Annals of Communications in Mathematics

AbstractIn this article, we establish several integral inequalities under new parametric primitive exponential-weighted integral inequality assumptions. The generality and versa- tility of these assumptions allow our results to extend and unify existing frameworks in the literature. In total, eight theorems are formulated. To ensure completeness and accessibil- ity, detailed proofs are given for all the theorems.

AbstractIn this article, we revisit a well-known result from the literature that can be considered a variant of the Hardy integral inequality. First, we present a counterexample to demonstrate the invalidity of the current formulation. We then revise the result by identifying and addressing a gap in the original proof. Finally, as an additional contribution, we derive a new integral inequality.

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.This article is devoted to five distinct integral inequalities of the Hardy-Hilbert  type, each possessing its own originality. In particular, we highlight new trigonometric variants that yield sharp upper bounds involving weighted integral norms of the underlying functions. Complete and rigorous proofs are provided in detail.

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.