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Zbl 1053.30006
Küstner, Reinhold
(Ruscheweyh, S.)
Mapping properties of hypergeometric functions and convolutions of starlike or convex functions of order $\alpha$.
(English)
[J] Comput. Methods Funct. Theory 2, No. 2, 597-610 (2002). ISSN 1617-9447

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. \par 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. \par 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.
[Dorin Blezu (Sibiu)]
MSC 2000:
*30C45 Special classes of univalent and multivalent functions
33C05 Classical hypergeometric functions

Keywords: Hadamard product; hypergeometric function; order of convexity; order of starlikness; convexity in direction of the imaginary axis; continued fraction of Gauss; g-fraction; Hausdorff sequence

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