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For p∈R , let Mp(a,b) denote the usual power mean of order p of positive real numbers a and b , and let A = M1 , G = M0 and H = M−1 . We prove that the inequalities M0(a,b) 1 3 [A(a,b)+G(a,b)+H(a,b)] M ln2 ln6 (a,b) and M− 6 (a,b) 1 2 [He(a,b)+H(a,b)] M ln2 ln6 (a,b) hold for all positive real numbers a and b , with strict inequality for a = b , and that the orders of power means involved are optimal. Mathematics subject classification (2010): 26E60, 26D15.

The optimal power mean bounds for two convex combinations of A-G-H means