The expression in question is:
$$ e^{2h(x+1)}(m-1)-me^{2h}+e^{h(1+x+y)}((1 - (-1 + 2 m) h (-1 + x + y)) $$
Graphing, it appears that this expression is always negative as long as $1>m>0$, and $y\neq x-1$. While I only need to show this for $0<x<1$ and $0<y<1$ and $h>0$, these don't seem to matter with respect to the expression being negative. Any tips or tricks would be greatly appreciated!