Mapping properties of hypergeometric functions and convolutions of starlike or convex functions of order \(\alpha\)
type: Item
instance of: scholarly article
title: Mapping properties of hypergeometric functions and convolutions of starlike or convex functions of order \(\alpha\)
zbMATH Open document ID: 1053.30006
DOI: 10.1007/BF03321867
author: Reinhold Küstner
published in: Computational Methods and Function Theory
publication date: 2004-09-27T00:00:00Z
review text: The authors determine the order of convexity of hypergeometric functions \(z\rightarrow F(a, b, c, z)\) as well as the order of starlikness of shifted hypergeometric functions \(z\rightarrow F(a, b, c, z)\), for certain ranges of the real parameters \(a, b\) and \(c\). As a consequence he obtains the sharp lower bound for the order of convexity of the convolution \((f \star g)(z):= \sum_{n=0}^{\infty} a_{n}b_{n}z^{n}\) when \(f(z)= \sum_{n=0}^{\infty} a_{n}z^{n}\) is convex of order \(\alpha \in [0, 1]\) and \(g(z)= \sum_{n=0}^{\infty} b_{n}z^{n}\) is convex of order \(\beta \in [0, 1]\), and likewise obtains the sharp lower bound for the order of starlikness of \(f\star g\) when \(f, g\) are starlike of order \(\alpha, \beta \in [1/2, 1]\), respectively. Further he obtains convexity in the direction of the imaginary axis for hypergeometric functions and for three ratios of hypergeometric functions as well as for the corresponding shifted expressions. In the proofs he uses the continued fraction of Gauss, a theorem of Wall which yields a characterization of Haussdorff moment sequences by means of (continued) g-fractions, and results of Merkes, Wirths and Pólya. Finally he states a subordination problem. This paper presents the main result of the author's Diploma thesis written at the University of Würzburg under the guidance of professor Stephan Ruscheweyh to whom the author is deeply indepted as well as to Richard Greiner.
reviewed by: Dorin Blezu
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 33C05
zbMATH DE Number: 2103486
zbMATH Keywords: Hadamard product
zbMATH Keywords: hypergeometric function
zbMATH Keywords: order of convexity
zbMATH Keywords: order of starlikness
zbMATH Keywords: convexity in direction of the imaginary axis
zbMATH Keywords: continued fraction of Gauss
zbMATH Keywords: g-fraction
zbMATH Keywords: Hausdorff sequence
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2103486