On new Hardy–Hilbert integral inequalities using primitives and homogeneous functions

This article presents new generalizations of Hardy–Hilbert-type integral inequalities involving primitives and various homogeneous functions. Unlike previous results, our approach enables two parameters to vary independently and introduces an adjustable homogeneity parameter. This improves the flexibility and applicability of the framework. We derive sharp integral inequalities for two distinct parameter regimes. Three comprehensive applications are provided to emphasize the importance of our main theorem.