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I am trying to prove the following: Let $g: \mathbb{R} \to \mathbb{R}$ be a measurable function which is not convex. Prove that there exists a random variable $f$ on some probability space such that $E|f| \le \infty$ and $-\infty \le E(g(f)) \le g(E(f)) \le \infty$.

I know the last inequality holds for concave functions and can be proved the same as for convex functions since if $g$ is concave, $-g$ is convex. But in this case since the function is not convex does this automatically imply that it is concave? How am I ensure that there exists such a random variable $f$ and $E|f| \le \infty$?

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    $\begingroup$ Jensen's inequality for convex functions holds with $\ge$ instead of $\le$ because you multiply by $-1$. Also, "not convex" is a much larger set than "concave": a function with an inflexion point is neither concave nor convex. $\endgroup$
    – mlc
    Mar 27 '17 at 16:16
  • $\begingroup$ Right the sign of the inequality is flipped for Jensen's Inequality for convex functions. So essentially we are trying to prove for some general non-convex function we can find a random variable such that Jensen's Inequality for concave functions holds? $\endgroup$
    – user75514
    Mar 27 '17 at 16:37
  • $\begingroup$ That's right. Assuming some regularity (e.g., the function is continuous), it should not be hard to define a r.v. with support over the domain where $g$ is concave. $\endgroup$
    – mlc
    Mar 27 '17 at 16:54
  • $\begingroup$ Can I just define f to be 1 on the set where g is concave and 0 everywhere else? $\endgroup$
    – user75514
    Mar 27 '17 at 16:59
  • $\begingroup$ What about the $Ef$ argument of $g(E(f))$? You know only $0 \le Ef \le 1$. $\endgroup$
    – mlc
    Mar 27 '17 at 17:03
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This answers makes additional assumptions (in italics). Let $g: \mathbb{R} \to \mathbb{R}$ be a measurable bounded, differentiable function which is not everywhere convex.

Under the stated assumptions, the function $g$ has at least one inflexion point where it goes from convex to concave (or vice versa). Wlog, suppose the first case and call such point $a$. Let $b$ the infimum of all inflexion points strictly greater than $a$; if there is none, let $b=\infty$. The function $g$ is concave on $[a,b]$.

Let $X$ be a r.v. uniformly distributed on $[a,b]$. The result follows by Jensen's inequality applied to the concave restriction of $g$ to the interval $[a,b]$.

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