Recent zbMATH articles in MSC 30Dhttps://www.zbmath.org/atom/cc/30D2022-05-16T20:40:13.078697ZWerkzeugA 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.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})].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.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.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 \(|z|\geq k\), \(k\leq 1\).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.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)].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)].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 \(\mathbb{D}\).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.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)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)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)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)