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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?

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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 $$ \int_0^1|p(t)|\,dt=\int_0^1t^n\,dt=\frac{1}{n+1}. $$

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