On extremal problems related to integral transforms of a class of analytic functions

Abstract

For γ ≥ 0 and β<1 given, let P_γ (β) denote the class of all analytic functions f in the unit disk with the normalization f(0)=f'(0)−1=0 and satisfying the condition Re{e^iφ(f'(z) +γzf"(z) − β)} > 0, z ε D, for some φ ∈ R. We shall investigate the integral transform V_λ(f)(z) = ∫₀¹ λ(t) f(tz)/t dt, where λ is a nonnegative real valued function normalized by ∫₀¹λ(t)dt = 1. From our main results we get conditions on the number β and the function λ such that V_λ(f) is starlike of order μ (0≤μ≤½ ) when f ∈ P_γ(β). As applications we study various choices of λ(t), related to classical integral transforms.