NEW GENERALIZATION OF YOUNG’S INEQUALITY FOR THREE INTEGRAL TERMS

Abstract

We know that in functional analysis, there is an inequality that is very well known, important, and very applicable. This inequality is Young's inequality. This inequality can also give birth to other well-known inequalities such as Holder’s inequality, Murkowski’s inequality, average inequality, etc. So many researchers are interested in discussing these inequalities. Many mathematicians try to improve these inequalities, reverse these inequalities, and infrequently generalize these inequalities. However, of all the generalizations of Young's inequality (in integral form) that have been produced, all of them are in the form of the sum of two integral terms. Through a literature study, in this note, we aim to obtain a new generalization of Young's inequality in the form of the sum of more than two integral terms, in particular, three integral terms. Then from the resulting generalization form, we will also construct the inverse. The results of this study are theorems that present a new generalization of Young's inequality for three integral terms and the reverse of a new generalization of Young's inequality for three integral terms. The theorems have been accompanied by their proofs so that the truth of these theorems can be accepted logically and systematically. From these results, it can be concluded that a new generalization of Young's inequality has been found and has a different form from the previous one. It is hoped that wider applications will be obtained with this new generalization.