Some Subclasses of q-Analytic Starlike Functions

Abstract

We define two new subclasses, $\mathcal{S}_{q}^{\ast }(\alpha )$ and $\mathcal{TS}_{q}^{\ast }(\alpha )$, of analytic univalent functions. We obtain a sufficient condition for analytic univalent functions to be in $\mathcal{S}_{q}^{\ast }(\alpha )$ and we prove that this condition is also necessary for the functions in the class $\mathcal{TS}_{q}^{\ast }(\alpha )$. We also obtain extreme points, distortion bounds, covering result, convex combination and convolution properties for the functions in the class $\mathcal{TS}_{q}^{\ast }(\alpha )$.