A generalization of starlike functions of order alpha

For every $q\in(0,1)$ and $0\le α<1$ 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.

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