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 + a2 z2 + ⋯ satisfying the condition| f′ (z) (frac(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 transformG (z) = Gfc (z) = frac(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.