Refining and extending two special Hardy-Hilbert-type integral inequalities

Numerous variants of the Hardy-Hilbert integral inequality have been developed in the literature. This article first presents refinements and extensions of two such variants. Both are characterised by a double integral with a ratio-power-minimum integrand controlled by two adjustable parameters. The results are obtained under mild assumptions on these parameters. The beta function evaluated at certain values appears naturally in the definition of the constant factors. These constants may be better than those already established when considering a classical setting. Several additional integral inequalities are then derived from our main results. The proofs are also innovative in some technical aspects. They are presented in detail to ensure clarity, to facilitate verification, and to encourage further research in this direction.