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SHARP BOUNDS OF SOME COEFFICIENT FUNCTIONALS OVER THE CLASS OF FUNCTIONS CONVEX IN THE DIRECTION OF THE IMAGINARY AXIS

Part of: Geometric function theory

Published online by Cambridge University Press:  29 January 2019

NAK EUN CHO Open the ORCID record for NAK EUN CHO [Opens in a new window] ,
BOGUMIŁA KOWALCZYK Open the ORCID record for BOGUMIŁA KOWALCZYK [Opens in a new window]  and
ADAM LECKO Open the ORCID record for ADAM LECKO [Opens in a new window]
NAK EUN CHO
Affiliation:
Department of Applied Mathematics, Pukyong National University, Busan 48513, Korea email necho@pknu.ac.kr
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
*
✉alecko@matman.uwm.edu.pl
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Abstract

We apply the Schwarz lemma to find general formulas for the third coefficient of Carathéodory functions dependent on a parameter in the closed unit polydisk. Next we find sharp estimates of the Hankel determinant $H_{2,2}$ and Zalcman functional $J_{2,3}$ over the class ${\mathcal{C}}{\mathcal{V}}$ of analytic functions $f$ normalised such that $\text{Re}\{(1-z^{2})f^{\prime }(z)\}>0$ for $z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, that is, the subclass of the class of functions convex in the direction of the imaginary axis.

Keywords

univalent functionsfunctions convex in the direction of the imaginary axisHankel determinantZalcman functionalcoefficientsCarathéodory class

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 100 , Issue 1 , August 2019 , pp. 86 - 96
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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Footnotes

The first-named author was supported by the Basic Science Research Program through the National Research Foundation of the Republic of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2016R1D1A1A09916450).

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