Computational Methods and Function Theory 2 (2002), No. 2, 597--610
Copyright Heldermann Verlag 2002
Reinhold Küstner
kustner@ano.univ-lille1.fr, Equipe ANO, U.F.R. de Mathématiques, Bâtiment M2--318, Université des Sciences et Technologies de Lille, 59655 Villeneuve d'Ascq cedex, France.
We determine the order of convexity of hypergeometric functions $z\mapsto F(a,b,c,z)$ as well as the order of starlikeness of shifted hypergeometric functions $z\mapsto zF(a,b,c,z)$, for certain ranges of the real parameters $a,b$ and $c$. As a consequence we obtain the sharp lower bound for the order of convexity of the convolution $(f*g)(z) := \sum_{n=0}^{\infty}a_nb_nz^n$ when $f(z) = \sum_{n=0}^{\infty} a_nz^n$ is convex of order $\alpha\in[0,1]$ and $g(z) = \sum_{n=0}^{\infty}b_nz^n$ is convex of order $\beta\in[0,1]$, and likewise we obtain the sharp lower bound for the order of starlikeness of $f*g$ when $f,g$ are starlike of order $\alpha,\beta\in[1/2,1]$, respectively. Further we obtain 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 we use the continued fraction of Gauss, a theorem of Wall which yields a characterization of Hausdorff moment sequences by means of (continued) $g$-fractions, and results of Merkes, Wirths and P\'{o}lya. Finally we state a subordination problem.
Keywords: Hadamard product, hypergeometric function, order of convexity, order of starlikeness, convexity in direction of the imaginary axis, continued fraction of Gauss, $g$-fraction, Hausdorff moment sequence.
MSC 2000: 30C45, 33C05.
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