Reverse inequalities on chaotically geometric mean via Specht ratio, II

In 1967, as a converse of the arithmetic-geometric mean inequality, Mond and Shisha gave an estimate of the difference between the arithmetic mean and the geometric one, which we call it the Mond-Shisha difference. As an application of Mond-Pečarić method, we show some order relations between the power means of positive operators on a Hilbert space. Among other, 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.