# Note on constants appearing in refined Young inequalities

## TL;DR

This paper presents a refined version of Young's inequality using Specht's ratio, offering a simpler proof, new properties, and connections to Kantorovich constant, advancing the understanding of inequalities in mathematical analysis.

## Contribution

The paper introduces a new refined Young inequality with Specht's ratio, providing an elementary proof, new properties, and a generalization framework.

## Key findings

- Refined Young inequality with Specht's ratio established
- New property of Specht's ratio discovered
- Alternative proof of previous refined Young inequality provided

## Abstract

In this short note, we give the refined Young inequality with Specht's ratio by only elementary and direct calculations. The obtained inequality is better than one previously shown by the author in 2012. In addition, we give a new property of Specht's ratio. These imply an alternative proof of the refined Young ineqaulity shown by author in 2012. We also give a remark on the relation to Kantorovich constant. Finally, we give a proposition for a corresponding general function.