Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-08T17:33:22.411Z Has data issue: false hasContentIssue false
  • English
  • Français

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 [email protected]
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 [email protected]
YOUNG JAE SIM
Affiliation:
Department of Mathematics, Kyungsung University, Busan 48434, Korea email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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).

References

Babalola, K. O., ‘On H 3(1) Hankel determinants for some classes of univalent functions’, in: Inequality Theory and Applications, Vol. 6 (ed. Cho, Y. J.) (Nova Science Publishers, New York, 2010), 1–7.Google Scholar
Bansal, D., Maharana, S. and Prajapat, J. K., ‘Third order Hankel determinant for certain univalent functions’, J. Korean Math. Soc. 52(6) (2015), 1139–1148.Google Scholar
Carathéodory, C., ‘Über den Variabilitatsbereich der Koeffizienten von Potenzreihen, die gegebene werte nicht annehmen’, Math. Ann. 64 (1907), 95–115.CrossRefGoogle Scholar
Cho, N. E., Kowalczyk, B., Kwon, O. S., Lecko, A. and Sim, Y. J., ‘The bounds of some determinants for starlike functions of order alpha’, Bull. Malay. Math. Sci. Soc., to appear, doi:10.1007/s40840-017-0476-x.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. Ineq. 11(2) (2017), 429–439.CrossRefGoogle Scholar
Duren, P. T., Univalent Functions (Springer, New York, 1983).Google Scholar
Goodman, A. W., Univalent Functions (Mariner, Tampa, FL, 1983).Google Scholar
Janteng, A., Halim, S. A. and Darus, M., ‘Coefficient inequality for a function whose derivative has a positive real part’, J. Inequal. Pure Appl. Math. 7(2) (2006), 1–5.Google Scholar
Janteng, A., Halim, S. A. and Darus, M., ‘Hankel determinant for starlike and convex functions’, Int. J. Math. Anal. 1(13) (2007), 619–625.Google Scholar
Lee, S. K., Ravichandran, V. and Supramanian, S., ‘Bound for the second Hankel determinant of certain univalent functions’, J. Ineq. Appl. 2013(281) (2013), 1–17.Google Scholar
Kwon, O. S., Lecko, A. and Sim, Y. J., ‘On the fourth coefficient of functions in the Carathéodory class’, Comp. Methods Funct. Theory, to appear; doi:10.1007/s40315-017-0229-8.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.Google Scholar
Libera, R. J. and Zlotkiewicz, E. J., ‘Coefficient bounds for the inverse of a function with derivatives in O’, Proc. Amer. Math. Soc. 87(2) (1983), 251–257.Google Scholar
Livingston, A. E., ‘The coefficients of multivalent close-to-convex functions’, Proc. Amer. Math. Soc. 21(3) (1969), 545–552.Google Scholar
Mishra, A. K. and Gochhayat, P. , ‘Second Hankel determinant for a class of analytic functions defined by fractional derivative’, Int. J. Math. Math. Sci. 2008 (2008), Article ID 153280, 1–10.Google Scholar
Pommerenke, C., ‘On the coefficients and Hankel determinant of univalent functions’, J. Lond. Math. Soc. 41 (1966), 111–122.Google Scholar
Pommerenke, C., Univalent Functions (Vandenhoeck & Ruprecht, Göttingen, 1975).Google Scholar
Prajapat, J. K., Bansal, D., Singh, A. and Mishra, A. K., ‘Bounds on third Hankel determinant for close-to-convex functions’, Acta Univ. Sapientae Math. 7(2) (2015), 210–219.Google Scholar
Raza, M. and Malik, S. N., ‘Upper bound of the third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli’, J. Inequal. Appl. 2013(412) (2013), 8 pages.Google Scholar
Shanmugam, G., Stephen, B. A. and Babalola, K. O., ‘Third Hankel determinant for 𝛼-starlike functions’, Gulf J. Math. 2(2) (2014), 107–113.CrossRefGoogle Scholar
Study, E., Vorlesungen über ausgewählte Gegenstände der Geometrie, Zweites Heft; Konforme Abbildung Einfach-Zusammenhängender Bereiche (Druck und Verlag von BG Teubner, Leipzig und Berlin, 1913).Google Scholar
Sudharsan, T. V., Vijayalaksmi, S. P. and Sthephen, B. A, ‘Third Hankel determinant for a subclass of analytic functions’, Malaya J. Math. 2(4) (2014), 438–444.Google Scholar
Zaprawa, P., ‘Third Hankel determinants for classes of univalent functions’, Medit. J. Math. 14(19) (2017), 1–10.Google Scholar