A generalization of starlike functions

Abstract

For every q. 0<q<1 we define a class of complex functions as the class of functions f, analytic on the open unit disc ℬ, f(0)=0, f′(0)=1 and |f(qz)| ⩽ |f(z)| on ℬ. This class is denoted by PS_q . We study this class and explore the relationships between this class and other classes of analytic functions. We find the function F _q that maximizes the coefficients of members of PS _q. We also prove that a basic hypergeomelric function Ф(a,b;c;q,rz)ε PS_q for certain values of r.

AMS No.:

• 30C45
• 33A70

^†Research partially supported by a grant from the National Science Foundation.

^†Research partially supported by a grant from the National Science Foundation.

Notes

^†Research partially supported by a grant from the National Science Foundation.