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.

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.