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Turán-Type Inequalities for Some Lommel Functions of the First Kind

Part of: Harmonic analysis in one variable Elementary classical functions

Published online by Cambridge University Press:  29 October 2015

Árpád Baricz  and
Stamatis Koumandos
Árpád Baricz
Affiliation:
Department of Economics, Babeş-Bolyai University, Cluj-Napoca 400591, Romania (bariczocsi@yahoo.com) Institute of Applied Mathematics, Óbuda University, 1034 Budapest, Hungary
Stamatis Koumandos
Affiliation:
Department of Mathematics and Statistics, The University of Cyprus, PO Box 20537, 1678 Nicosia, Cyprus (skoumand@ucy.ac.cy)
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Abstract

In this paper certain Turán-type inequalities for some Lommel functions of the first kind are deduced. The key tools in our proofs are the infinite product representation for these Lommel functions of the first kind, a classical result of Pólya on the zeros of some particular entire functions, and the connection of these Lommel functions with the so-called Laguerre–Pólya class of entire functions. Moreover, it is shown that in some cases Steinig's results on the sign of Lommel functions of the first kind combined with the so-called monotone form of l’Hospital's rule can be used in the proof of the corresponding Turán-type inequalities.

Keywords

Lommel functions of the first kindzeros of Lommel functions of the first kindTurán-type inequalitiesmonotone form of l’Hospital's ruleinfinite product representationLaguerre–Pólya class

MSC classification

Secondary: 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$ 33B10: Exponential and trigonometric functions 42A05: Trigonometric polynomials, inequalities, extremal problems
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 3 , August 2016 , pp. 569 - 579
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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