Mapping Properties for Convolutions Involving Hypergeometric Functions

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 = ∫ z 0 (fμ(t)/t)dt∗f(z) for f ∈ Rη(β).