Doing some tests with Maple I "guessed" the following inequality with exponential function (for $x\geq 0$)
$$ x\exp(-x^2/4)(\exp(x)+\exp(-x)) \leq 1000 \exp(-x).$$
Is there an easy proof?
Can one improve the "constant" $1000$?
One can probably give a very ugly proof as follows.
It suffices to show that $$x\exp(-x^2/4) \exp(x) \leq 500 \exp(-x).$$ This inequality holds for $x=0$. The maximum value $M$ of the LHS can be calculated explicitly and one can show that the RHS is bigger than $M$ for $0\leq x \leq x_1$, where $x_1$ is some explicit real number. Then, we just compute the derivatives and we show that they satisfy a certain inequality. (This becomes messy.)