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This problem is from page 25 of concentration inequalities a nonasymptotic theory of independence

$X$ is a random variable and $X^{'}$ is independent copy of $X$

q is integer and $q\geq1$

It is said that the following inequality was implied by the convexity of $x^{2q}$

$$\mathbb{E} \big[ \left(X-X^{'}\right)^{2q} \big]\leq 2^{2q-1} \left( \mathbb{E}X^{2q}+ \mathbb{E}X^{'2q} \right) $$

However, I do not know how to deduce it. The first guess is using Minkowski inequality to get terms of $\mathbb{E}X^{2q}$ and $\mathbb{E}X^{'2q}$. The Jensen ineqaulity will give inequality of reverse direction, I am wondering how convexity is applied. Any suggestion is welcomed.

Rikeijin
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    https://math.stackexchange.com/questions/1102727/classic-c-r-inequality-in-l-r-space-exyr-leq-c-rexreyr –  Nov 16 '18 at 23:38
  • Thx. I am wondering how $$(a+b)^r\leqslant a^r+b^r\mbox{ if }0\lt r\leqslant 1 $$ is deduced from concavity of $ x^r$. I can see it throguh Bernoulli inequality but it's fine to find a unified way to do it through Jensen's inequality. – Rikeijin Nov 17 '18 at 10:17
  • https://math.stackexchange.com/questions/318649/square-root-of-a-sum-bound –  Nov 17 '18 at 16:44

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