Jensen's formula

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, ..., a_n are the zeros of ƒ in the interior of D repeated according to multiplicity, and ƒ(0) ≠ 0. Jensen's formula states that

${\displaystyle \log |f(0)|=\sum _{k=1}^{n}\log \left({\frac {|a_{k}|}{r}}\right)+{\frac {1}{2\pi }}\int _{0}^{2\pi }\log |f(re^{i\theta })|\,d\theta .}$

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

${\displaystyle \log |f(0)|={\frac {1}{2\pi }}\int _{0}^{2\pi }\log |f(re^{i\theta })|\,d\theta ,}$

which is the mean-value property of the harmonic function ${\displaystyle \log |f(z)|}$.

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

${\displaystyle {\frac {1}{2\pi }}\int _{0}^{2\pi }\log |f(re^{i\theta })|\;d\theta -\log |f(0)|=\int _{0}^{r}{\frac {n(t)}{t}}\;dt}$

where ${\displaystyle n(t)}$ denotes the number of zeros of ${\displaystyle f}$ in the disc of radius ${\displaystyle t}$ centered at the origin.

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

${\displaystyle f(z)=z^{l}{\frac {g(z)}{h(z)}},}$

where g and h are analytic functions in D having zeros at ${\displaystyle a_{1},\ldots ,a_{n}\in \mathbb {D} \setminus \{0\}}$ and ${\displaystyle b_{1},\ldots ,b_{m}\in \mathbb {D} \setminus \{0\}}$ respectively, then Jensen's formula for meromorphic functions states that

${\displaystyle \log \left|{\frac {g(0)}{h(0)}}\right|=\log \left|r^{m-n}{\frac {a_{1}\ldots a_{n}}{b_{1}\ldots b_{m}}}\right|+{\frac {1}{2\pi }}\int _{0}^{2\pi }\log |f(re^{i\theta })|\,d\theta .}$

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

${\displaystyle {\frac {1}{\log(R/r)}}\log {\frac {M}{|f(z_{0})|}}.}$

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.

Jensen's formula is a consequence of the more general Poisson–Jensen 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, ..., a_n located in the interior of the unit disk, then for every ${\displaystyle z_{0}=r_{0}e^{i\varphi _{0}}}$ in the unit disk the Poisson–Jensen formula states that

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

Here,

${\displaystyle P_{r}(\omega )=\sum _{n\in \mathbb {Z} }r^{|n|}e^{in\omega }}$

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

${\displaystyle \log |f(z_{0})|={\frac {1}{2\pi }}\int _{0}^{2\pi }P_{r_{0}}(\varphi _{0}-\theta )\log |f(e^{i\theta })|\,d\theta ,}$

which is the Poisson formula for the harmonic function ${\displaystyle \log |f(z)|}$.

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• Jensen, J. (1899), "Sur un nouvel et important théorème de la théorie des fonctions", Acta Mathematica (in French), 22 (1): 359–364, doi:10.1007/BF02417878, ISSN 0001-5962, JFM 30.0364.02, MR 1554908
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