# Jensen's formula

In the mathematical field known as complex analysis, Jensen's formula, introduced by Johan Jensen (1899), relates the average magnitude of an analytic function on a circle with the number of its zeros inside the circle. It forms an important statement in the study of entire functions.

## The statement

Suppose that ƒ is an analytic function in a region in the complex plane which contains the closed disk D of radius r about the origin, a1, a2, ..., an are the zeros of ƒ in the interior of D repeated according to multiplicity, and ƒ(0)  0. Jensen's formula states that

$\log |f(0)|=\sum _{k=1}^{n}\log \left({\frac {|a_{k}|}{r}}\right)+{\frac {1}{2\pi }}\int _{0}^{2\pi }\log |f(re^{i\theta })|\,d\theta .$ This formula establishes a connection between the moduli of the zeros of the function ƒ inside the disk D and the average of log |f(z)| on the boundary circle |z| = r, and can be seen as a generalisation of the mean value property of harmonic functions. Namely, if f has no zeros in D, then Jensen's formula reduces to

$\log |f(0)|={\frac {1}{2\pi }}\int _{0}^{2\pi }\log |f(re^{i\theta })|\,d\theta ,$ which is the mean-value property of the harmonic function $\log |f(z)|$ .

An equivalent statement of Jensen's formula that is frequently used is

${\frac {1}{2\pi }}\int _{0}^{2\pi }\log |f(re^{i\theta })|\;d\theta -\log |f(0)|=\int _{0}^{r}{\frac {n(t)}{t}}\;dt$ where $n(t)$ denotes the number of zeros of $f$ in the disc of radius $t$ centered at the origin.

Jensen's formula may be generalized for functions which are merely meromorphic on D. Namely, assume that

$f(z)=z^{l}{\frac {g(z)}{h(z)}},$ where g and h are analytic functions in D having zeros at $a_{1},\ldots ,a_{n}\in \mathbb {D} \backslash \{0\}$ and $b_{1},\ldots ,b_{m}\in \mathbb {D} \backslash \{0\}$ respectively, then Jensen's formula for meromorphic functions states that

$\log \left|{\frac {g(0)}{h(0)}}\right|=\log \left|r^{m-n}{\frac {a_{1}\ldots a_{n}}{b_{1}\ldots b_{m}}}\right|+{\frac {1}{2\pi }}\int _{0}^{2\pi }\log |f(re^{i\theta })|\,d\theta .$ Jensen's formula can be used to estimate the number of zeros of analytic function in a circle. Namely, if f is a function analytic in a disk of radius R centered at z0 and if |f| is bounded by M on the boundary of that disk, then the number of zeros of f in a circle of radius r<R centered at the same point z0 does not exceed

${\frac {1}{\log(R/r)}}\log {\frac {M}{|f(z_{0})|}}.$ Jensen's formula is an important statement in the study of value distribution of entire and meromorphic functions. In particular, it is the starting point of Nevanlinna theory.

## Poisson–Jensen formula

Jensen's formula is a consequence of the more general PoissonJensen formula, which in turn follows from Jensen's formula by applying a Möbius transformation to z. It was introduced and named by Rolf Nevanlinna. If f is a function which is analytic in the unit disk, with zeros a1, a2, ..., an located in the interior of the unit disk, then for every $z_{0}=r_{0}e^{i\varphi _{0}}$ in the unit disk the PoissonJensen formula states that

$\log |f(z_{0})|=\sum _{k=1}^{n}\log \left|{\frac {z_{0}-a_{k}}{1-{\bar {a}}_{k}z_{0}}}\right|+{\frac {1}{2\pi }}\int _{0}^{2\pi }P_{r_{0}}(\varphi _{0}-\theta )\log |f(e^{i\theta })|\,d\theta .$ Here,

$P_{r}(\omega )=\sum _{n\in \mathbb {Z} }r^{|n|}e^{in\omega }$ is the Poisson kernel on the unit disk. If the function f has no zeros in the unit disk, the Poisson-Jensen formula reduces to

$\log |f(z_{0})|={\frac {1}{2\pi }}\int _{0}^{2\pi }P_{r_{0}}(\varphi _{0}-\theta )\log |f(e^{i\theta })|\,d\theta ,$ which is the Poisson formula for the harmonic function $\log |f(z)|$ .

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