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THE SHARP BOUND FOR THE HANKEL DETERMINANT OF THE THIRD KIND FOR CONVEX FUNCTIONS

Part of: Geometric function theory

Published online by Cambridge University Press:  04 April 2018

BOGUMIŁA KOWALCZYK ,
ADAM LECKO Open the ORCID record for ADAM LECKO [Opens in a new window]  and
YOUNG JAE SIM
BOGUMIŁA KOWALCZYK
Affiliation:
Department of Complex Analysis, Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, ul. Słoneczna 54, 10-710 Olsztyn, Poland email b.kowalczyk@matman.uwm.edu.pl
ADAM LECKO*
Affiliation:
Department of Complex Analysis, Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, ul. Słoneczna 54, 10-710 Olsztyn, Poland email alecko@matman.uwm.edu.pl
YOUNG JAE SIM
Affiliation:
Department of Mathematics, Kyungsung University, Busan 48434, Korea email yjsim@ks.ac.kr
*
✉alecko@matman.uwm.edu.pl
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Abstract

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We prove the sharp inequality $|H_{3,1}(f)|\leq 4/135$ for convex functions, that is, for analytic functions $f$ with $a_{n}:=f^{(n)}(0)/n!,~n\in \mathbb{N}$, such that

$$\begin{eqnarray}Re\bigg\{1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\bigg\}>0\quad \text{for}~z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\},\end{eqnarray}$$
where $H_{3,1}(f)$ is the third Hankel determinant
$$\begin{eqnarray}H_{3,1}(f):=\left|\begin{array}{@{}ccc@{}}a_{1} & a_{2} & a_{3}\\ a_{2} & a_{3} & a_{4}\\ a_{3} & a_{4} & a_{5}\end{array}\right|.\end{eqnarray}$$

Keywords

univalent functionsconvex functionsCarathéodory functionsHankel determinant

MSC classification

Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 3 , June 2018 , pp. 435 - 445
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

This work was partially supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science, ICT and Future Planning) (No. NRF-2017R1C1B5076778).

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