Mohd Nazran Mohammed PAUZI,Maslina DARUS,Saibah SIREGAR.The Starlikeness of Analytic Functions of Koebe Type with Complex Order[J].数学研究及应用,2018,38(6):586~596
The Starlikeness of Analytic Functions of Koebe Type with Complex Order
The Starlikeness of Analytic Functions of Koebe Type with Complex Order

DOI：10.3770/j.issn:2095-2651.2018.06.005

 作者 单位 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

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.

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.