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International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 17, Pages 1083-1091

Mapping properties for convolutions involving hypergeometric functions

1Department of Mathematics, Pohang University of Science Technology, Kyungbuk, Pohang 790-784, Korea
2Department of Mathematics, College of Natural Sciences, Pusan National University, Pusan 609-735, Korea

Received 4 March 2002

Copyright © 2003 Hindawi Publishing Corporation. 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.


For μ0, we consider a linear operator Lμ:AA defined by the convolution fμf, where fμ=(1μ)z2F1(a,b,c;z)+μz(z2F1(a,b,c;z)). Let φ(A,B) denote the class of normalized functions f which are analytic in the open unit disk and satisfy the condition zf/f(1+Az)/1+Bz, 1A<B1, and let Rη(β) denote the class of normalized analytic functions f for which there exits a number η(π/2,π/2) such that Re(eiη(f(z)β))>0, (β<1). The main object of this paper is to establish the connection between Rη(β) and φ(A,B) involving the operator Lμ(f). Furthermore, we treat the convolution I=0z(fμ(t)/t)dtf(z) for fRη(β).