Criteria for monotonicity of operator means

Abstract

Let $\{{\psi }_r{\}}_{r>0}$ and $\{{\varphi }_r{\}}_{r>0}$ be the families of operator monotone functions on $[0,\infty )$ satisfying ${\psi }_r\left(x^{r}g\right(x\left)\right)=x^r,$${\varphi }_r\left(x^{r}g\right(x\left)\right)=x^{r}h\left(x\right)$, where $g$ and $h$ are continuous and $g$ is increasing. Suppose ${\sigma }_{{\psi }_a}$ and ${\sigma }_{{\varnothing }_r}$ are the corresponding operator connections. We will show that if A${\sigma }_{{\psi }_a}B\ge 1(a>0)$, then $A^r{\sigma }_{{\psi }_r}B$ and $A^r{\sigma }_{{\varnothing }_r}B$ are both increasing for $r\ge a$, and then we will apply this to the geometric operator means to get a simple assertion from which many operator inequalities follow.