New results on composed variants of the Young integral inequality

In this article, we contribute to the theory of integral inequalities by developing new composed variants of the Young integral inequality. The novelties include some convexity assumptions on the composition function, allowing the use of original tools in this context, such as the Jensen integral inequality. Our extensions and improvements are illustrated by an example of particular interest. It can be described as a "power composed" Young integral inequality, which has sharper bounds than those established in the literature. Another notable novelty is the consideration of a "multiple integral composed" Young integral inequality. In addition, some open problems are raised, bringing a new perspective to an old topic.