Suppose $x,y\in [0,\pi]$, does the inequality $$f(x,y) = x+y-x\,e^{\sin^2x-\sin^2y}-y\,e^{\sin^2y-\sin^2x} \leq 0$$ always hold?
I have been trying to prove the following integral inequality $$\int_0^{\pi}x\,e^{\sin^2x}\int_0^{\pi}\,e^{-\sin^2x} \geq \frac{\pi^3}2,$$ and it seems to me that if one can prove $f(x,y) \leq 0$ for any $x,y\in [0,\pi]$, then $$0 \geq \int_0^{\pi}\int_0^{\pi}f(x,y)dx\,dy = \frac{\pi^3}2-\int_0^{\pi}x\,e^{\sin^2x}\int_0^{\pi}\,e^{-\sin^2x}.$$ But I got stuck proving the first inequality. I have tried to differentiate $f$ and got nothing. I wonder whether the first inequality is right or not? If it is right, can anyone prove it for me? Thanks in advance!
It is not true. Let\begin{align}g(y)&=f(\pi,y)\\&=\pi+y-ye^{\sin ^2(y)}-\pi e^{-\sin ^2(y)}.\end{align}Then $g(0)=g'(0)=0$, but $g''(0)=2\pi>0$. So, $g(y)>0$ on some interval $(0,\varepsilon)$. Or you can check that$$g\left(\frac\pi4\right)=\frac{5 \pi }{4}-\frac{\pi }{\sqrt{e}}-\frac{\sqrt{e} \pi }{4}\approx0.73.$$