On an Integral Transform of a Class of Analytic Functions

Abstract

For α, γ ≥ 0 and β < 1, let 𝒲β(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 𝒲β(1,0) are close‐to‐convex and hence univalent for 0 ≤ β < 1. For f ∈ 𝒲β(α, γ), we consider the integral transform , where λ is a nonnegative real‐valued integrable function satisfying the condition . The aim of present paper is, for given δ < 1, to find sharp values of β such that (i) Vλ(f) ∈ 𝒲δ(1,0) whenever f ∈ 𝒲β(α, γ) and (ii) Vλ(f) ∈ 𝒲δ(α, γ) whenever f ∈ 𝒲β(α, γ).