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THE SHARP BOUND OF THE THIRD HANKEL DETERMINANT FOR SOME CLASSES OF ANALYTIC FUNCTIONS
Kowalczyk, Bogumila,Lecko, Adam,Lecko, Millenia,Sim, Young Jae Korean Mathematical Society 2018 대한수학회보 Vol.55 No.6
In the present paper, we have proved the sharp inequality ${\mid}H_{3,1}(f){\mid}{\leq}4$ and ${\mid}H_{3,1}(f){\mid}{\leq}1$ for analytic functions f with $a_n:=f^{(n)}(0)/n!$, $n{\in}{\mathbb{N}},$, such that $$Re\frac{f(z)}{z}>{\alpha},\;z{\in}{\mathbb{D}}:=\{z{\in}{\mathbb{C}}:{\mid}z{\mid}<1\}$$ for ${\alpha}=0$ and ${\alpha}=1/2$, respectively, where $$H_{3,1}(f):=\left|{\array{{\alpha}_1&{\alpha}_2&{\alpha}_3\\{\alpha}_2&{\alpha}_3&{\alpha}_4\\{\alpha}_3&{\alpha}_4&{\alpha}_5}}\right|$$ is the third Hankel determinant.
The sharp bound of the third Hankel determinant for some classes of analytic functions
Bogumi la Kowalczyk,Adam Lecko,Millenia Lecko,심영재 대한수학회 2018 대한수학회보 Vol.55 No.6
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.