On a generalized Pachpatte inequality through a convexity-based approach

In this article, we extend a key integral inequality established by Pachpatte in 2002 by introducing a new convexity-based approach. Specifically, we incorporate a general convex function to create a flexible framework that can be adapted to various mathematical contexts. The resulting techniques are original and reusable, offering potential for further innovations in related analytical frameworks. We present several examples to illustrate the theory and demonstrate the versatility of the approach.