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Which power-means are majorized by (a^2/b+b^2/a)/2?

Source: Germany 2024, Problem 6

June 14, 2024
algebra proposedalgebrainequalitiesinequalities proposedMean Inequality Chain

Problem Statement

Decide whether there exists a largest positive integer nn such that the inequality a2b+b2a2≥an+bn2n\frac{\frac{a^2}{b}+\frac{b^2}{a}}{2} \ge \sqrt[n]{\frac{a^n+b^n}{2}} holds for all positive real numbers aa and bb. If such a largest positive integer nn exists, determine it.
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