a*-families of analytic functions

Abstract

Using convolutions, a new family of analytic functions is introduced. This family, called a*-family, serves in certain situations to unify the study of many previously well known classes of analytic functions like multivalent convex, starlike, close-to-convex or prestarlike functions, functions starlike with respect to symmetric points and other such classes related to the class of univalent or multivalent functions. A necessary and sufficient condition on the Taylor series coefficients so that an analytic function with negative coefficients is in an a*-family is obtained and sharp coefficents bound for functions in such a family is deduced. The extreme points of an a*-family of functions with negative coefficients are completely determined. Finally, it is shown that Zmorvic conjecture is true if the concerned families consist of functions with negative coefficients.

Keywords

• univalent functions
• multivalent functions
• convolution
• p-valent starlike functions
• p-valent close-to-convex functions
• p-valent prestarlike funations
• starlike functions with respect to symmetric points
• coefficients bound
• extreme points etc.