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 $\alpha, 0\le\alpha<1$.
Categories: math.CV