Abstract
This article presents a unified framework that extends the scope of two existing theorems on multivariate Hardy-Hilbert-type integral inequalities. Key to this extension is the use of two additional adjustable parameters that increase flexibility and generality. The framework also has the originality of including the incomplete lower gamma function in the integral definitions governed by a parameter. Detailed proofs are given, mainly based on the Laplace transform, the generalized Young inequality, the generalized Hölder integral inequality and changes of variables. This article thus provides a new comprehensive foundation for future research in generalized multivariate integral inequalities.