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 $\, \mathcal \,$ 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

:$N(z) = C + D z + \int_ \bigg(\frac - \frac \bigg) \operatorname \mu(\lambda), \quad z \in \mathcal,$

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

:$\int_ \frac < \infty.$

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

:$C = \Re \big( N(i) \big) \qquad \text \qquad D = \lim_ \frac$

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

:$\mu \big( (\lambda_1, \lambda_2 ] \big) = \lim_ \lim_ \frac \int_^ \Im \big( N(\lambda + i \varepsilon) \big) \operatorname \lambda.$

A very similar representation of functions is also called the Poisson representation.See for example Section 4, "Poisson representation" in De Branges gives a form for functions whose ''real'' part is non-negative in the upper half-plane.

Examples

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

*$z^p\text 0 \le p \le 1$

*$-z^p\text -1 \le p \le 0$

:::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 $i(z/i)^p ~\text~-1\le p\le 1$.

*A sheet of $\ln(z)$ such as the one with $f(1)=0$.

*$\tan(z)$ (an example that is surjective but not injective).

* A Möbius transformation

::$z \mapsto \frac$

: is a Nevanlinna function if (sufficient but not necessary) $\overline d - b \overline$ is a positive real number and $\Im (\overline d ) = \Im (\overline c) = 0$. 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: $\frac$

* $1 + i + z$ and $i + \operatorname^$ are examples which are entire functions. The second is neither injective nor surjective.
* If is a self-adjoint operator in a Hilbert space and $f$ is an arbitrary vector, then the function

::$\langle (S-z)^ f, f \rangle$

: is a Nevanlinna function.

* If $M(z)$ and $N(z)$ are both Nevanlinna functions, then the composition $M \big( N(z) \big)$ is a Nevanlinna function as well.

Importance in operator theory

Nevanlinna functions appear in the study of Operator monotone functions.

References

General

*
*
* {{cite book , title=Topics in Hardy Classes and Univalent Functions , author=Marvin Rosenblum and James Rovnyak , isbn=3-7643-5111-X , year=1994

Complex analysis