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.

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.