TITLE: Integral with absolute value of the derivative
QUESTION [6 upvotes]: I'm trying to estimate this integral $\int_0^1 t |p'(t)|dt$ using this value $\int_0^1 |p(t)|dt$; here $p $ is a real polynomial.
This means, I am looking for an $M>0$ such that $$\int_0^1 |t p'(t)|dt \le M \cdot \int_0^1 |p(t)|dt$$
I've been thinking about integration by parts but I don't know how to do that with an absolute value involved.
Could you help me with that?

REPLY [2 votes]: The inequality is not true. Let $p(t)=t^n$. Then
$$
\int_0^1t\,|p'(t)|\,dt=n\int_0^1t^n\,dt=\frac{n}{n+1},
$$
while
$$
\int_0^1|p(t)|\,dt=\int_0^1t^n\,dt=\frac{1}{n+1}.
$$