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\sqrt{ab}- 2ab/(a + b) \le p ( (a + b)/2} -\sqrt{ab} ) , in p for any a,b>0

Source: Czech-Polish-Slovak Junior Match 2015, Team p5 CPSJ

March 19, 2020
inequalitiesminimumminalgebra

Problem Statement

Find the smallest real constant pp for which the inequality holds ab−2aba+b≤p(a+b2−ab)\sqrt{ab}- \frac{2ab}{a + b} \le p \left( \frac{a + b}{2} -\sqrt{ab}\right) with any positive real numbers a,ba, b.
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