Bull. Korean Math. Soc. 2018; 55(6): 1859-1868
Online first article May 2, 2018 Printed November 30, 2018
https://doi.org/10.4134/BKMS.b171122
Copyright © The Korean Mathematical Society.
Bogumila Kowalczyk, Adam Lecko, Millenia Lecko, Young Jae Sim
University of Warmia and Mazury in Olsztyn, University of Warmia and Mazury in Olsztyn, Rzeszow University of Technology, Kyungsung University
In the present paper, we have proved the sharp inequality $|H_{3,1}(f)|$ $\le 4$ and $|H_{3,1}(f)|\le 1$ for analytic functions $f$ with $a_n:=f^{(n)}(0)/n!,\ n\in\mathbb{N},$ such that $$\mathrm{Re}\, \frac{f(z)}{z}> \alpha,\quad z\in\mathbb{D}:=\{z \in\mathbb{C} : |z|<1\}$$ for $\alpha=0$ and $\alpha=1/2,$ respectively, where \begin{equation*} H_{3,1}(f):= \begin{vmatrix} a_1 & a_2 & a_3 \\ a_2 & a_3 & a_4 \\ a_3 & a_4 & a_5 \end{vmatrix} \end{equation*} is the third Hankel determinant.
Keywords: univalent functions, Caratheodory functions, Hankel determinant
MSC numbers: Primary 30C45
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