I am interested in approximating the minimum of
$$\dfrac{1-c}{1-\dfrac{(a-b)^3}{(1-a)^2} - \dfrac{(b-c)^3}{(a-b)^2} - \dfrac{c^3}{b^2}}$$ Subject to $0 < \frac{a-b}{1-a} < \frac{b-c}{a-b} < \frac{c}{b} < 1$ and $0\leq a,b,c \leq 1$. The minimum is $\approx 0.82$ numerically, but not sure how to show this. In particular, I'm interested in a lower bound.