The purpose of the present paper is to investigate some interesting properties on generalized convolutions of functions for the classes HP ∗(α),HS(α) andHC(α). Further, an application of the convolution on certain integral operator are mentioned. AMS 2010 Mathematics Subject Classification : 30C45, 26D15.
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 $.
Berzelius failed to make use of Faraday's electrochemical laws in his laborious determination of equivalent weights.