Commun. Korean Math. Soc. 2019; 34(2): 451-464
Online first article October 15, 2018 Printed April 1, 2019
https://doi.org/10.4134/CKMS.c180106
Copyright © The Korean Mathematical Society.
Oh Sang Kwon, Young Jae Sim
Kyungsung University; Kyungsung University
In the present paper, we find the sharp bound for the fourth coefficient of starlike functions $f$ which are normalized by $f(0)=0=f'(0)-1$ and satisfy the following two-sided inequality: $$ 1+ \frac{\gamma-\pi}{2\sin\gamma} < \mathfrak{R} \left\{ \frac{zf'(z)}{f(z)} \right\} < 1 + \frac{\gamma}{2\sin\gamma}, \quad z\in\mathbb{D}, $$ where $\mathbb{D}:=\{ z\in\mathbb{C}: |z|<1 \}$ is the unit disk and $\gamma$ is a real number such that $\pi/2 \leq \gamma <\pi$. Moreover, the sharp bound for the fifth coefficient of $f$ defined above with $\gamma$ in a subset of $[\pi/2,\pi)$ also will be found.
Keywords: coefficient estimate, starlike function, vertical strip domain
MSC numbers: Primary 30C45, 30C80
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