Let $$F(x,y)=\dfrac{f\left(\frac{x}{x+y}\right)+f\left(\frac{y}{x+y}\right)}{x+y},$$ where $f>0$ is a concave function. Using brute force computation (computer based proof) with $f(x)=\frac{1-x}{2-x}$, I know that:
$$a>b>c>d>0\implies F(a,b)+F(c,d)>F(a,c)+F(b,d)\text{ and }$$ $$ F(a,b)+F(c,d)>F(a,d)+F(b,c).$$
Unfortunately because of the denominator, $F$ fails to be concave. Any suggestions or hints on how to try to prove these inequalities for a general concave $f$? Or for $f(x)=\frac{1-x}{2-x}$ but with pencil and paper methods?
