Annals of Communications in Mathematics

AbstractHardy-Hilbert-type integral inequalities lie at the heart of mathematical analysis. They have been the subject of much research. In this article, we make a contribution to the field by examining two new two-parameter modifications of the classical Hardy-Hilbert integral inequality. We derive the closed-form expression of the optimal constant for each modification. We also present supplementary results, including one-function and primitive variants. All proofs are provided in full, with each step justified, to ensure the article is self-contained.

ABSTRACT. In this article, we present a new generalized version of the Hardy integral inequality. It has the property of depending on an auxiliary function. Thanks to this function, numerous variants are examined. The theory is complemented by two secondary results,  one showing that the main inequality can be improved under additional assumptions, and another giving a valuable lower bound for the main integral term. Several examples  are given for illustration.