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If $X_n$ is a martingales with $sup E|X_n|^p<\infty$ where $p>1$, How can show that $$E^p|X_n|\leq E |X_n|^p$$

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    $\begingroup$ The martingale property is irrelevant. This is a standard application of Holder's inequality: en.wikipedia.org/wiki/H%C3%B6lder's_inequality. $\endgroup$ – dsaxton Dec 2 '16 at 20:54
  • $\begingroup$ Yes, this is a Holder's inequality with $Y=1$ and $q=\dfrac{p}{p-1}$. thanks @dsaxton $\endgroup$ – amin roshani Dec 2 '16 at 20:59
  • $\begingroup$ Taking the pth root of both sides reduces this to the duplicate question (assuming a generous interpretation of the notation, which is ambiguous). $\endgroup$ – whuber Dec 2 '16 at 21:23