I am looking at the following problem. I have a function $f(x)$ with support on $[0, \infty)$. Furthermore, $f(x)$ is bounded between 0 and 1, monotonically increasing and concave everywhere.
Separately, there is a r.v. $X$ with some unknown density $p(x)$ which is also defined on $[0, \infty)$. We know that $E[X]=m$.
I would like to maximize $E[f(X)]$. Clearly it is bounded by $f(m)$ regardless of the distribution of $X$ because of Jensen’s inequality. But can I do better (obtain a lower upper bound) by finding a specific pdf for $X$ which maximizes $E[f(X)]$ subject to the constraints on $X$? Something along the lines of how the exponential distribution has maximum entropy for the class of distributions with support on the positive real line and known mean, except instead of entropy I am maximizing the expectation of $f(x)$. Any ideas or direction would be much appreciated. Thanks!
Wouldn't the distribution that just puts all the mass at $m$ give you $E[f(X)] = f(m)$? (Or if you don't want to have a mass point, you could come arbitrarily close to this by putting almost all the mass arbitrarily close to $m$.)