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Jensen's inequality states that if $\mu$ is a probability measure on $X$, $\phi$ is convex, and $f$ is a real-valued function, then

$$ \int \phi(f) \, d\mu \geq \phi\left(\int f \, d\mu\right).$$

Is there a variant on this inequality for complex-valued functions? Namely, if $\phi$ is a function from $\mathbb C$ to itself such that

$$\left|\phi\left(\frac{z+w}{2}\right)\right| \leq \frac{|\phi(z)|}{2} + \frac{|\phi(w)|}{2}$$

whenever $z,w\in \mathbb C$, and $f$ is a complex valued function on $X$, can we conclude that

$$\left|\int \phi(f) \,d\mu\right| \geq \left|\phi\left(\int f \, d\mu\right)\right|\ ?$$

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If $\phi$ is convex then $|\phi|$ is convex, but you are not assuming convexity at all just in the middle it is more puzzling. – checkmath May 14 '12 at 0:37
Certainly we have $\displaystyle\int|\phi(f)|\,d\mu \ge \left|\int \phi(f)\,d\mu\right|$, and we also have Jensen's inequality applied to $|\phi|$, namely $\displaystyle\int|\phi(f)|\,d\mu\ge\left|\phi\left(\int f\,d\mu\right)\right|$. So it looks as if the proposed inequality is stronger than Jensen's inequality for real functions. I wonder if that's a sort of hint about where to look for counterexamples, if any exist? – Michael Hardy May 14 '12 at 0:43

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