As the title says, I am trying to ascertain whether the following is true: Suppose $a,b,c,d\in \mathbb{R}^+$ are such that $a + b < c + d$, then it is also true that $e^{-a} + e^{-b} > e^{-c} + e^{-d}$.
This is a simplified version of a problem I am working on and I have failed in making in any useful progress except realize that Convexity (e.g., Arithmetic Mean - Geometric Mean Inequality) doesn't seem to help me.
$2+2<1+4$ but
$$.27\approx\frac{2}{e^2}<\frac{1}{e}+\frac{1}{e^4}\approx.39$$