Starlike and Prestarlike Hypergeometric Functions

A linear operator is defined which acts on an analytic function in the open unit disk by forming its Hadamard product with an incomplete beta function. The operator is shown to be convenient in discussing starlike, convex, and prestarlike functions. It is applied to the study of certain classes of hypergeometric functions which constitute dense subsets in the classes of starlike functions of order $\alpha $, convex functions of order $\alpha $, and prestarlike functions of order $\alpha $. Integral representations of the functions in these classes are obtained from the integral representation of the starlike functions of order $\alpha $.