  For every $q\in(0,1)$ and $0\le \alpha<1$ we define a class of analytic
functions, the so-called $q$-starlike functions of order $\alpha$, on the open
unit disk. We study this class of functions and explore some inclusion
properties with the well-known class of starlike functions of order $\alpha$.
The paper is also devoted to the discussion on the Herglotz representation
formula for analytic functions $zf'(z)/f(z)$ when $f(z)$ is $q$-starlike of
order $\alpha$. As an application we also discuss the Bieberbach conjecture
problem for the $q$-starlike functions of order $\alpha$. Further application
includes the study of the order of $q$-starlikeness of the well-known basic
hypergeometric functions introduced by Heine.
