Univalence and starlikeness of certain transforms defined by convolution of analytic functions

Let U(λ)U(λ) denote the class of all analytic functions f in the unit disk Δ   of the form f(z)=z+a2z2+⋯f(z)=z+a2z2+⋯ satisfying the condition|f′(z)(zf(z))2−1|⩽λ,z∈Δ. In this paper we find conditions on λ   and on c∈Cc∈C with Rec⩾0≠cRec⩾0≠c such that for each f∈U(λ)f∈U(λ) satisfying (z/f(z))∗F(1,c;c+1;z)≠0(z/f(z))∗F(1,c;c+1;z)≠0 for all z∈Δz∈Δ the transformG(z)=Gfc(z)=z(z/f(z))∗F(1,c;c+1;z),z∈Δ, is univalent or starlike. Here F(a,b;c;z)F(a,b;c;z) denotes the Gauss hypergeometric function and ∗ denotes the convolution (or Hadamard product) of analytic functions on Δ.