A Theoretical Extension of the Hardy-Hilbert Integral Inequality to Three Dimensions with Five Parameters

In this article, we extend the theory of the Hardy-Hilbert integral inequality by developing a novel three-dimensional variant. This variant depends on three norm parameters and five adjustable parameters, including two power parameters that influence the structure of the denominator in a novel way. We determine the exact conditions under which the inequality holds for these parameters, together with the expression of the constant factor in terms of beta functions. Two illustrative examples are presented, and two additional results are derived, including a lower bound. Thus, this work adds to a classical topic in analysis by exploring an original extension in both dimensionality and parametric complexity.