I was looking at an integrated form of Jensen's inequality for holomorphic functions: $$ \ln|f(0)|\le\frac1{\pi}\int_0^1\int_0^{2\pi}\ln|f(re^{i\theta})|r\text{d}\theta\text{d}r $$ (which can be found by multiplying both sides of the traditional inequality by $\int_0^1r\text{d}r$). When exactly is the integrand integrable? It seems like every typical text fails to mention the case where $f$ has a zero somewhere in the unit disc.
As an example, I was analyzing the function $f(z) = z + \frac12$. The integrand is $\frac12\ln(r^2 + r\cos\theta + 1/4)$, which is zero when $\theta = 0$ and $r = -\frac12$. How do we know that this function is, indeed, integrable at that point?
By extension of that, precisely when is the integrand actually integrable?
If $f$ is analytic in a neighbourhood of the closed unit disk and not identically zero, the integrand is always integrable. Note that if $f$ has a zero of order $k$ at $p$, $|f(z)| \sim c |z-p|^k$ as $z \to p$, so $\ln |f(z)| \sim k \ln |z-p|$. A logarithmic singularity is integrable. It might not be integrable (even without zeros) if $f$ is just analytic in the open unit disk.