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Punishing Factors for Angles

Punishing Factors for Angles Let Ω and П be two simply connected domains in the complex plane C which are not equal to the whole plane C We consider functions f: Ω → П II analytic in Ω and we get estimates for $|f^{(n)}(z)|,z\in \Omega$ , which are sharp in the following sense. Let λΩ(z) and λП(w) denote the reciprocal of the conformal radius of Ω in z and of П in w, respectively. Inequalities of the type $${|f^{(n)}(z)|\over n!}\leq M_{n}(z,\Omega,\Pi){(\lambda_{\Omega}(z))^{n}\over \lambda_{\Pi}(f(z))^{'}}\qquad z\in \Omega$$ , are considered where $M_{n}(z,\Omega,\Pi)$ does not depend on f and represents the smallest value possible at this place. We especially consider cases where Ω or П is an angular domain H α with opening angle απ, 1 ≤ α ≤ 2. We determine M n(z, Δ, H α) where Δ denotes the unit disk. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Punishing Factors for Angles

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References (22)

Publisher
Springer Journals
Copyright
Copyright © 2003 by Heldermann  Verlag
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/BF03321030
Publisher site
See Article on Publisher Site

Abstract

Let Ω and П be two simply connected domains in the complex plane C which are not equal to the whole plane C We consider functions f: Ω → П II analytic in Ω and we get estimates for $|f^{(n)}(z)|,z\in \Omega$ , which are sharp in the following sense. Let λΩ(z) and λП(w) denote the reciprocal of the conformal radius of Ω in z and of П in w, respectively. Inequalities of the type $${|f^{(n)}(z)|\over n!}\leq M_{n}(z,\Omega,\Pi){(\lambda_{\Omega}(z))^{n}\over \lambda_{\Pi}(f(z))^{'}}\qquad z\in \Omega$$ , are considered where $M_{n}(z,\Omega,\Pi)$ does not depend on f and represents the smallest value possible at this place. We especially consider cases where Ω or П is an angular domain H α with opening angle απ, 1 ≤ α ≤ 2. We determine M n(z, Δ, H α) where Δ denotes the unit disk.

Journal

Computational Methods and Function Theory – Springer Journals

Published: Mar 7, 2013

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