On the Logarithmic Concavity of the Power Means

In this paper, we prove that the power mean (or the quasi-arithmetic mean) of two positive real numbers satisfies logarithmic concavity. The power mean is a one-parameter mean which interpolates arithmetic, geometric and harmonic means. Our main result is an extension of the inequality by Jameson. Then we give counterexamples for which we cannot extend our main result to weighted power means case and multivariate power means.