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 conditionRe {ei φ{symbol} (f′ (z) + γ z f″ (z) - β)} > 0, z ∈ D, for some φ{symbol} ∈ R. We shall investigate the integral transformVλ (f) (z) = underover(∫, 0, 1) λ (t) frac(f (t z), t) d t, where λ is a nonnegative real valued function normalized by ∫01 λ (t) d t = 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.