16T20:40:13.078697Z"Lubinsky, D. S."https://www.zbmath.org/authors/?q=ai:lubinsky.doron-s"Sidi, A."https://www.zbmath.org/authors/?q=ai:sidi.avramThe aim of the paper under review is the study of some biorthogonal polynomials that arise in certain topics of Numerical Analysis and Approximation Theory such as Numerical Integration and Convergence Acceleration.
Throughout their work the authors discuss the most general form of biorthogonality, that is, involving two families of polynomials orthogonal with respect to certain measures.They provide a survey of these polynomials, biorthogonal with respect different measures such as the logarithm or exponentials.
The authors also discuss the positivity of the weights in the interpolatory quadrature formulas generated by these biorthogonal polynomials. Finally they show the application of the potential theory, powerful tool in so many problems involving polynomials, to the study of some topics of biorthogonal polynomials such as aymptotics and zero distributions.
Reviewer: María-José Cantero (Zaragoza)On conditions of the completeness of some systems of Bessel functions in the space \(L^2 ((0;1); x^{2p} dx)\)https://www.zbmath.org/1483.420232022-05-16T20:40:13.078697Z"Khats, R. V."https://www.zbmath.org/authors/?q=ai:khats.r-vIn this paper the author gives necessary and sufficient conditions for the system \(\{x^{-p-1}\sqrt{x\rho_k}J_\nu(x\rho_k): k \in \mathbb{N}\}\) to be complete in the weighted space \(L^2((0,1), x^{2p} dx)\). Here \(J_\nu\) is the first kind Bessel function of index \(\nu \geq \frac{1}{2}\), \(p \in \mathbb{R}\) and \(\rho_k : k \in \mathbb{N}\) is an arbitrary sequence of distinct nonzero complex numbers.
The fact that \(\rho_k\) can be arbitrary had already been considered by \textit{B. V. Vynnyts'kyi} and \textit{R. V. Khats'} [Eurasian Math. J. 6, No. 1, 123--131 (2015; Zbl 1463.30015)]. In the present paper, he gives new conditions which depend only on properties of the \(\rho_k\).
Reviewer: Ursula Molter (Buenos Aires)Riesz means in Hardy spaces on Dirichlet groupshttps://www.zbmath.org/1483.430062022-05-16T20:40:13.078697Z"Defant, Andreas"https://www.zbmath.org/authors/?q=ai:defant.andreas"Schoolmann, Ingo"https://www.zbmath.org/authors/?q=ai:schoolmann.ingoIn a recent series of papers the authors have developed a theory of Hardy spaces of general Dirichlet series, closely connected with harmonic analysis on groups. Given a frequency \(\lambda = (\lambda_{n})_{n}\) (i.e., strictly increasing and unbounded), they introduced in [\textit{A. Defant} and \textit{I. Schoolmann}, J. Fourier Anal. Appl. 25, No. 6, 3220--3258 (2019; Zbl 1429.43004)] the notion of \(\lambda\)-Dirichlet group (which defines a family of characters \((h_{\lambda_{n}})_{n}\)). For such a group \(G\), they also defined the Hardy space \(H_{p}^{\lambda}(G)\) for \(1 \leq p \leq \infty\). Here they deal with the convergence of the Riesz means for functions in these spaces.
Given \(f \in H_{1}^{\lambda}(G)\), the first \((\lambda,k)\)-Riesz sum of length \(x >0\) is defined as
\[
R^{\lambda,k}_{x}(f) = \sum_{\lambda_{n}<x} \hat{f}(h_{\lambda_{n}}) \Big( 1 - \frac{\lambda_{n}}{x} \Big)^{k} h_{\lambda_{n}} \,.
\]
The main result of the paper shows that, for every \(k>0\), the expression
\[
R^{\lambda,k}_{\max}(f) (\omega) = \sup_{x >0} \big\vert R^{\lambda,k}_{x}(f) (\omega) \big\vert \,,
\]
for \(f \in H_{1}^{\lambda}(G)\) and \(\omega \in G\), defines a bounded sublinear operator
\[
R^{\lambda,k}_{\max} : H_{1}^{\lambda}(G) \to L_{1,\infty}(G)
\]
and
\[
R^{\lambda,k}_{\max} : H_{p}^{\lambda}(G) \to L_{p}(G) \text{ for } 1 < p \leq \infty .
\]
As a consequence, \(R^{\lambda,k}_{x}(f)(\omega)\) converges (in \(x\)) to \(f(\omega)\) for almost every \(\omega\).
When horizontal translations are considered, the situation improves. It is shown that for \(u,k > 0\), there exists a constant \(C=C(u,k)\) so that for every frequency \(\lambda\), all \(1 \leq p \leq \infty\) and \(f \in H_{p}(G)^{\lambda}\) we have
\[
\bigg( \int_{G} \sup_{x >0} \Big\vert \sum_{\lambda_{n} < x} \hat{f} (h_{\lambda_{n}}) e^{-u\lambda_{n}} \Big( 1 - \frac{\lambda_{n}}{x} \Big)^{k} h_{\lambda_{n}} (\omega) \Big\vert^{p} d \omega \bigg)^{1/p} \leq C \Vert f \Vert_{p} \,.
\]
Note that in this case the inequality holds even for \(p=1\), and that the constant does not depend on \(p\).
One of the main tools to prove the main result is a maximal Hardy-Littlewood operator, adapted to this setting. If \((G, \beta)\) is a Dirichlet group and \(f \in L_{1}(G)\), then for almost every \(\omega \in G\) the function defined by \(f_{\omega}(t) = f(\omega \beta(t))\) is locally integrable on \(\mathbb{R}\). It is proved that the adapted Hardy-Littlewood maximal operator, given by
\[
\overline{M}(f) (\omega) = \sup_{\genfrac{}{}{0pt}{2}{I \subset \mathbb{R}}{\text{interval}}} \frac{1}{\vert I \vert} \int_{I} \vert f_{\omega} (t) \vert dt
\]
defines a sublinear bounded operator \(\overline{M}: L_{1}(G) \to L_{1,\infty}(G)\) and \(\overline{M}: L_{p}(G) \to L_{p}(G)\) for \(1 < p \leq \infty\).
It is known that, for \(1 < p < \infty\) and any frequency \(\lambda\), the sequence \((h_{\lambda_{n}})\) is a Schauder basis of \(H_{p}^{\lambda}(G)\) and, therefore the Riesz means of any function \(f\) converge (in norm) to \(f\). Here it is proved that this is also the case for \(p=1\), that is
\[
\lim_{x \to \infty} \big\Vert R^{\lambda, k}_{x}(f) - f \Vert_{1} =0
\]
for every \(k>0\) and every \(f \in H_{1}^{\lambda}(G)\).
Applications of all these are given to general Dirichlet series and to almost periodic functions.
Reviewer: Pablo Sevilla Peris (Valencia)The factorisation property of \(\ell^\infty(X_k)\)https://www.zbmath.org/1483.460142022-05-16T20:40:13.078697Z"Lechner, Richard"https://www.zbmath.org/authors/?q=ai:lechner.richard"Motakis, Pavlos"https://www.zbmath.org/authors/?q=ai:motakis.pavlos"Müller, Paul F. X."https://www.zbmath.org/authors/?q=ai:muller.paul-f-x"Schlumprecht, Thomas"https://www.zbmath.org/authors/?q=ai:schlumprecht.thomasSummary: In this paper we consider the following problem: let \(X_k\) be a Banach space with a normalised basis \((e_{(k, j)})_j\), whose biorthogonals are denoted by \((e_{(k,j)}^*)_j\), for \(k\in\mathbb{N}\), let \(Z=\ell^\infty(X_k:k\in\mathbb{N})\) be their \(\ell^\infty\)-sum, and let \(T:Z\to Z\) be a bounded linear operator with a large diagonal, i.e.,
\[
\inf\limits_{k,j}\left|e^*_{(k,j)}(T(e_{(k,j)})\right|>0.
\]
Under which condition does the identity on \(Z\) factor through \(T\)? The purpose of this paper is to formulate general conditions for which the answer is positive.Invariant spaces of entire functionshttps://www.zbmath.org/1483.460212022-05-16T20:40:13.078697Z"Krivosheev, A. S."https://www.zbmath.org/authors/?q=ai:krivosheev.aleksandr-sergeevich"Krivosheeva, O. A."https://www.zbmath.org/authors/?q=ai:krivosheeva.o-aLet \(D\subset\mathbb{C}\) be a convex domain and let \(H(D)\) be the space of holomorphic functions on \(D\) endowed with the compact open topology. The paper under review deals with the following problem: Let \(W\) be an invariant subspace of the differentiation operator on \(H(D)\). Which conditions ensure that all functions of \(W\) can be extended to entire functions? This problem naturally arises from the problem of expanding convergence domains of exponential series and their special cases, power series and Dirichlet series. \(W\) is assumed to satisfy \textit{spectral synthesis}, i.e., the closure of the span of the eigenvectors of the differentiation operator in \(H(D)\) is the whole \(W\). The following subset of the unit circle \(\mathbb{T}\) is defined,
\[
J(D)= \Bigl\{\omega\in \mathbb T: \ \sup_{z\in D}\text{Re}\,z\omega=+\infty \Bigr\}.
\]
Let \(\Delta:=\{\lambda_k: k\in\mathbb{N}\}\) be the sequence of eigenvalues of the differentiation operator acting on \(W\). Let \(\Xi(\Delta):=\{\overline{\lambda}/|\lambda|: \lambda\in \Delta\} \). The main theorem asserts that the continuation problem has a positive solution when \(\Xi(\Delta)\subset J(D)\). This result was known only under the assumption that \(J(D)\) is open in \(\mathbb{T}\).
Reviewer: Enrique Jordá (Alicante)RKH spaces of Brownian type defined by Cesàro-Hardy operatorshttps://www.zbmath.org/1483.460232022-05-16T20:40:13.078697Z"Galé, José E."https://www.zbmath.org/authors/?q=ai:gale.jose-e"Miana, Pedro J."https://www.zbmath.org/authors/?q=ai:miana.pedro-j"Sánchez-Lajusticia, Luis"https://www.zbmath.org/authors/?q=ai:sanchez-lajusticia.luisSummary: We study reproducing kernel Hilbert spaces introduced as ranges of generalized Cesàro-Hardy operators, in one real variable and in one complex variable. Such spaces can be seen as formed by absolutely continuous functions on the positive half-line (or paths of infinite length) of fractional order, in the real case. A theorem of Paley-Wiener type is given which connects the real setting with the complex one. These spaces are related with fractional operations in the context of integrated Brownian processes. We give estimates of the norms of the corresponding reproducing kernels.On the Fock kernel for the generalized Fock space and generalized hypergeometric serieshttps://www.zbmath.org/1483.460242022-05-16T20:40:13.078697Z"Park, Jong-Do"https://www.zbmath.org/authors/?q=ai:park.jong-doSummary: In this paper, we compute the reproducing kernel \(B_{m, \alpha}(z, w)\) for the generalized Fock space \(F_{m, \alpha}^2(\mathbb{C})\). The usual Fock space is the case when \(m=2\). We express the reproducing kernel in terms of a suitable hypergeometric series \({}_1 F_q\). In particular, we show that there is a close connection between \(B_{4, \alpha}(z, w)\) and the error function. We also obtain the closed forms of \(B_{m, \alpha}(z, w)\) when \(m=1,2/3,1/2\). Finally, we also prove that \(B_{m, \alpha}(z, z)\sim e^{\alpha |z|^m} |z|^{m-2}\) as \(|z|\longrightarrow\infty\).Rough traces of \textit{BV} functions in metric measure spaceshttps://www.zbmath.org/1483.460352022-05-16T20:40:13.078697Z"Buffa, Vito"https://www.zbmath.org/authors/?q=ai:buffa.vito"Miranda, Michele jun."https://www.zbmath.org/authors/?q=ai:miranda.michele-junSummary: Following a Maz'ya-type approach, we adapt the theory of rough traces of functions of bounded variation (\textit{BV}) to the context of doubling metric measure spaces supporting a Poincaré inequality. This eventually allows for an integration by parts formula involving the rough trace of such functions. We then compare our analysis with the study done in a recent work by
\textit{P.~Lahti} and \textit{N.~Shanmugalingam} [J. Funct. Anal. 274, No.~10, 2754--2791 (2018; Zbl 1392.26016)],
where traces of \textit{BV} functions are studied by means of the more classical Lebesgue-point characterization, and we determine the conditions under which the two notions coincide.Orlicz-Sobolev inequalities and the doubling conditionhttps://www.zbmath.org/1483.460362022-05-16T20:40:13.078697Z"Korobenko, Lyudmila"https://www.zbmath.org/authors/?q=ai:korobenko.lyudmilaSummary: In [\textit{L. Korobenko} et al., Proc. Am. Math. Soc. 143, No.~9, 4017--4028 (2015; Zbl 1325.35055)] it has been shown that a \((p,q)\) Sobolev inequality with \(p>q\) implies the doubling condition on the underlying measure. We show that even weaker Orlicz-Sobolev inequalities, where the gain on the left-hand side is smaller than any power bump, imply doubling. Moreover, we derive a condition on the quantity that should replace the radius on the righ-hand side (which we call ``superradius''), that is necessary to ensure that the space can support the Orlicz-Sobolev inequality and simultaneously be non-doubling.Norms of composition operators on the \(H^2\) space of Dirichlet serieshttps://www.zbmath.org/1483.470452022-05-16T20:40:13.078697Z"Brevig, Ole Fredrik"https://www.zbmath.org/authors/?q=ai:brevig.ole-fredrik"Perfekt, Karl-Mikael"https://www.zbmath.org/authors/?q=ai:perfekt.karl-mikaelSummary: We consider composition operators \(\mathscr{C}_\varphi\) on the Hardy space of Dirichlet series \(\mathscr{H}^2\), generated by Dirichlet series symbols \(\varphi \). We prove two different subordination principles for such operators. One concerns affine symbols only, and is based on an arithmetical condition on the coefficients of \(\varphi \). The other concerns general symbols, and is based on a geometrical condition on the boundary values of \(\varphi \). Both principles are strict, in the sense that they characterize the composition operators of maximal norm generated by symbols having given mapping properties. In particular, we generalize a result of \textit{J. H. Shapiro} [Monatsh. Math. 130, No. 1, 57--70 (2000; Zbl 0951.47026)] on the norm of composition operators on the classical Hardy space of the unit disc. Based on our techniques, we also improve the recently established upper and lower norm bounds in the special case that \(\varphi(s) = c + r 2^{- s} \). A~number of other examples are given.Some essentially normal weighted composition operators on the weighted Bergman spaceshttps://www.zbmath.org/1483.470472022-05-16T20:40:13.078697Z"Fatehi, Mahsa"https://www.zbmath.org/authors/?q=ai:fatehi.mahsa"Shaabani, Mahmood Haji"https://www.zbmath.org/authors/?q=ai:shaabani.mahmood-hajiSummary: First of all, we obtain a necessary and sufficient condition for a certain operator \(T_{w}C_{\varphi}\) to be compact on \(A^{2}_{\alpha}\). Next, we give a short proof for Proposition 2.5 which was proved by \textit{B. D. MacCluer} et al. [Complex Var. Elliptic Equ. 58, No. 1, 35--54 (2013; Zbl 1285.47031)]. Then, we characterize the essentially normal weighted composition operators \(C_{\psi, \varphi}\) on the weighted Bergman spaces \(A^{2}_{\alpha}\), when \(\varphi \in \mathrm{LFT} (\mathbb D)\) is not an automorphism and \(\psi \in H^\infty\) is continuous at a point \(\zeta\) which \(\varphi\) has a finite angular derivative. After that, we find some non-trivially essentially normal weighted composition operators, when \(\varphi \in \mathrm{LFT} (\mathbb D)\) is not an automorphism. In the last section, for \(\varphi \in \mathrm{AUT} (\mathbb D)\) and \(\psi \in {A} (\mathbb D)\), we characterize the essentially normal weighted composition operators \(C_{\psi, \varphi}\) on \(A^{2}_{\alpha}\) and investigate some essentially normal weighted composition operators \(C_{\psi, \varphi}\) on \(H^2\) and \(A^{2}_{\alpha}\). Finally, we find some non-trivially essentially normal weighted composition operators \(C_{\psi, \varphi}\) on \(H^2\) and \(A^{2}_{\alpha}\), when \(\varphi \in \mathrm{AUT} (\mathbb D)\) and \(\psi \in {A} (\mathbb D)\).Differences of generalized weighted composition operators from the Bloch space into Bers-type spaceshttps://www.zbmath.org/1483.470492022-05-16T20:40:13.078697Z"Liu, Xiaosong"https://www.zbmath.org/authors/?q=ai:liu.xiaosong"Li, Songxiao"https://www.zbmath.org/authors/?q=ai:li.songxiaoSummary: We study the boundedness and compactness of the differences of two generalized weighted composition operators acting from the Bloch space to Bers-type spaces.Libera operator on mixed norm spaces \(H_{\nu}^{p,q,\alpha}\) when \(0 < p < 1\)https://www.zbmath.org/1483.470602022-05-16T20:40:13.078697Z"Jevtić, Miroljub"https://www.zbmath.org/authors/?q=ai:jevtic.miroljub"Karapetrović, Boban"https://www.zbmath.org/authors/?q=ai:karapetrovic.bobanSummary: Results from [\textit{M. Pavlović}, ``Definition and properties of the libera operator on mixed norm spaces'', Sci. World J. 2014, Article ID 590656, 15 p. (2014; \url{doi:10.1155/2014/590656})] on Libera operator acting on mixed norm spaces \(H_{\nu}^{p,q,\alpha}\), for \(1 \leq p \leq \infty\), are extended to the case \(0 < p < 1\).\(C\)-selfadjointness of the product of a composition operator and a maximal differentiation operatorhttps://www.zbmath.org/1483.470632022-05-16T20:40:13.078697Z"Shaabani, Mahmood Haji"https://www.zbmath.org/authors/?q=ai:shaabani.mahmood-haji"Fatehi, Mahsa"https://www.zbmath.org/authors/?q=ai:fatehi.mahsa"Hai, Pham Viet"https://www.zbmath.org/authors/?q=ai:pham-viet-hai.Summary: Let \(\varphi\) be an automorphism of \(\mathbb{D}\). In this paper, we consider the operator \(C_\varphi D_{\psi_0,\psi_1}\) on the Hardy space \(H^2\) which is the product of composition and the maximal differential operator. We characterize these operators which are \(C\)-selfadjoint with respect to some conjugations \(C\). Moreover, we find all Hermitian operators \(C_\varphi D_{\psi_0,\psi_1}\), when \(\varphi\) is a rotation.Norm of some operators from logarithmic Bloch-type spaces to weighted-type spaceshttps://www.zbmath.org/1483.470642022-05-16T20:40:13.078697Z"Stević, Stevo"https://www.zbmath.org/authors/?q=ai:stevic.stevoSummary: Operator norm of weighted composition operators from the iterated logarithmic Bloch space \(\mathcal{B}_{{\log}_k} , k\in \mathbb{N}\), or the logarithmic Bloch-type space \(\mathcal{B}_{{\log}^{\beta}},\beta \in (0,1)\) to weighted-type spaces on the unit ball are calculated. It is also calculated norm of the product of differentiation and composition operators among these spaces on the unit disk.Anti-de Sitter geometry and Teichmüller theoryhttps://www.zbmath.org/1483.530022022-05-16T20:40:13.078697Z"Bonsante, Francesco"https://www.zbmath.org/authors/?q=ai:bonsante.francesco"Seppi, Andrea"https://www.zbmath.org/authors/?q=ai:seppi.andreaSummary: The aim of this chapter is to provide an introduction to Anti-de Sitter geometry, with special emphasis on dimension three and on the relations with Teichmüller theory, whose study has been initiated by the seminal paper of Geoffrey Mess in 1990. In the first part we give a broad introduction to Anti-de Sitter geometry in any dimension. The main results of Mess, including the classification of maximal globally hyperbolic Cauchy compact manifolds and the construction of the Gauss map, are treated in the second part. Finally, the third part contains related results which have been developed after the work of Mess, with the aim of giving an overview on the state-of-the-art.
For the entire collection see [Zbl 1470.57002].A weighted Trudinger-Moser inequality on a closed Riemann surface with a finite isometric group actionhttps://www.zbmath.org/1483.580042022-05-16T20:40:13.078697Z"Yang, Jie"https://www.zbmath.org/authors/?q=ai:yang.jie.4|yang.jie.3|yang.jie.1|yang.jie.2Summary: Let \((\Sigma, g)\) be a closed Riemann surface, \(G\) be a finite isometric group acting on \((\Sigma, g)\) and \(H^{1, 2}(\Sigma)\) be the standard Sobolev space. Taking a positive smooth function \(f\) which is \(G\)-invariant, we define a function space \(\mathcal{H}_f^G\) by
\[
\mathcal{H}_f^G=\left\{ u\in H^{1,2}(\Sigma)\left| u(\sigma(x))=u(x), \int_\Sigma uf dv_g=0,\, \forall x\in \Sigma ,\, \forall \sigma \in G \right.\right\}.
\]
Using blow-up analysis, we prove that for any \(\alpha <\lambda_1^f\), the supremum
\[
\sup_{u\in\mathcal{H}_f^G, \int_\Sigma |\nabla_g u|^2fdv_g-\alpha \int_\Sigma u^2fdv_g\le 1}\int_\Sigma e^{4\pi \ell u^2f}dv_g
\]
is attained, where \(\lambda_1^f\) is the first eigenvalue of the \(f\)-Laplacian \(\Delta_f=-\operatorname{div}_g(f\nabla_g)\) on the space \(\mathcal{H}_f^G\), \(\ell =\min_{x\in \Sigma}\sharp G(x)\) and \(\sharp G(x)\) denotes the number of all distinct points of \(G(x)\). Moreover, we consider the case of higher order eigenvalues. Our results generalized those of \textit{Y. Yang} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 65, No. 3, 647--659 (2006; Zbl 1095.58005); J. Differ. Equations 258, No. 9, 3161--3193 (2015; Zbl 1339.46041)] and \textit{Y. Fang} and \textit{Y. Yang} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 20, No. 4, 1295--1324 (2020; Zbl 1471.30005)].Formalizing basic quaternionic analysishttps://www.zbmath.org/1483.684892022-05-16T20:40:13.078697Z"Gabrielli, Andrea"https://www.zbmath.org/authors/?q=ai:gabrielli.andrea"Maggesi, Marco"https://www.zbmath.org/authors/?q=ai:maggesi.marcoSummary: We present a computer formalization of quaternions in the HOL Light theorem prover. We give an introduction to our library for potential users and we discuss some implementation choices.
As an application, we formalize some basic parts of two recently developed mathematical theories, namely, slice regular functions and Pythagorean-hodograph curves.
For the entire collection see [Zbl 1369.68009].A novel approach to the computation of one-loop three- and four-point functions. I: The real mass casehttps://www.zbmath.org/1483.811122022-05-16T20:40:13.078697Z"Guillet, J. Ph"https://www.zbmath.org/authors/?q=ai:guillet.j-ph"Pilon, E."https://www.zbmath.org/authors/?q=ai:pilon.eric"Shimizu, Y."https://www.zbmath.org/authors/?q=ai:shimizu.yuji|shimizu.yasuhiro|shimizu.yasushi|shimizu.yuuko|shimizu.yoshiaki|shimizu.yuya|shimizu.yusuke|shimizu.yoshifumi-r|shimizu.yuichi|shimizu.yukiko-s|shimizu.yuma|shimizu.yoshinori|shimizu.yuuki|shimizu.yasutaka|shimizu.yoshimasa|shimizu.yoshimitsu|shimizu.yosuke|shimizu.yasuyuki|shimizu.yoshiyuki|shimizu.youichiro|shimizu.yuki"Zidi, M. S."https://www.zbmath.org/authors/?q=ai:zidi.m-sThe article presents a novel method for the computation of certain correlation functions in perturbative quantum field theory, namely the three and four point functions evaluated at one loop. It is the first part of a planned series of three articles.
The computation of Feynman diagrams is the most important theoretical tool in the search for new phenomena in high energy physics. At the high precision frontier theoretical computations are matched with experimental results. The continuous increase of experimental accuracy demands an equal development of theoretical methods. While numerous computations can be done numerically, there are structural limitations in computing power. Hence the need of analytical results, even partial.
The series of articles intends to present a method to evaluate certain generalized one-loop type N-point Feynman-type integrals, which enter as building blocks of more complicated amplitudes. The present paper focuses on three and four point functions, where the integrals have the form of a certain rational function integrated over a certain simplex-like domain. In particular the authors present their methods using as a proof of concept the one loop amplitudes in a scalar theory. The method efficiently trivializes the integrals over Feynman parameters as boundary terms, using a Stokes type identity, and obtains all the necessary analytic continuations in a systematic fashion. The paper consider the case with real internal masses, postponing the cases of complex or zero masses to two subsequent articles.
This method is illustrated in detail during the course of the paper, with technical proofs and results explained in 6 appendices. The paper is very clearly written and easy to read, and contains several interesting manipulations of Feynman integrals. The results and the methods explained are likely to prove quite useful in the evaluation of higher loop amplitudes. The paper is mostly aimed to expert in the field and assumes that the reader is already familiar with the computations of loop amplitudes.
Reviewer: Michele Cirafici (Trieste)Co-spherical electronic configuration of the helium-like atomic systemshttps://www.zbmath.org/1483.811622022-05-16T20:40:13.078697Z"Liverts, Evgeny Z."https://www.zbmath.org/authors/?q=ai:liverts.evgeny-zSummary: The properties of a special configuration of a helium-like atomic system, when both electrons are on the surface of a sphere of radius \(r\), and angle \(\theta\) characterizes their positions on sphere, are investigated. Unlike the previous studies, \(r\) is considered as a quantum mechanical variable but not a parameter. It is important that the ``co-spherical'' and the ``collinear'' configuration are coincident in two points. For \(\theta=0\) one obtains the state of the electron-electron coalescence, whereas the angle \(\theta=\pi\) characterizes the \textbf{e-n-e} configuration when the electrons are located at the ends of the diameter of sphere with the nucleus at its center. The Pekeris-like method representing a fully three-body variational technique is used for the expedient calculations. Some interesting features of the expectation values representing the basic characteristics of the ``co-spherical'' electronic configuration are studied. The unusual properties of the expectation values of the operators associated with the kinetic and potential energy of the two-electron atom/ion possessing the ``co-spherical'' configuration are found. Refined formulas for calculations of the two-electron Fock expansion by the Green's function approach are presented. The model wave functions of high accuracy describing the ``co-spherical'' electronic configuration are obtained. All results are illustrated in tables and figures.Spacetimes with continuous linear isotropies. I: Spatial rotationshttps://www.zbmath.org/1483.830062022-05-16T20:40:13.078697Z"MacCallum, M. A. H."https://www.zbmath.org/authors/?q=ai:maccallum.malcolm-a-hSummary: The weakest known criterion for local rotational symmetry (LRS) in spacetimes of Petrov type D is due to \textit{S. W. Goode} and \textit{J. Wainwright} [ibid. 18, 315--331 (1986; Zbl 0584.53029)]. Here it is shown, using methods related to the Cartan-Karlhede procedure, to be equivalent to local spatial rotation invariance of the Riemann tensor and its first derivatives. Conformally flat spacetimes are similarly studied and it is shown that for almost all cases the same criterion ensures LRS. Only for conformally flat accelerated perfect fluids are three curvature derivatives required to ensure LRS, showing that Ellis's original condition for that case is necessary as well as sufficient.Spacetimes with continuous linear isotropies. III: Null rotationshttps://www.zbmath.org/1483.830202022-05-16T20:40:13.078697Z"MacCallum, M. A. H."https://www.zbmath.org/authors/?q=ai:maccallum.malcolm-a-hSummary: It is shown that in many of the possible cases local null rotation invariance of the curvature and its first derivatives is sufficient to ensure that there is an isometry group \(G_r\) with \(r\ge 3\) acting on (a neighbourhood of) the spacetime and containing a null rotation isotropy. The exceptions where invariance of the second derivatives is additionally required to ensure this conclusion are Petrov type N Einstein spacetimes, spacetimes containing ``pure radiation'' (a Ricci tensor of Segre type [(11,2)]), and conformally flat spacetimes with a Ricci tensor of Segre type [1(11,1)] (a ``tachyon fluid'').
For Parts I and II, see [the author, ibid. 53, No. 6, Paper No. 57, 21 p. (2021; Zbl 1483.83006); ibid. 53, No. 6, Paper No. 61, 12 p. (2021; Zbl 1483.83019)].