Young's integral inequality with upper and lower bounds

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.

Keywords

• Young's inequality
• monotone functions
• Pochhammer lower factorial
• difference equations
• time scales