Theoretical Results on a Special Two-parameter Trivariate Hilbert-type Integral Inequality

Hilbert integral inequalities of various types have been widely studied in mathematics. There is still room for research in this area, especially in the trivariate case, which has received less attention than the bivariate case. In this article, we contribute to this topic by studying a special trivariate Hilbert-type integral inequality. It has the property of depending on three functions and several adjustable parameters. We derive four theoretical results under different assumptions, the first three of which establish upper bounds, while the last one establishes a lower bound. These inequalities provide mathematical tools useful for solving three-dimensional problems involving complex integrals.