The Starlikeness of Analytic Functions of Koebe Type with Complex Order

Abstract:

      For $\alpha > 0$, $\lambda>0$ and $\beta,\eta \in \mathbb{R} $, we consider the $M(\alpha,\lambda)_{b}$ of normalized analytic $\alpha-\lambda$ convex functions defined in the open unit disc $\U$. In this paper we investigate the class $M(\alpha,\lambda,\beta,\eta)_b$, with $f_{b}:= \frac{z }{(1-z^{n})^{b}}$ being Koebe type. By making use of Jack's Lemma as well as several differential and other inequalities, the authors derive sufficient conditions for starlikeness of the class $M(\alpha,\lambda,\beta,\eta)_b$ of $n$-fold symmetric analytic functions of Koebe type. Relevant connections of the results presented here with those given in earlier works are also indicated.