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

For γ 0 and β <1 given, let Pγ (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 eiφ f (z)+γ zf (z) −β > 0, z∈ D, for some φ ∈ R. We shall investigate the integral transform Vλ(f )(z) = 1 0 λ(t) f (tz) t dt, where λ is a nonnegative real valued function normalized by 1 0 λ(t) dt = 1. From our main results we get conditions on the number β and the function λ such that Vλ(f ) is starlike of order μ (0 μ 1/2) when f ∈ Pγ (β). As applications we study various choices of λ(t), related to classical integral transforms. © 2007 Elsevier Inc. All rights reserved