Young's integral inequality with upper and lower bounds

TL;DR

This paper presents a reformulation of Young's integral inequality with bounds on the remainder, improving its applicability across various time scales and including difference equations and piecewise-monotone functions.

Contribution

It introduces new bounds for Young's inequality that enhance understanding and application across continuous, discrete, and piecewise functions.

Findings

Improved bounds for Young's integral inequality on all time scales

Explicit results for difference equations

Extensions to piecewise-monotone functions

Abstract

Young's integral inequality is reformulated with upper and lower bounds for the remainder. The new inequalities improve Young's integral inequality on all time scales, such that the case where equality holds becomes particularly transparent in this new presentation. The corresponding results for difference equations are given, and several examples are included. We extend these results to piecewise-monotone functions as well.