Two New General Integral Results Related to the Hilbert Integral Inequality

In this article, we generalize two integral results from the literature. The first result concerns a flexible double integral inequality, considering a specific form for the integrated function and a double integral as a lower or upper bound. Several examples are discussed, as well as some of its indirect connections with the Hilbert integral inequality. The second result also gives a double integral inequality, but with the product of the square root of simple integrals, following the spirit of the Hilbert integral inequality. Several theoretical and numerical examples are discussed. Both of our results have the property of being dependent on several adjustable functions and parameters, thus offering a wide range of applications.