Close-to-convexity properties of Gaussian hypergeometric functions

Let A be the class of normalized analytic functions in the unit disk Δ. Let φ(z) be either zF(a,b;c;z) or (cab)[F(a,b;c;z) − 1], where F(a,b;c;z) denotes the classical hypergeometric function. The purpose of this paper is to study close-to-convexity (and hence univalency) of φ(z) in the unit disc. More generally, we find conditions on a,b,c and β such that φ satisfies Re eiη((1 − z)φ′(z) − β) > 0 for all z ∈ Δ and for some real η ∈ (−12π, 12π).