Bohr Radius for Subordination and K-quasiconformal Harmonic Mappings

By Zhihong Liu, Saminathan Ponnusamy. Bulletin of the Malaysian Mathematical Sciences Society, 2019

68 citations recorded by the source corpus.

The present article concerns the Bohr radius for K-quasiconformal sense-preserving harmonic mappings f=h+g¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=h+\overline{g}$$\end{document} in the unit disk D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {D}$$\end{document} for which the analytic part h is subordinated to some analytic function φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document}, and the purpose is to look into two cases: when φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} is convex, or a general univalent function in D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {D}$$\end{document}. The results state that if h(z)=∑n=0∞anzn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(z) =\sum _{n=0}^{\infty }a_n z^n$$\end{document} and g(z)=∑n=1∞bnzn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(z)=\sum _{n=1}^{\infty }b_n z^n$$\end{document}, then ∑n=1∞(|an|+|bn|)rn≤dist(φ(0),∂φ(D))forr≤r∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n=1}^{\infty }(|a_n|+|b_n|)r^n\le {{\text {dist}}}(\varphi (0),\partial \varphi (\mathbb {D})) \quad \text{ for } r\le r^* \end{aligned}$$\end{document}and give estimates for the largest possible r∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^*$$\end{document} depending only on the geometric property of φ(D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (\mathbb {D})$$\end{document} and the parameter K. Improved versions of the theorems are given for the case when b1=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_1 = 0$$\end{document} and corollaries are drawn for the case when K→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\rightarrow \infty $$\end{document}.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c94c1fc176de11455954ae2738335cf682237eec

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Citations in the corpus

Bohr phenomenon for certain subclasses of harmonic mappings (2020, background): “For more intriguing aspects of Bohr phenomenon we refer to [5, 11, 12, 19, 26, 31] and the references therein.”
Multidimensional analogues of refined Bohr’s inequality (2020, background): “Several other aspects and generalizations of Bohr’s inequality may be obtained from [8, 10, 18, 21, 27, 32, 33, 34, 35], the monograph of Kresin and Maz’ya [28], and the references therein.”
Bohr radius for certain classes of starlike and convex univalent functions (2020, background): “Bohr phenomenon for harmonic mappings has also been extensively studied by several authors (see [4, 11, 17, 23, 24, 26, 30]).”