# An inequality about expectation [duplicate]

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

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