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In this article, we introduce new variations of the Hardy-Hilbert integral inequality, incorporating weight functions that depend on sums, products, absolute differences of variables, and a power parameter. Some of these inequalities involve the primitives of the main functions, as well as original weight functions of logarithmic and exponential types, along with several adjustable parameters. Additionally, we extend our analysis to a three-dimensional setting, deriving related results of potential significance. The optimality of certain constants in the obtained inequalities is also established. By formulating these new variations, we expand the family of Hardy-Hilbert-type integral inequalities, with possible applications in various branches of mathematical analysis.
Christophe Chesneau (Thu,) studied this question.