A generalization of starlike functions of order alpha

For every q∈(0,1) and 0≤ α<1 we define a class of analytic functions, the so-called q-starlike functions of order $α$, 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 $α$. 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 $α$. As an application we also discuss the Bieberbach conjecture problem for the q-starlike functions of order $α$. Further application includes the study of the order of q-starlikeness of the well-known basic hypergeometric functions introduced by Heine.