A Lower bounds for exponential function How to find a "nonnegative" lower bounds for 

$e^{uX-u^2Y} + e^{-uX-u^2Y} - 2 $?

where $u&gt;0$, $Y \geq 0$ and $X \neq0$

I know I can bound above function by using the fact that $e^{uX} + e^{-uX} &gt; 2$; i.e.,

$ (e^{uX} + e^{-uX})e^{-u^2Y} - 2 &gt; 2(e^{-u^2Y} -1) $ but this yields negative value.
It looks to me like that expression can indeed become negative. Fix $u$ and $X$ and let $Y$ grow very large. The expression approaches $-2$. I would say your greatest lower bound is $-2$. You can see that this is the case because the expression cannot ever equal $-2$. If you set the expression equal to $-2$, you will get $e^{-u^2Y}\cosh uX = 0$ which is impossible.
It looks to me like that expression can indeed become negative. Fix $u$ and $X$ and let $Y$ grow very large. The expression approaches $-2$. I would say your greatest lower bound is $-2$. You can see that this is the case because the expression cannot ever equal $-2$. If you set the expression equal to $-2$, you will get $e^{-u^2Y}\cosh uX = 0$ which is impossible.