Ky Fan inequalities for bivariate means

Abstract

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