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 https://doi.org/10.4134/BKMS.b171122 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.
