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Chesneau, Christophe
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Chemistry Control & Systems Engineering Engineering Mathematics Physics

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General inequalities of the Hilbert integral type using the method of switching to polar coordinates Chesneau, Christophe
Hilbert Journal of Mathematical Analysis Vol. 3 No. 1 (2024): Hilbert J. Math. Anal.
Publisher : KOMUNITAS Analisis Matematika INDONESIA

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.62918/hjma.v3i1.30

Abstract

Various inequalities of the Hilbert integral type have been established in the literature using different methods. Among them, the classical Hilbert integral inequality was proved in an elegant way by David C. Ullrich in 2013. It consists in using the method of switching to polar coordinates after some thorough integral manipulations. Despite its effectiveness, this method seems to have been under-studied for more in the topic. In this paper we rehabilitate it somewhat and show how it can be used to prove new general inequalities of the Hilbert integral type, including some with multiple tuning parameters. Particular examples of interest are also discussed.
A new integral inequality depending on the Lobachevskii function Chesneau, Christophe
Hilbert Journal of Mathematical Analysis Vol. 3 No. 2 (2025): Hilbert J. Math. Anal.
Publisher : KOMUNITAS Analisis Matematika INDONESIA

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.62918/hjma.v3i2.38

Abstract

This paper investigates a new logarithmic variant of the Hilbert integral inequality, beyond the standard homogeneous assumption. To this end, several theorems and propositions are proved, each of which gives new integral inequalities of independent interest. Remarkably, the Lobachevskii function appears quite naturally in some proofs, and forms an important part of the upper bound obtained. Several sharp examples are developed and discussed. A brief overview of the Lobachevskii function is given in the appendix.
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