  In this paper, for every $q\in(0,1)$, we obtain the Herglotz representation
theorem and discuss the Bieberbach type problem for the class of $q$-convex
functions of order $\alpha, 0\le\alpha<1$. In addition, we discuss the
Fekete-szeg\"o problem and the Hankel determinant problem for the class of
$q$-starlike functions, leading to couple of conjectures for the class of
$q$-starlike functions of order $\alpha, 0\le\alpha<1$.
