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Estimates of the Derivatives of Meromorphic Maps from Convex Domains into Concave Domains

Estimates of the Derivatives of Meromorphic Maps from Convex Domains into Concave Domains Let ω be a proper convex subdomain of the complex plane C. Let further π1 ⊂ ℂ be a compact convex set containing more than one point and π = \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}${\rm}\overline C\ \Pi_1$\end{document}. We denote by RΩ (z) and Rπ (ω) the conformal radius of Ω at z and of π at finite points ω, respectively. We are concerned with the set A(Ω, π) of functions f: Ω → π meromorphic on Ω. We prove that for n ≥ 2, f ∈ A(Ω, π), z∈Ω and f(z) finite the inequalities \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${|f^{(n)}(z)|\over n!}{(R_\Omega(z))^n\over R_\Pi(f(z))} \leq {(1+p)^{n-2}\over p^{n-1}} {\sum^n_{k=0}} p^k$$\end{document}are valid, where p is a measure for the distance between f(z) and the point at infinity.We give examples showing that equality is possible in this estimate. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Estimates of the Derivatives of Meromorphic Maps from Convex Domains into Concave Domains

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Publisher
Springer Journals
Copyright
Copyright © Heldermann  Verlag 2008
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/bf03321674
Publisher site
See Article on Publisher Site

Abstract

Let ω be a proper convex subdomain of the complex plane C. Let further π1 ⊂ ℂ be a compact convex set containing more than one point and π = \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}${\rm}\overline C\ \Pi_1$\end{document}. We denote by RΩ (z) and Rπ (ω) the conformal radius of Ω at z and of π at finite points ω, respectively. We are concerned with the set A(Ω, π) of functions f: Ω → π meromorphic on Ω. We prove that for n ≥ 2, f ∈ A(Ω, π), z∈Ω and f(z) finite the inequalities \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${|f^{(n)}(z)|\over n!}{(R_\Omega(z))^n\over R_\Pi(f(z))} \leq {(1+p)^{n-2}\over p^{n-1}} {\sum^n_{k=0}} p^k$$\end{document}are valid, where p is a measure for the distance between f(z) and the point at infinity.We give examples showing that equality is possible in this estimate.

Journal

Computational Methods and Function TheorySpringer Journals

Published: May 1, 2008

Keywords: Convex domain; concave domain; nth derivative; conformal radius; subordination; 30C80; 30C55; 30C20

References