Hypergeometric transforms of functions with derivative in a half plane

Let A be the class of normalized analytic functions in the unit disk Δ, F(a,b;c;z) and Φ(a;c;z) denote respectively, the Gaussian and confluent hypergeometric functions. Let R(β) = [lcub]ƒ ∈ A: ∃η ∈ R[rcub] such that Re [eiη(ƒ′(z) − β)]>0, z ∈ Δ[rcub]. For ƒ ∈ A, we define the hypergeometric transforms Va,b;c(ƒ) and Ua;c(ƒ) by the convolution Va,b;c(ƒ) := zF(a,b;c;z) ∗ ƒ(z) and Ua;c(ƒ) := zΦ(a;c;z) ∗ ƒ(z), respectively. The main aim of this paper is to find conditions on β1, β2 and the parameters (a,b,c) such that each of the operators Va,b;c(ƒ) and Ua;c(ƒ) maps R(β1) into R(β2). We also find conditions such that the fu (cab)[F(a,b;c;z)−1] or (ca)[Φ(a;c;z) − 1] is in R(β).