Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T09:20:10.554Z Has data issue: false hasContentIssue false
  • English
  • Français

SECOND HANKEL DETERMINANT OF LOGARITHMIC COEFFICIENTS OF CONVEX AND STARLIKE FUNCTIONS

Part of: Geometric function theory

Published online by Cambridge University Press:  30 September 2021

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]
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 e-mail: 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
*
e-mail: alecko@matman.uwm.edu.pl
Get access Rights & Permissions [Opens in a new window]

Abstract

We begin the study of Hankel matrices whose entries are logarithmic coefficients of univalent functions and give sharp bounds for the second Hankel determinant of logarithmic coefficients of convex and starlike functions.

Keywords

starlike functionconvex functionCarathéodory functionHankel determinantlogarithmic coefficient

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 105 , Issue 3 , June 2022 , pp. 458 - 467
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alexander, J. W., ‘Functions which map the interior of the unit circle upon simple regions’, Ann. of Math. (2) 17(1) (1915), 12–22.CrossRefGoogle Scholar
Ali, M. F. and Vasudevarao, A., ‘On logarithmic coefficients of some close-to-convex functions’, Proc. Amer. Math. Soc. 146 (2018), 1131–1142.Google Scholar
Ali, M. F., Vasudevarao, A. and Thomas, D. K., ‘On the third logarithmic coefficients of close-to-convex functions’, in: Current Research in Mathematical and Computer Sciences II (ed. Lecko, A.) (UWM, Olsztyn, 2018), 271–278.Google Scholar
Carathéodory, C., ‘Über den Variabilitatsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen’, Math. Ann. 64 (1907), 95–115.Google Scholar
Cho, N. E., Kowalczyk, B., Kwon, O. S., Lecko, A. and Sim, Y. J., ‘The bound of the Hankel determinant for strongly starlike functions of order alpha’, J. Math. Inequal. 11(2) (2017), 429–439.Google Scholar
Cho, N. E., Kowalczyk, B., Kwon, O. S., Lecko, A. and Sim, Y. J., ‘On the third logarithmic coefficient in some subclasses of close-to-convex functions’, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 114 (2020), Article no. 52, 14 pages.Google Scholar
Cho, N. E., Kowalczyk, B. and Lecko, A., ‘Sharp bounds of some coefficient functionals over the class of functions convex in the direction of the imaginary axis’, Bull. Aust. Math. Soc. 100 (2019), 86–96.CrossRefGoogle Scholar
Choi, J. H., Kim, Y. C. and Sugawa, T., ‘A general approach to the Fekete–Szegö problem’, J. Math. Soc. Japan 59 (2007), 707–727.CrossRefGoogle Scholar
Duren, P. T., Univalent Functions (Springer, New York, 1983).Google Scholar
Girela, D., ‘Logarithmic coefficients of univalent functions’, Ann. Acad. Sci. Fenn. Math. 25 (2000), 337–350.Google Scholar
Goodman, A. W., Univalent Functions (Mariner, Tampa, FL, 1983).Google Scholar
Kowalczyk, B., Lecko, A. and Sim, Y. J., ‘The sharp bound for the Hankel determinant of the third kind for convex functions’, Bull. Aust. Math. Soc. 97 (2018), 435–445.CrossRefGoogle Scholar
Kumar, U. P. and Vasudevarao, A., ‘Logarithmic coefficients for certain subclasses of close-to-convex functions’, Monatsh. Math. 187(3) (2018), 543–563.CrossRefGoogle Scholar
Lecko, A. and Partyka, D., ‘A revised proof of starlikeness’, in: International Conference on 60 Years of Analytic Functions in Lublin, In Memory of Our Professors and Friends (eds. Krzyż, J. G., Lewandowski, Z. and Szapiels, W.) (Innovatio Press, Lublin, 2012), 85–95.Google Scholar
Libera, R. J. and Zlotkiewicz, E. J., ‘Early coefficients of the inverse of a regular convex function’, Proc. Amer. Math. Soc. 85(2) (1982), 225–230.CrossRefGoogle Scholar
Libera, R. J. and Zlotkiewicz, E. J., ‘Coefficient bounds for the inverse of a function with derivatives in $\boldsymbol{\mathcal{P}}$ ’, Proc. Amer. Math. Soc. 87(2) (1983), 251–257.Google Scholar
Milin, I. M., Univalent Functions and Orthonormal Systems (Nauka, Moscow, 1971) (in Russian); English translation, Translations of Mathematical Monographs, 49 (American Mathematical Society, Providence, RI, 1977).Google Scholar
Pommerenke, C., Univalent Functions (Vandenhoeck and Ruprecht, Göttingen, 1975).Google Scholar
Study, E., Vorlesungen über ausgewählte Gegenstände der Geometrie, Zweites Heft; Konforme Abbildung Einfach-Zusammenhängender Bereiche (Druck und Verlag von B. G. Teubner, Leipzig–Berlin, 1913).Google Scholar
Thomas, D. K., ‘On logarithmic coefficients of close to convex functions’, Proc. Amer. Math. Soc. 144 (2016), 1681–1687.CrossRefGoogle Scholar