Univalence and starlikeness of certain transforms defined by convolution of analytic functions ✩

Let U(λ) denote the class of all analytic functions f in the unit disk Δ of the form f (z) = z+a2z2+··· satisfying the condition f (z) z f (z) 2 − 1 λ, z ∈ Δ. In this paper we find conditions on λ and on c ∈ C with Re c 0 = c such that for each f ∈ U(λ) satisfying (z/f (z)) ∗ F(1, c;c + 1;z) = 0 for all z ∈ Δ the transform G(z) = Gc f (z) = z (z/f (z)) ∗ F(1, c;c + 1;z) , z∈ Δ, is univalent or starlike. Here F(a,b;c;z) denotes the Gauss hypergeometric function and ∗ denotes the convolution (or Hadamard product) of analytic functions on Δ. © 2007 Elsevier Inc. All rights reserved.