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