An equivalent form of Young's inequality with upper bound

TL;DR

This paper presents an equivalent form of Young's inequality that includes an upper bound for the remainder, clarifying the conditions for equality and enhancing understanding of the inequality's structure.

Contribution

The paper introduces a new formulation of Young's inequality with an upper bound, making the equality conditions more transparent and providing a deeper insight into the inequality's properties.

Findings

New equivalent form of Young's inequality with an upper bound

Clearer characterization of equality cases

Enhanced understanding of the inequality's structure

Abstract

Young's integral inequality is complemented with an upper bound to the remainder. The new inequality turns out to be equivalent to Young's inequality, and the cases in which the equality holds become particularly transparent in the new formulation.