Univalency of weighted integral transforms of certain functions

For β < 1 and γ ⩾ 0 , let P γ ( β ) denote the class of all normalized analytic functions f in the unit disc such that Re e i η ( 1 - γ ) f ( z ) z + γ f ′ ( z ) - β > 0 , z ∈ Δ , for some η ∈ R . For f ∈ P γ ( β ) , we consider the integral transform V λ ( f ) ( z ) = ∫ 0 1 λ ( t ) f ( tz ) t d t , where λ ( t ) is a real-valued nonnegative weight function so that ∫ 0 1 λ ( t ) d t = 1 . The main aim of this paper is to find conditions such that V λ ( f ) ∈ P 1 ( α ) whenever f ∈ P γ ( β ) for γ ⩾ 1 . We also obtain conditions such that V λ ( f ) ∈ P 1 ( β ′ ) whenever f ∈ P 0 ( β ) for various choices of λ ( t ) . As a useful consequence, we find conditions for certain hypergeometric functions to be univalent.