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This article demonstrates an improved version of the Hardy-Hilbert integral inequality, highlighting its constant factor and implications in determining other variants.
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.
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Christophe Chesneau (2025) studied this question.
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