Non-commutative arithmetic-geometric mean inequality

In this paper we consider a non-commutative analogue of the arithmetic-geometric mean inequality

$$a^{r}b^{1-r}+(r-1)b\geq ra$$

for two positive numbers $a,b$ and for $r> 1$. We show that under certain assumptions the non-commutative analogue of $a^{r}b^{1-r}$ which satisfies this inequality is unique and equal to the $r$-mean. The case $0<r<1$ is also considered. In particular, we give a new characterization of the geometric mean.