  Recht and R\'{e} introduced the noncommutative arithmetic geometric mean
inequality (NC-AGM) for matrices with a constant depending on the degree $d$
and the dimension $m$. In this paper we prove AGM inequalities with a
dimension-free constant for general operators. We also prove an order version
of the AGM inequality under additional hypothesis. Moreover, we show that our
AGM inequality almost holds for many examples of random matrices .
