TUYMAADA 2013 J3

For every positive real numbers $a$ and $b$ prove the inequality $$\sqrt{ab} \leq \dfrac{1}{3} \sqrt{\dfrac{a^2+b^2}{2}}+\dfrac{2}{3} \dfrac{2}{\dfrac{1}{a}+\dfrac{1}{b}}.$$

A. Khabrov