On a Hardy-Type Integral Inequality under Convexity and Submultiplicativity Assumptions

The focus of this article is a Hardy-type integral inequality published in 2012, which has the particularity of combining convexity and submultiplicativity assumptions. A revision is carried out in several steps. First, a counterexample to this result is given, questioning its validity. Then an alternative statement is proposed, together with a detailed proof. Among the facts revealed is the relaxation of the assumption on a key parameter of the integral norm. Finally, several examples of the revised result are presented, showing a new modification of the Hardy integral inequality. Their originality takes advantage of the combined convexity and submultiplicativity assumptions, which remain an underexplored framework.