Annals of Communications in Mathematics

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.

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.