Study of some new one-parameter modifications of the Hardy-Hilbert integral inequality

One of the pillars of mathematical analysis is the Hardy-Hilbert integral inequality. In this article, we advance the theory by introducing several new modifications of this inequality. They have the property of incorporating an adjustable parameter and different power functions, allowing for greater flexibility and broader applicability. Notably, one modification has a logarithmic structure, offering a distinctive extension to the classical framework. For the main results, the optimality of the corresponding constant factors is shown. Additional integral inequalities of various forms and scopes are also established. Thus, this work contributes to the ongoing development of Hardy-Hilbert-type inequalities by presenting new generalizations and providing rigorous mathematical justification for each result.