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Results in Mathematics

, 73:89 | Cite as

On Hankel Determinant \({{\varvec{H}}}_\mathbf{2}{} \mathbf{(3)}\) for Univalent Functions

  • Paweł Zaprawa
Open Access
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Abstract

In this paper we consider the Hankel determinant \(H_2(3) = a_3a_5 - a_4{}^2\) defined for the coefficients of a function f which belongs to the class \(\mathcal {S}\) of univalent functions or to its subclasses: \(S^*\) of starlike functions, \(\mathcal {K}\) of convex functions and \(\mathcal {R}\) of functions whose derivative has a positive real part. Bounds of \(|H_2(3)|\) for these classes are found; the bound for \(\mathcal {R}\) is sharp. Moreover, the sharp results for starlike functions and convex functions for which \(a_2=0\) are obtained. It is also proved that \(\max \{|H_2(3)|: f\in \mathcal {S}\}\) is greater than 1.

Keywords

Univalent functions starlike functions convex functions Hankel determinant 

Mathematics Subject Classification

30C50 

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mechanical EngineeringLublin University of TechnologyLublinPoland

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