The Starlikeness of Analytic Functions of Koebe Type with Complex Order
Received:March 21, 2018  Revised:September 01, 2018
Key Word: analytic function   subordination   superordination   Koebe type
Fund ProjectL:Supported by UKM (Grant No.GUP-2017-064).
 Author Name Affiliation Mohd Nazran Mohammed PAUZI School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Selangor, DE, Bangi UKM, 43600, Malaysia Maslina DARUS School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Selangor, DE, Bangi UKM, 43600, Malaysia Saibah SIREGAR Faculty of Engineering and Life Sciences, Universiti Selangor, Selangor, DE, UNISEL, 45600, Malaysia
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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.