Let $X$ be a random variable, and let $f$ be a concave function.

Are there any known lower bounds for (or methods of lower bounding) $\mathbb{E}[f(X)]$?

Jensen's inequality only gives an upper bound on the above.

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    $\begingroup$ An obvious lower bound is $\text{E}[f(X)]\geqslant \inf_{x\in S} f(x)$ where $S$ is the support of the random variable $X$. $\endgroup$ – StijnDeVuyst Nov 27 '16 at 13:03
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    $\begingroup$ That bound given by @Stijn is the best possible in general: since there is no assumption connecting $X$ to $f$, $X$ could have all its probability concentrated in an interval $[\inf f, \inf f+\epsilon)$ for arbitrarily small positive $\epsilon$. $\endgroup$ – whuber Nov 27 '16 at 15:37

I don't know if this is helpful, but for any function $f$ set $Z \equiv f(X)$. Then by Jensen's Inequality

$$E(Z^2) > [E(Z)]^2 \implies \sqrt {E(Z^2)} > \big |E(Z)\big |$$

$$\implies -\sqrt {E(Z^2)} < E(Z) < \sqrt {E(Z^2)} $$

$$ \implies -\sqrt {E([f(X)]^2)} < E[f(X)] < \sqrt {E([f(X)]^2)} $$


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