Sufficiency for Gaussian hypergeometric functions to be uniformly convex

Let F(a, b; c; z) be the classical hypergeometric function and f be a normalized analytic functions defined on the unit disk 𝒰. Let an operator Ia,b;c(f) be defined by [Ia,b;c(f)](z) = zF(a, b; c; z)*f(z). In this paper the authors identify two subfamilies of analytic functions ℱ1 and ℱ2 and obtain conditions on the parameters a, b, c such that f ∈ ℱ1 implies Ia,b;c(f) ∈ ℱ2.