If x,y and z are non-negative reals prove that:

$\displaystyle \frac{\frac{x \sqrt y + y \sqrt z + z \sqrt x}{3} + \frac{y \sqrt x + z \sqrt y + x \sqrt z}{3}}{2} \le \sqrt{\left(\frac{x+y}{2}\right) \left( \frac{y+z}{2} \right) \left(\frac{z+x}{2}\right)}$