Jensen's formula
In the mathematical field known as complex analysis, Jensen's formula, introduced by , 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.

Formal statement 

Suppose that  is an analytic function in a region in the complex plane which contains the closed disk  of radius  about the origin,  are the zeros of  in the interior of  (repeated according to their respective multiplicity), and that . 

Jensen's formula states that

This formula establishes a connection between the moduli of the zeros of the function  inside the disk  and the average of
 on the boundary circle , and can be seen as a generalisation of the mean value property of harmonic functions. Namely, if  has no zeros in , then Jensen's formula reduces to

which is the mean-value property of the harmonic function .

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

where  denotes the number of zeros of  in the disc of radius  centered at the origin.

Proof

Applications 

Jensen's formula can be used to estimate the number of zeros of an analytic function in a circle. Namely, if  is a function analytic in a disk of radius  centered at  and if  is bounded by  on the boundary of that disk, then the number of zeros of  in a circle of radius 
 centered at the same point  does not exceed

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, and it often appears in proofs of Hadamard factorization theorem, which requires an estimate on the number of zeros of an entire function.

Generalizations
Jensen's formula may be generalized for functions which are merely meromorphic on . Namely, assume that
 
where  and  are analytic functions in  having zeros at 
and 

respectively, then Jensen's formula for meromorphic functions states that

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 . It was introduced and named by Rolf Nevanlinna. If  is a function which is analytic in the unit disk, with zeros
 located in the interior of the unit disk, then for every  in the unit disk the Poisson–Jensen formula states that

Here,

is the Poisson kernel on the unit disk.
If the function  has no zeros in the unit disk, the Poisson-Jensen formula reduces to

which is the Poisson formula for the harmonic function .

References

Sources 
 
 
 

Theorems in complex analysis