International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 4, Pages 765-773
doi:10.1155/S0161171299227652

Sufficiency for Gaussian hypergeometric functions to be uniformly convex

Yong Chan Kim1 and S. Ponnusamy2

1Department of Mathematics, Yeungnam University, 214-1, Daedong, Gyongsan 712-749, Korea
2Department of Mathematics, University of Helsinki, P. O. Box 4, Hallitskatu 15, Helsinki FIN-00014, Finland

Received 10 July 1997; Revised 27 October 1997

Copyright © 1999 Yong Chan Kim and S. Ponnusamy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let F(a,b;c;z) be the classical hypergeometric function and f be a normalized analytic functions defined on the unit disk 𝒰. Let an operator Ia,b;c(f) be defined by [Ia,b;c(f)](z)=zF(a,b;c;z)*f(z). In this paper the authors identify two subfamilies of analytic functions ℱ1 and ℱ2 and obtain conditions on the parameters a,b,c such that f∈ℱ1 implies Ia,b;c(f)∈ℱ2.