47A30, 47A63.

In 1967, as a converse of the arithmetic-geometric mean inequality, Mond and Shisha gave an estimate of the difference between the arithmtic mean and the geometric one, which we call it the Mond-Shisha difference. As an application of Mond-Pecaric method, we show some order relations between the power means of positive operators on a Hilbert space. Among others, we show that the upper bound of the difference between the arithmetic mean and the chaotically geometric one of positive operators coincides with the Mond-Shisha difference. ;