Unifying Hardy-Hilbert-type integral inequalities

The Hardy-Hilbert integral inequality is a classic result in mathematical analysis that has inspired numerous generalizations and modifications. In this article, we present two comprehensive frameworks that unify many of these developments. Our approach introduces kernel functions that take into account both the maximum and the product of the variables, controlled by three independent, adjustable parameters. Although the kernels are primarily inhomogeneous and somewhat complicated, they offer much greater generality. We provide detailed proofs, together with thorough discussions and context within the wider mathematical literature. In addition, several intermediate integral results emerge naturally from our framework and may serve as useful tools for further exploration.