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Suppose $\phi$ is a real function on $\mathbb{R}$ such that $$\phi \Bigl(\int_0^1f(x)\,dx\Bigr)\leq \int_0^1 \phi(f)\,dx$$ for every real bounded measurable function $f$. Prove that $\phi$ is then convex.

This one is in Rudin's Real and Complex Analysis, page 74, and, to be quite honest, I am unsure what my first step to solve this problem should even be. To help think of a general methodology, I attempted this similar question, from page 71 of the same book:

Assume $\phi$ is a continuous real valued function on $(a,b)$ such that $$\phi\Bigl(\frac{x+y}{2}\Bigr)\leq\frac{\phi(x)+\phi(y)}{2}$$ for all $x, y\in (a,b).$ Prove that $\phi$ is then convex (the conclusion does not follow if continuity is omitted from the hypothesis).

I take it that the latter question is just a special case of the former (assuming $\phi$ is bounded on $(a,b)$), so I attempted to solve the latter problem first. My attempt was to note that \begin{align*} \phi\big(\lambda x + (1-\lambda)y\big) &\leq \phi\Bigl(\frac{2\lambda x + 2(1-\lambda)y}{2}\Bigr) \\ &\leq \frac{1}{2}\Bigl(\phi\bigl(2\lambda x\bigr)+\phi \bigl(2(1-\lambda)y\bigr)\Bigr) \\ & \leq \phi\bigl(\lambda x\bigr)+\phi\bigl((1-\lambda)y\bigr) \end{align*} with the latter inequality justified simply by applying the hypothesis again twice to the previous expression.

From here, however, I am stuck; I cannot justify "popping out" the $\lambda$ and the $(1-\lambda)$ in the last inequality via some bound afforded by the hypothesis and/or the continuity of $\phi$.

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    $\begingroup$ You could check out this question. It has a lot of non-standard proofs in it and the question itself might give you a hint at the standard proof. $\endgroup$ – Milo Brandt Dec 26 '14 at 3:32
  • $\begingroup$ Meelo, thanks. Are you aware of any solution to the first part of my question, or would that depend on solving the second part of my question? $\endgroup$ – Darrin Dec 26 '14 at 3:36
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    $\begingroup$ Well, solving the second part would be good for the first part, though not strictly necessary - informally, if $f$ were a measure with value $\lambda$ for $\{x\}$ and $(1-\lambda)$ for $\{y\}$, then the integral will collapse to the definition of convexity. (Formally, you'd have to actually know about the formalisms of this topic, which I don't) $\endgroup$ – Milo Brandt Dec 26 '14 at 3:46
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Consider $f$ such that $f(x) = x_1$ for $0\leq x \leq \alpha$ and $f(x) = x_2 $ for $\alpha < x \leq 1$. So the Jensen's inequality for integrals will be

$$\phi\left(\alpha x_1 + (1- \alpha)x_2 \right) \leq \alpha \phi(x_1) + (1-\alpha)\phi(x_2)$$

by changing values of $x_1$ and $x_2$, we have the result.

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