A generalization of starlike functions

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 PSq . 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 PSq for certain values of r. AMS No.: 30C4533A70 †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.