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Strange inequality with Max function, AM and GM

Source: 2023 Brazil Ibero TST P3

August 28, 2023
inequalitiesInequalityarithmetic meangeometric meanmax

Problem Statement

Show that for all positive real numbers a,b,ca, b, c, we have that a+b+c3−abc3≤max⁡{(a−b)2,(b−c)2,(c−a)2}\frac{a+b+c}{3}-\sqrt[3]{abc} \leq \max\{(\sqrt{a}-\sqrt{b})^2, (\sqrt{b}-\sqrt{c})^2, (\sqrt{c}-\sqrt{a})^2\}
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