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\frac{a + b}{2} \ge \sqrt[a+b]{a^b b^a}

Source: 1963-64 Germany R4 12.1

October 12, 2024
algebrainequalities

Problem Statement

Prove that for all positive, entirely rational numbers aa and bb always holds a+b2≥ab⋅baa+b.\frac{a + b}{2} \ge \sqrt[a+b]{a^b \cdot b^a}. When does the equal sign hold?
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