Close-to-convexity properties of Gaussian hypergeometric functions

Abstract

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 ( c ab )[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 e iη ((1 − z) φ′( z) − β) > 0 for all z ∈ Δ and for some real η ∈ ( −1 2 π, 1 2 π) .