Jensen's formula

Not to be confused with Jensen's inequality.

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 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, ..., an located in the interior of the unit disk, then for every z_0=r_0e^{i\varphi_0} in the unit disk the Poisson–Jensen 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^{i n\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)|.

References