Some General Classes of $q$-Starlike Functions Associated with the Janowski Functions

Abstract

By making use of the concept of basic (or $q$-) calculus, various families of
$q$-extensions of starlike functions of order $\alpha$ in the open unit disk
$\mathbb{U}$ were introduced and studied from many different
viewpoints and perspectives. In this paper, we first investigate
the relationship between various known classes of $q$-starlike functions
which are associated with the Janowski functions.
We then introduce and study a new
subclass of $q$-starlike functions which involves the celebrated Janowski functions.
We also derive several properties of such families of $q$-starlike functions
with negative coefficients including (for example) distortion theorems.