Geometric Properties of an Integral Operator

Abstract

Let P γ (α, β), denote the class of all normalized analytic functions f defined in the unit disc E = {z : |z| < 1} such thatfor all z ∈ E and η ∈ R, where β < 1, α ≥ 0 and 0 ≤ γ ≤ 1. For a real-valued nonnegative function λ with the normalization 1 0 λ(t)dt = 1, we consider the integral operatorsand a class S δ (ν) of normalized analytic functions f which satisfy the conditionfor δ < ν ≤ 1+δ and 0 ≤ δ < 1. The aim of this paper is to find the sharp value of β so that the operator V λ,α ( f ) carries P γ (α, β) into S δ (α). Some interesting applications for different choices of λ are discussed.