Mapping properties for convolutions involving hypergeometric functions

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).