General Two- and Three-Dimensional Integral Inequalities Based a Change of Variables Methodology

This article establishes new general two- and three-dimensional integral inequalities. The first result involves four functions: two main functions defined on the positive real line and two auxiliary functions defined on the unit interval. As a significant contribution, the upper bound obtained is quite simple; it is expressed only as the product of the unweighted integral norms of these functions. The main ingredient of the proof is an original change of variables methodology. The article also presents a three-dimensional extension of this result. This higher-dimensional version uses a similar structure but with nine functions: three main functions defined on the positive real line and six auxiliary functions defined on the unit interval. It retains the simplicity and sharpness of the upper bound. Both results open up new directions for applications in analysis. This claim is supported by various examples, including some based on power, logarithmic, trigonometric, and exponential functions, as well as some secondary but still general integral inequalities.