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I'm faced with the following problem: I have to lower bound the expected value of the n-th root of an arbitrary distributed real random variable using its expected value. So I'm looking for something that has a similar form as the Jensen inequalty but goes the other way around.

I can assume the variable satisfies 0< X< 2 so I thought I could lower bound the root by a line but that approximation is to strong.

Does any one know a way of lower bounding the expected value of a root?

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    If you rule out the estimate $\sqrt{x}\geq \frac{x}{\sqrt{2}}$, then please define when an approximation is not too strong.2011-10-07
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    To strong was in referring to my application. I tried to bound an error probability and I ended up with a value >1. So I was hoping for any other approximation that is not strictly worse. btw something for $0 would also help me, but I guess that does not really make a difference.2011-10-07
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    I seems that, in order to get a helpful answer, you should describe what you actually want to achieve.2011-10-07
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    I want an inequality of the form $\mathbb{E}[x^\frac{1}{r}] < f(\mathbb{E}[x])$ for x arbitrarily distributed between 0 and 2.2011-10-10
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    I assume you mean the converse inequality? Since you don't make further requirements, Didier Piau's answer contains a solution.2011-10-10

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