# Ky Fan inequalities for bivariate means

## TL;DR

This paper establishes sufficient conditions under which certain bivariate means satisfy Ky Fan inequalities, contributing to the theoretical understanding of inequalities involving symmetric, homogeneous means.

## Contribution

It provides new theoretical conditions ensuring Ky Fan inequalities hold for classes of bivariate, homogeneous, symmetric means.

## Key findings

- Derived sufficient conditions for Ky Fan inequalities to hold.
- Applied inequalities to specific classes of means.
- Enhanced understanding of mean inequalities in mathematical analysis.

## Abstract

In this note we give sufficient conditions for bivariate, homogeneous, symmetric means $M$ and $N$ to satisfy Ky Fan inequalities $$\frac{M}{M'}<\frac{N}{N'}\qquad\text{ and }\quad \frac{1}{M}-\frac{1}{M'}<\frac{1}{N}-\frac{1}{N'}.$$