The sharp bound of the third Hankel determinant for some classes of analytic functions
Bull. Korean Math. Soc. 2018 Vol. 55, No. 6, 1859-1868
Published online May 2, 2018
Printed November 30, 2018
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
Abstract : 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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