Consider $x$, $y$, $z$, $w$, all in $\left(0,1\right)$. Suppose $\frac{1-z}{w}<\frac{1-x}{y}<1<\frac{z}{1-w}<\frac{x}{1-y}$. I want to prove $\frac{w\left(1-x\right)-y\left(1-z\right)}{x+y-1}<1$.
Numerical simulation suggests this inequality is correct.
Here is what I have tried. $\frac{1-x}{y}<1$ implies $x+y-1>0$. Then the inequality is equivalent to $\left(1+w\right)\left(1-x\right)<y\left(2-z\right)$, which does not look like the right way of proving it...
Edit: Thank @River Li for pointing out the error. The inequality was incorrect.
