On an Integral Transform of a Class of Analytic Functions

Abstract EN

For α,γ≥0 and β<1, let ?β(α,γ) denote the class of all normalized analytic functions f in the open unit disc E={z:|z|<1} such that ℜeiϕ((1-α+2γ)(f(z)/z)+(α-2γ)f′(z)+γzf′′(z)-β)>0, z∈E for some ϕ∈ℝ.

It is known (Noshiro (1934) and Warschawski (1935)) that functions in ?β(1,0) are close-to-convex and hence univalent for 0≤β<1.

For f∈?β(α,γ), we consider the integral transform F(z)=Vλ(f)(z):=∫01λ(t)(f(tz)/t)dt, where λ is a nonnegative real-valued integrable function satisfying the condition ∫01λ(t)dt=1.

The aim of present paper is, for given δ<1, to find sharp values of β such that (i) Vλ(f)∈?δ(1,0) whenever f∈?β(α,γ) and (ii) Vλ(f)∈?δ(α,γ) whenever f∈?β(α,γ).