New integral formulas with applications to integral inequalities

In this article, we derive new integral formulas involving a ratio function, a maximum function, and three adjustable parameters. Two of these parameters control the maximum function in di erent ways. The arctangent function plays a central role in the resulting expressions. These formulas are then used to construct new and varied types of integral inequalities. In particular, we present weighted Hölder-type integral inequalities, as well as new Hardy-Hilbert-type integral inequalities. Their novelty lies mainly in the inclusion of the maximum function and the two parameters governing it. Detailed proofs are given.