The Hardy integral inequality is a key result in analysis and plays an important role in many areas of mathematics. It has been proved in various ways, including a direct one combining a change of variables and the Minkowski integral inequality. However, the approach of using change of variables perspectives to deepen the understanding of the Hardy integral inequality and to extend its scope seems to be underexplored. This paper fills this gap in two complementary ways. First, using change of variables techniques, we show that the ”optimal” constant factor of the Hardy integral inequality can be significantly improved under basic assumptions on the main function. Second, using similar techniques, we generalize the Hardy integral inequality using three intermediate functions. A new adaptable integral inequality is thus introduced. Several examples are given to support the theoretical results.
Christophe Chesneau (Fri,) studied this question.