Mathematical analysis demonstrates logarithmic concavity in bivariate power means, indicating boundaries where weighted and multivariate generalizations fail.
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.
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Takeaki Yamazaki (2026) studied this question.
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