Bohr’s phenomenon for the classes of Quasi-subordination and K-quasiregular harmonic mappings

By Ming-Sheng Liu, Saminathan Ponnusamy, Jun Wang. RACSAM, 2020

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In this paper, we investigate the Bohr radius for K-quasiregular 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} such that the translated analytic part h(z)-h(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(z)-h(0)$$\end{document} is quasi-subordinate to some analytic function. The main aim of this article is to extend and to establish sharp versions of four recent theorems by Liu and Ponnusamy (Bull Malays Math Sci Soc 42:2151–2168, 2019) and, in particular, we settle affirmatively the two conjectures proposed by them. Furthermore, we establish two refined versions of Bohr’s inequalities and determine the Bohr radius for the derivatives of analytic functions associated with quasi-subordination.

Subfields: Applied Mathematics, Geometry and Topology

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