How can I show that a particular expression is positive?

Question

I have the expression $G(m) = \frac{\mu}{2e[F(\mu)-F(0)]} - \frac{\int_{-\infty}^\mu F(c)^e}{F(\mu)^e}, e\in(0,1],\mu>0,m>\geq 0,$ and $F(c) = \mu^{-1}\int_0^\mu 1/(1+exp(-m(c-x-1))dx$, and I want to prove whether it is nonnegative for all $m$. I've already simplified the original expression quite a bit, simulated a wide range of values that show that $G\geq 0$, but I keep coming up against dead ends trying to proceed further. I've tried e.g. showing $\lim_{m\rightarrow\infty}G(m)=0$ and then that $G$ is nonincreasing in $m$ but cannot say definitively that this is so. Are there any ways I could further simplify this expression, or potential methods to proceed with proving this?