On the equality and comparison problem of a class of mean values

Abstract

Let ${I\subset \mathbb {R}}$ be a nonvoid open interval. A function ${K:I^2\to I}$ is called an M-conjugate mean if there exists ${(p,q)\in [0,1]^2}$ and a continuous strictly monotone real valued function ${\varphi}$ on I such that

$$K(x,y)=\varphi^{-1}(p\varphi(x)+q\varphi(y)+(1-p-q)\varphi(M(x,y)))=:M_ \varphi^{(p,q)}(x,y)$$

holds for all ${x,y\in I}$. In this paper, we investigate the equality and comparison problem in the class of M-conjugate means, in the case when

$$M(x,y):=\min\{x,y\}\quad (x,y\in I)$$

.