New Hardy-Hilbert-type integral inequalities involving special inhomogeneous kernel functions

In this article, new Hardy-Hilbert-type integral inequalities are established. Our main result is based on a special inhomogeneous two-parameter kernel function. It is of the ratio power form, and has the property of involving a product term which perturbs the standard homogeneity property. We then use this result to derive new weighted integral norm inequalities and other Hardy-Hilbert-type integral inequalities. They are also defined with inhomogeneous kernel functions, but with innovative power and logarithmic forms. Some of them are obtained by treating an adjustable parameter as a variable and integrating with respect to it, which remains an original technique of proof. The article concludes with an at-tempt to unify some new and old Hardy-Hilbert-type integral inequalities. Due to the mathematical complexity, the optimality of the final result remains an open question, giving some new perspectives to a classical topic.