Entropy/Variance inequality

Question

The following inequality is sometimes used as a building block to prove log Sobolev inequalities. Does anyone have a simple proof of it? $$ x\log x + y\log y - (x+y)\log \frac{x+y}{2}\leq (\sqrt x-\sqrt y)^2,\qquad x,y>0. $$

Answer 1

Let $y=t^2x$, where $t\geq1$.

Hence, we need to prove that $f(t)\geq0$,

where $f(t)=(t-1)^2-2t^2\ln{t}+(t^2+1)\ln{\frac{t^2+1}{2}}$.

We have $f'''(t)=\frac{4(t^2-1)}{t(t^2+1)^2}\geq0$. Hence, $f''(t)=2-4\ln{t}+2\ln\frac{t^2+1}{2}-\frac{4}{t^2+1}\geq0$.

Hence, $f'(t)=2t\ln\frac{t^2+1}{2}+2t-4t\ln{t}-2\geq0$ and $f(t)\geq f(1)=0$.

Done!