- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnlilne Submission ㆍMy Manuscript - For Reviewers - For Editors
 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.b171122Published online 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 Full-Text :