KöMaL 2014 A616

Prove that $\left(\frac{1+a}2\right)^{2x(x+y)} \left(\frac{1+b}2\right)^{2y(x+y)} \ge a^{x^2} b^{y^2} \left(\frac{a+b}2\right)^{2xy}$ holds for all real numbers $a,b>0$ and $x$ , $y$ .