A New Proof of Young’s Inequality Using Multivariable Optimization

Abstract

In its standard form, Young's inequality for products is a mathematical inequality about the product of two numbers and it allows us to estimate a product of two terms by a sum of the same terms raised to a power and scaled. This inequality, though very simple, has attracted researchers working in many fields of mathematics due to its applications. Apart from the above standard form, there are numerous refinements and variants of Young’s inequality in the literature. Some of these variants are Young’s inequality for arbitrary products, Young’s inequality for increasing functions, Young’s inequality for convolutions, Young’s inequality for integrals, Young’s inequality for matrices, trace version of Young’s inequality, determinant version of Young’s inequality, and so on. The present study examines three variants of Young’s inequality, namely the standard Young’s inequality, Young’s inequality for increasing functions and Young’s inequality for arbitrary products. There are various proofs for these three variants in the literature. For example, just like several other classical important inequalities, these inequalities can be deduced from Jensen’s inequality. The objective of this article is to provide a new alternative proof for each of them. The significance of the article lies in its attempts to open a new direction of poof so that the same approach could be applied to other useful inequalities. The proofs to be presented are based on the methods of multivariable optimization theory.