Lower bound expected value of n-th root 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?
 A: If $n>1$, there cannot exist a positive $c_n$ such that $\mathrm E(X^{1/n})\geqslant c_n\mathrm E(X)^{1/n}$ for every $[0,2]$ valued random variable $X$. To see this assume that $X=2$ with probability $p$ and $X=0$ with probability $1-p$. Then one asks that $p2^{1/n}\geqslant c_n(2p)^{1/n}$, hence $c_n\leqslant p^{1-1/n}$. When $p\to0^+$, one gets $c_n\leqslant0$ as soon as $n>1$.
On the other hand, since $X\leqslant2$ almost surely, $X^{1/n}\geqslant2^{-1+1/n}X$ almost surely, hence $\mathrm E(X^{1/n})\geqslant 2^{-1+1/n}\mathrm E(X)$. Likewise, for every positive $k\leqslant n$, $X^{1/n}\geqslant2^{1/n-1/k}\,X^{1/k}$ almost surely, hence $\mathrm E(X^{1/n})\geqslant 2^{1/n-1/k}\,\mathrm E(X^{1/k})$. 
A: For $X \ge 0$ you can estimate it from below using higher moments via Markov and Paley-Zygmund inequalities. Set $a = \varepsilon \sqrt{\mathbb{E}X^2}$ for $0 < \varepsilon < 1$. Then,
$$
\frac{\mathbb{E}X}{a} \ge \mathbb{P}(X \ge a) = \mathbb{P}(X^2 \ge a^2)
= \mathbb{P}(X^2 \ge \varepsilon^2 \mathbb{E}X^2) \ge (1 - \varepsilon^2)^2 \frac{(\mathbb{E}X^2)^2}{\mathbb{E}X^4}.
$$ 
By bringing $a$ to the other side, we obtain
$$
\mathbb{E}X \ge \varepsilon (1 - \varepsilon^2)^2 \frac{(\mathbb{E}X^2)^{5/2}}{\mathbb{E}X^4}
$$
Optimizing over $\varepsilon$ grants maximum at $\varepsilon = \sqrt 5 - 2$ and final bound
$$
\mathbb{E}X \ge 16(\sqrt 5 - 2 )^3 \frac{(\mathbb{E}X^2)^{5/2}}{\mathbb{E}X^4}
\ge \frac{(\mathbb{E}X^2)^{5/2}}{5 \mathbb{E}X^4}
$$
Same logic applies to $X^{1/n}$ and $n$-th power in probability.
