Geometric Properties of Partial Sums of Univalent Functions

The $n$th partial sum of an analytic function $f(z)=z+\sum_{k=2}^\infty a_k z^k$ is the polynomial $f_n(z):=z+\sum_{k=2}^n a_k z^k$. A survey of the univalence and other geometric properties of the $n$th partial sum of univalent functions as well as other related functions including those of starlike, convex and close-to-convex functions are presented

A generalization of starlike functions of order alpha

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.Comment: 13 pages, 4 figures, submitted to a journa