Nevanlinna function
In mathematics, in the field of complex analysis, a Nevanlinna function  is a complex function which is an analytic function on the open upper half-plane  and has non-negative imaginary part. A Nevanlinna function maps the upper half-plane to itself or to a real constant, but is not necessarily injective or surjective. Functions with this property are sometimes also known as Herglotz, Pick or R functions.

Integral representation

Every Nevanlinna function  admits a representation

where  is a real constant,  is a non-negative constant,  is the upper half-plane, and  is a Borel measure on  satisfying the growth condition
 

Conversely, every function of this form turns out to be a Nevanlinna function. 
The constants in this representation are related to the function  via

and the Borel measure  can be recovered from  by employing the Stieltjes inversion formula (related to the inversion formula for the Stieltjes transformation):

A very similar representation of functions is also called the Poisson representation.

Examples
Some elementary examples of Nevanlinna functions follow (with appropriately chosen branch cuts in the first three). ( can be replaced by  for any real number .)

These are injective but when  does not equal 1 or −1 they are not surjective and can be rotated to some extent around the origin, such as .

A sheet of  such as the one with .

 (an example that is surjective but not injective).

 A Möbius transformation

 is a Nevanlinna function if (sufficient but not necessary)  is a positive real number and . This is equivalent to the set of such transformations that map the real axis to itself. One may then add any constant in the upper half-plane, and move the pole into the lower half-plane, giving new values for the parameters. Example: 

  and  are examples which are entire functions. The second is neither injective nor surjective.
 If  is a self-adjoint operator in a Hilbert space and  is an arbitrary vector, then the function

 is a Nevanlinna function.

 If  and  are both Nevanlinna functions, then the composition  is a Nevanlinna function as well.

Importance in operator theory

Nevanlinna functions appear in the study of Operator monotone functions.

References

General
 
  
 

Complex analysis