Univalency of weighted integral transforms of certain functions

For β<1 and γ⩾0, let Pγ(β) denote the class of all normalized analytic functions f in the unit disc such thatReeiη(1-γ)f(z)z+γf′(z)-β>0,z∈Δ,for some η∈R. For f∈Pγ(β), we consider the integral transformVλ(f)(z)=∫01λ(t)f(tz)tdt,where λ(t) is a real-valued nonnegative weight function so that ∫01λ(t)dt=1. The main aim of this paper is to find conditions such that Vλ(f)∈P1(α) whenever f∈Pγ(β) for γ⩾1. We also obtain conditions such that Vλ(f)∈P1(β′) whenever f∈P0(β) for various choices of λ(t). As a useful consequence, we find conditions for certain hypergeometric functions to be univalent.