A Certain Class of q-Close-to-Convex Functions of Order α

Abstract

For every 0<q<1 and 0 <= alpha<1, we introduce a class of analytic functions f on the open unit disc  with the standard normalization  f(0)=0=f'(0)-1 and satisfying

 |(1/(1-alpha))(z(D_qf)(z) /h(z)-alpha) 1/(1-q) |<= 1/(1-q)

 where h is a q-starllike function. This class is denoted by Kq(alpha), so called the class of q-close-to-convex- functions of order alpha. In this paper, we study some geometric properties of this class. In addition, we consider the famous Bieberbach conjecture problem on coefficients for the class Kq(alpha). We also find some sufficient conditions for the function to be in Kq(alpha) for some particular choices of the functions h. Finally, we provide some applications on q-analogue of Gaussian hypergeometric function.