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Mapping properties of hypergeometric functions and convolutions of starlike or convex functions of order $\alpha$. (English) Zbl 1053.30006

The authors determine the order of convexity of hypergeometric functions $z\to F\left(a,b,c,z\right)$ as well as the order of starlikness of shifted hypergeometric functions $z\to F\left(a,b,c,z\right)$, 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 $\left(f☆g\right)\left(z\right):={\sum }_{n=0}^{\infty }{a}_{n}{b}_{n}{z}^{n}$ when $f\left(z\right)={\sum }_{n=0}^{\infty }{a}_{n}{z}^{n}$ is convex of order $\alpha \in \left[0,1\right]$ and $g\left(z\right)={\sum }_{n=0}^{\infty }{b}_{n}{z}^{n}$ is convex of order $\beta \in \left[0,1\right]$, and likewise obtains the sharp lower bound for the order of starlikness of $f☆g$ when $f,g$ are starlike of order $\alpha ,\beta \in \left[1/2,1\right]$, 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.

##### MSC:
 30C45 Special classes of univalent and multivalent functions 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$