The punishing factors for convex pairs are \(2^{n-1}\)
type: Item
instance of: scholarly article
title: The punishing factors for convex pairs are \(2^{n-1}\)
zbMATH Open document ID: 1147.30014
DOI: 10.4171/RMI/516
author: Karl-Joachim Wirths
author: F. G. Avkhadiev
published in: Revista Matemática Iberoamericana
publication date: 2008-05-29T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmi/1204128303
full work available at URL: https://eudml.org/doc/43617
review text: Let \(\Omega\) and \(\Pi\) be simply connected proper subdomains of the complex plane, \(f:\Omega\rightarrow\Pi\) be a holomorphic function, \(\lambda_\Omega(z)\) and \(\lambda_\Pi(z)\) be the density of the Poincaré metric with curvature \(K=-4\) at \(z\in\Omega\) and \(w\in\Pi,\) respectively. The authors prove that if \(\Omega\) and \(\Pi\) are convex then \[ \frac{|f^{(n)}(z)|}{n!}\leq 2^{n-1}\frac{(\lambda_\Omega(z))^n}{\lambda_\Pi(f(z))} \] and if \(\Omega\) is convex and \(\Pi\) linearly accessible then \[ \frac{|f^{(n)}(z)|}{(n+1)!}\leq 2^{n-2}\frac{(\lambda_\Omega(z))^n}{\lambda_\Pi(f(z))} \] for \(n=2,3,\ldots\) and \(z\in\Omega.\) Even more \(2^{n-1}\) and \(2^{n-2}\) are the best possible for any admissible pair \((\Omega,\Pi).\) The title of the paper is motivated by the fact that the quotient \(\frac{(\lambda_\Omega(z))^n}{\lambda_\Pi(f(z))}\) reflects the influence of the position of the points \(z\) and \(f(z)\) in \(\Omega\) and \(\Pi\) on the \(f^{(n)}(z)\) and the coefficients \(2^{n-1}\) and \(2^{n-2}\) are factors that ``punish'' bad behavior of \(\Omega\) and \(\Pi\) at the boundary.
reviewed by: Nikola Tuneski
Mathematics Subject Classification ID: 30C50
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30D50
zbMATH DE Number: 5281233
zbMATH Keywords: convex domain
zbMATH Keywords: linear accessible domain
zbMATH Keywords: Taylor coefficients
zbMATH Keywords: convex functions
zbMATH Keywords: close-to-convex functions
zbMATH Keywords: inverse functions
zbMATH Keywords: bounded functions
MaRDI profile type: Publication
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OpenAlex ID: W2051250235
description: scientific article; zbMATH DE number 5281233