Jensen's formula

Remarks

Divide $f$ by the Blaschke factors $r(z - a_j)/(r^2 - \overline{a_j} z)$ and multiply by the analogous factors for the poles. The quotient $g$ has no zeros or poles on the closed disc, and $\abs{g} = \abs{f}$ on the circle $\abs{z} = r$. Then $\log \abs{g}$ is harmonic, and its mean value property gives the formula, since each factor has modulus $\abs{a_j}/r$ or $\abs{b_k}/r$ at $0$. For example, take $f(z) = z - a$ with $0 < \abs{a} < 1$ and $r = 1$. The integral is $0$, because $\abs{e^{i\theta} - a} = \abs{1 - \overline{a} e^{i\theta}}$ and $\log \abs{1 - \overline{a} z}$ is harmonic on the closed unit disc with value $0$ at $0$; both sides then equal $\log \abs{a}$. The formula remains true when $f$ has zeros or poles on the circle, since the integral still converges.

Going deeper

For a student

For a holomorphic $f$, the formula gives $\sum_j \log(r/\abs{a_j}) \le \log M(r) - \log \abs{f(0)}$, where $M(r)$ is the maximum of $\abs{f}$ on $\abs{z} = r$. This bounds the number of zeros in terms of the growth, which is the first step of the Hadamard factorization theorem. The sum can be written as $\int_0^r n(t)/t \, dt$, where $n(t)$ counts the zeros in $\abs{z} \le t$. In this form the formula is the starting point of Nevanlinna theory and of its first main theorem.