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THE SHARP BOUND OF THE THIRD HANKEL DETERMINANT FOR SOME CLASSES OF ANALYTIC FUNCTIONS

  • Kowalczyk, Bogumila (Department of Complex Analysis Faculty of Mathematics and Computer Science University of Warmia and Mazury in Olsztyn) ;
  • Lecko, Adam (Department of Complex Analysis Faculty of Mathematics and Computer Science University of Warmia and Mazury in Olsztyn) ;
  • Lecko, Millenia (Rzeszow University of Technology Faculty of Mathematics and Applied Physics Department of Nonlinear Analysis) ;
  • Sim, Young Jae (Department of Mathematics Kyungsung University)
  • Received : 2017.12.29
  • Accepted : 2018.03.16
  • Published : 2018.11.30

Abstract

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.

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

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