---
tag: 497K99BQ4Y
kind: statement
role: theorem
name: Jensen's formula
msc: ["30D20", "30D35", "31A05"]
uses: [0NSD1TPHR0, 0CEGEASY49, 0WPQYW8MTE, 0W5NJQ8MKY, 9N6R5ADJXT]
about: [0NSD1TPHR0, 0CEGEASY49]
sources:
  - url: https://en.wikipedia.org/wiki/Jensen%27s_formula
same_as:
  wikidata: Q1048466
  wikipedia: Jensen's formula
---

## Statement

Let $r > 0$, and let $f$ be a [[meromorphic function|0NSD1TPHR0]] on an open set that contains the closed disc $\abs{z} \le r$.
Assume that $f(0) \ne 0$, that $0$ is not a pole of $f$, and that $f$ has no zeros and no poles on the circle $\abs{z} = r$.
Let $a_1, \dots, a_m$ be the [[zeros|0CEGEASY49]] and $b_1, \dots, b_n$ the [[poles|0WPQYW8MTE]] of $f$ in the disc $\abs{z} < r$, each repeated according to its multiplicity.
Then
$$\log \abs{f(0)} = \frac{1}{2\pi} \int_0^{2\pi} \log \abs{f(r e^{i\theta})} \, d\theta - \sum_{j=1}^{m} \log \frac{r}{\abs{a_j}} + \sum_{k=1}^{n} \log \frac{r}{\abs{b_k}}.$$

## Remarks

Divide $f$ by the [[Blaschke factors|9N6R5ADJXT]] $r(z - a_j)/(r^2 - \overline{a_j} z)$ 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|0W5NJQ8MKY]], 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.

## Explanations

### L1 human

Jensen's formula relates the size of a function on a circle to its zeros inside the circle.
For a function without zeros or poles, $\log \abs{f}$ is harmonic and its value at the centre is its average on the circle.
Each zero inside the disc pulls the value at the centre down, and each pole pushes it up, by an exact amount.
So a function that is not large on the circle cannot have many zeros inside.

### L2 human

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|KYSR7NHNFQ]].
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|1EHH63VNAD]] and of its [[first main theorem|QJWCFAK4AK]].