International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 17, Pages 1083-1091
doi:10.1155/S0161171203203021
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 J. A Kim and K. H. Shon. 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
For μ≥0, we consider a linear operator Lμ:A→A defined by the convolution fμ∗f,
where fμ=(1−μ)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,
−1≤A<B≤1, 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)dt ∗f(z) for f∈Rη(β).