Improvement of a known variant of the Hardy-Hilbert integral inequality

Over the past few decades, numerous variants of the Hardy-Hilbert integral inequality have been established. In this article, we present an improved version of a notable variant that combines the power, maximum, and absolute difference functions of the variables. The constant factor has an elegant form and is closely related to the beta function. A complete and detailed proof is provided. We also discuss how our result can be used to determine other variants.