# Moment and tail probability inequality

How can I prove the following inequality:

If $r>p$, then $$\left\|X\right\|_{p}\le\left(\frac{r}{r-p}\right)^{1/p}\left\|X\right\| _{r,\infty},$$ where $$\left\|X\right\|_p=\left(\operatorname{E}\left|X\right|^p\right)^{1/p}\qquad\text{and}\qquad\left\|X\right\| _{r,\infty}=\left(\sup _{t>0}\ t^r\Pr\left(\left|X\right|>t\right) \right)^{1/r}.$$

I found this inequality on page 10 of the book by Ledoux and Talagrand.

Thank you for your answers!

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## 1 Answer

The RHS is the upper bound of $\displaystyle\|X\|_p^p=\mathrm E(|X|^p)=p\int_0^{+\infty}x^{p-1}\,\mathrm P(|X|\geqslant x)\,\mathrm dx$ that one gets using the upper bound $\mathrm P(|X|\geqslant x)\leqslant\min\{1,\|X\|_{r,\infty}^rx^{-r}\}$ for every $x\gt0$.

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