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I came across the following inequality and I would like to know if it is true $$(x^xy^{-y})^\frac{1}{y-x}(x-y)^2\leq x+y,$$ which is the same as proving $$e^{-\frac{x\log x-y\log y}{x-y}}(x-y)^2\leq x+y.$$

When I do the simulations it seems it really holds, for any $x,y\geq 0\ $, ($x\ne y$).

I tried to use geometric-arithmetic mean inequality, i.e.,

$$(x^xy^{-y})^\frac{1}{y-x}=\left(\left(\frac{1}{x}\right)^{-x}\left(\frac{1}{y}\right)^{y}\right)^\frac{1}{y-x}\leq \frac{-x\frac{1}{x}+y\frac{1}{y}}{y-x}=0\leq x+y,$$ but obviously this is a wrong approach ("weights" $-x$, $-y$, can not be negative).

Do you have any idea how to prove it? Thanks.

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Note that your formula show that your inequality is equivalent to $$2\log |x-y|\leq \log (x+y)+\frac{x\log x-y\log y}{x-y}$$ Wlog, we may suppose that $x>y>0$, put $x=y+u$.

We have $$x\log x=(y+u)\log(y+u)=y\log y+y\log (1+u/y)+u\log u+u\log(1+y/u)$$ and

$$\log (x+y)=\log (2y+u)=\log u+\log (1+2y/u)$$

Hence we have to show: $$2\log u\leq \log u+\log (1+2y/u)+(y/u)\log (1+u/y)+\log u+\log(1+y/u)$$ or

$$0\leq \log (1+2y/u)+(y/u)\log (1+u/y)+\log(1+y/u)$$ and this is clear.

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