Consider a polynomial $p(x)$ where $p(x)>0$ for $x\in(0,1)$ and $p(0)=0$. Let $s(x)$ be an increasing analytic function such that $s(0)=0$ and $s(1)=1$. I am interested to bound the following integral: $\int_{0}^1 p'(s(x))(s'(x))^2 dx$.
Note that I can rewrite the above integral to the form: $\int_{0}^1 \frac{d p(s(x))}{dx} s'(x) dx$