Turán type inequalities for some Lommel functions of the first kind

Baricz, Árpád and Koumandos, Stamatis (2016) Turán type inequalities for some Lommel functions of the first kind. Proceedings of the Edinburgh Mathematical Society (Series 2). pp. 1-11. ISSN 0013-0915

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Abstract

In this paper certain Tur\'an 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 G. P\'olya on the zeros of some particular entire functions, and the connection of these Lommel functions with the so-called Laguerre-P\'olya class of entire functions. Moreover, it is shown that in some cases J. 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 Tur\'an type inequalities.

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