Suppose
- $f(v)=-dF(u)/du$
- $f(1)=f(-1)=0$
- $F(-1)=F(1)$
- $g(v):=2(F(v)-F(-1))=-2\int_{-1}^v f(u)\, du$,
- $\mu^2:=-f'(-1)=-f'(1)$
In particular, $g'(v)=-2f(v)$ and $g(1)=0$.
Now, it is claimed that by the definition of $g$ it is "quite standard" to get the estimate $$ f(v)g^{-1/2}(v)=\mu(1+O(1-v)), $$
And, to be honest, I have literally no idea how to see this!