I've been trying for 2 days to prove this inequality, and I've run out of ideas. How to do it?
$$\frac{(f+c)^{f+c}}{f^f}>\frac{(1-f)^{1-f}}{(1-f-c)^{1-f-c}} , 1/2<f<1, 0<c<1-f$$
So far I've tried endless combinations of power manipulation to see if I could cancel some things which i know are bigger than 1, for instance. But I always end up with something left that I can't prove to be bigger or smaller than 1.
Hint
As $x\log x$ is convex for $x> 0$, we may use Karamata's Inequality and just show $(f+c, 1-f-c) \succ (f, 1-f)$, which is true for the conditions which imply $\frac12< f< c+f< 1$.