Univalency of weighted integral transforms of certain functions

For β < 1 and γ {greater than or slanted equal to} 0, let Pγ ( β ) denote the class of all normalized analytic functions f in the unit disc such that{A formula is presented}for some η ∈ R. For f ∈ Pγ ( β ), we consider the integral transform{A formula is presented}where λ ( t ) is a real-valued nonnegative weight function so that ∫01 λ ( t ) d t = 1. The main aim of this paper is to find conditions such that Vλ ( f ) ∈ P1 ( α ) whenever f ∈ Pγ ( β ) for γ {greater than or slanted equal to} 1. We also obtain conditions such that Vλ ( f ) ∈ P1 ( β′ ) whenever f ∈ P0 ( β ) for various choices of λ ( t ). As a useful consequence, we find conditions for certain hypergeometric functions to be univalent. © 2005 Elsevier B.V. All rights reserved.