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(a-b)^2/8a < (a+b)/2 - \sqrt{ab} < (a-b)^2/8b for a > b > 0,

Source: 1985 Swedish Mathematical Competition p1

March 28, 2021
inequalitiesalgebraradical

Problem Statement

If a>b>0a > b > 0, prove the inequality (a−b)28a<a+b2−ab<(a−b)28b.\frac{(a-b)^2}{8a}< \frac{a+b}{2}- \sqrt{ab} < \frac{(a-b)^2}{8b}.
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