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Let B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ {\mathcal {B}} $$\end{document} be the class of analytic functions f 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} such that |f(z)|<1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ |f(z)|<1 $$\end{document} for all z∈D\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ z\in {\mathbb {D}} $$\end{document}. If f∈B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ f\in {\mathcal {B}} $$\end{document} is 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}$$ f(z)=\sum _{n=0}^{\infty }a_nz^n $$\end{document}, then |∑n=0Nanzn|<1holdsfor|z|<1/2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ |\sum _{n=0}^{N}a_nz^n|<1\;\; \text{ holds } \text{ for }\;\; |z|<{1}/{2} $$\end{document} and the radius 1/2 is best possible for the class B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ {\mathcal {B}} $$\end{document}. This inequality is called the Rogosinski inequality and the corresponding radius is called the Rogosinski radius. Let H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ {\mathcal {H}} $$\end{document} be the class of harmonic functions 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}$$ f=h+{\bar{g}} $$\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}, where h and g are analytic 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}. Let PH0(α)={f=h+g¯∈H:Re(h′(z)-α)>|g′(z)|with0≤α<1,g′(0)=0,z∈D}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {P}}_{{\mathcal {H}}}^{0}(\alpha )=\{f=h+{\overline{g}} \in {\mathcal {H}} : {\text {Re}}(h^{\prime }(z)-\alpha )>|g^{\prime }(z)|\; \text{ with }\; 0\le \alpha <1,\; g^{\prime }(0)=0,\; z \in {\mathbb {D}}\} $$\end{document} be the subclass of close-to-convex harmonic mappings. In this paper, in view of the Euclidean distance, we obtain the sharp Bohr–Rogosinski radius in terms of area measure Sr\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ S_r $$\end{document}, Jacobian Jf(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ J_f(z) $$\end{document} of the functions in the class PH0(α)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ {\mathcal {P}}_{{\mathcal {H}}}^{0}(\alpha ) $$\end{document}.
Computational Methods and Function Theory – Springer Journals
Published: Mar 16, 2022
Keywords: Analytic, univalent, harmonic functions; Starlike, convex, close-to-convex functions; Coefficient estimates; Growth theorem; Bohr radius; Bohr–Rogosisnki radius; Primary 30H05; 30B10 Secondary 30A10; 30C80
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