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Bohr Phenomenon for Certain Close-to-Convex Analytic Functions

Bohr Phenomenon for Certain Close-to-Convex Analytic Functions We say that a class G\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {G}}$$\end{document} of analytic functions f of the form f(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}$$f(z)=\sum _{n=0}^{\infty } a_{n}z^{n}$$\end{document} in the unit disk D:={z∈C:|z|<1}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {D}}:=\{z\in {\mathbb {C}}: |z|<1\}$$\end{document} satisfies a Bohr phenomenon if for the largest radius Rf<1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_{f}<1$$\end{document}, the following inequality ∑n=1∞|anzn|≤d(f(0),∂f(D))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \sum \limits _{n=1}^{\infty } |a_{n}z^{n}| \le d(f(0),\partial f({\mathbb {D}}) ) \end{aligned}$$\end{document}holds for |z|=r≤Rf\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|z|=r\le R_{f}$$\end{document} and for all functions f∈G\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f \in {\mathcal {G}}$$\end{document}. The largest radius Rf\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_{f}$$\end{document} is called Bohr radius for the class G\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {G}}$$\end{document}. In this article, we obtain the Bohr radius for certain subclasses of close-to-convex analytic functions. We establish the Bohr phenomenon for certain analytic classes Sc∗(ϕ),Cc(ϕ),Cs∗(ϕ),Ks(ϕ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {S}}_{c}^{*}(\phi ),\,{\mathcal {C}}_{c}(\phi ),\, {\mathcal {C}}_{s}^{*}(\phi ),\, {\mathcal {K}}_{s}(\phi )$$\end{document} and obtain the radius Rf\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_{f}$$\end{document} such that the Bohr phenomenon for these classes holds for |z|=r≤Rf\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|z|=r\le R_{f}$$\end{document}. As a consequence of these results, we obtain several interesting corollaries about the Bohr phenomenon for the aforesaid classes. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Bohr Phenomenon for Certain Close-to-Convex Analytic Functions

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Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-021-00412-6
Publisher site
See Article on Publisher Site

Abstract

We say that a class G\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {G}}$$\end{document} of analytic functions f of the form f(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}$$f(z)=\sum _{n=0}^{\infty } a_{n}z^{n}$$\end{document} in the unit disk D:={z∈C:|z|<1}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {D}}:=\{z\in {\mathbb {C}}: |z|<1\}$$\end{document} satisfies a Bohr phenomenon if for the largest radius Rf<1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_{f}<1$$\end{document}, the following inequality ∑n=1∞|anzn|≤d(f(0),∂f(D))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \sum \limits _{n=1}^{\infty } |a_{n}z^{n}| \le d(f(0),\partial f({\mathbb {D}}) ) \end{aligned}$$\end{document}holds for |z|=r≤Rf\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|z|=r\le R_{f}$$\end{document} and for all functions f∈G\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f \in {\mathcal {G}}$$\end{document}. The largest radius Rf\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_{f}$$\end{document} is called Bohr radius for the class G\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {G}}$$\end{document}. In this article, we obtain the Bohr radius for certain subclasses of close-to-convex analytic functions. We establish the Bohr phenomenon for certain analytic classes Sc∗(ϕ),Cc(ϕ),Cs∗(ϕ),Ks(ϕ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {S}}_{c}^{*}(\phi ),\,{\mathcal {C}}_{c}(\phi ),\, {\mathcal {C}}_{s}^{*}(\phi ),\, {\mathcal {K}}_{s}(\phi )$$\end{document} and obtain the radius Rf\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_{f}$$\end{document} such that the Bohr phenomenon for these classes holds for |z|=r≤Rf\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|z|=r\le R_{f}$$\end{document}. As a consequence of these results, we obtain several interesting corollaries about the Bohr phenomenon for the aforesaid classes.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Sep 1, 2022

Keywords: Starlike; Convex; Close-to-convex; Quasi-convex functions; Conjugate points; Symmetric points; Subordination; Majorant series; Bohr radius; Primary 30C45; 30C50; 30C80

References