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Let $X\in L_p(P), p>1$. Is the following result true? $$E[\lvert X\rvert I(\lvert X\rvert>C)]\leq C^{1-p}E\lvert X\rvert^p.$$ where $C>0$.

It can be found in the proof of Corollary A.1 (pp. 278) of Hall & Heyde's book.

My attempt

By Markov's inequality, $E[I(\lvert X\rvert>C)]=P(\lvert X\rvert>C)\leq C^{-p}E\lvert X\rvert^p$. But $\lvert X\rvert I(\lvert X\rvert>C)> CI(\lvert X\rvert>C)$, and I can't use Markov's result to bound $E[\lvert X\rvert I(\lvert X\rvert>C)]$. How to proceed?

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From Holder's and Markov's inequalities

$$E(\lvert X \rvert I(\lvert X \rvert>C))\leq \lVert X\rVert_p (P(\lvert X\rvert>C))^{1-1/p}\leq \lVert X\rVert_p [E(\lvert X \rvert^p/C^p)]^{1-1/p}=E\lvert X \rvert^pC^{1-p}$$ the desired result. The powers confused me a bit...

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