Jensen's formula via Blaschke factors

Let \(f\) be holomorphic in a neighborhood of the closed unit disc, suppose \(f(0)\ne0\), and let \(z_1,\ldots,z_N\) be its zeros in \(\mathbb D\), repeated according to multiplicity. For \(|\alpha|<1\), the Blaschke factor \[ \psi_\alpha(z)=\frac{\alpha-z}{1-\overline\alpha z} \] has one simple zero at \(\alpha\), no pole in the closed disc, and satisfies \[ |\psi_\alpha(e^{i\theta})|=1, \qquad \psi_\alpha(0)=\alpha. \] Indeed, for \(|z|=1\), \[ |\alpha-z|=|z|\,|\overline z\alpha-1|=|1-\overline\alpha z|. \]

Define \[ g(z)=\frac{f(z)}{\prod_{j=1}^N\psi_{z_j}(z)}. \] The zeros of the denominator cancel exactly the zeros of \(f\), so \(g\) is holomorphic and nowhere zero in a neighborhood of the closed disc. Hence \(\log|g|\) is harmonic in the disc. The mean-value property at \(0\) gives \[ \log|g(0)|=\frac1{2\pi}\int_0^{2\pi}\log|g(e^{i\theta})|\,d\theta. \] On the unit circle all Blaschke factors have modulus \(1\), so the right side is \[ \frac1{2\pi}\int_0^{2\pi}\log|f(e^{i\theta})|\,d\theta. \] At the origin, \[ |g(0)|=\frac{|f(0)|}{\prod_{j=1}^N|z_j|}. \] Therefore \[ \boxed{ \log|f(0)| =\sum_{j=1}^N\log|z_j| +\frac1{2\pi}\int_0^{2\pi}\log|f(e^{i\theta})|\,d\theta}, \] which is Jensen’s formula in the unit disc.