Recent zbMATH articles in MSC 30Ahttps://www.zbmath.org/atom/cc/30A2022-05-16T20:40:13.078697ZWerkzeugAnalog 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.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)].\(L^r\) 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.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.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.