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 $$\, \mathcal{H} \,$$ and has a non-negative imaginary part. A Nevanlinna function maps the upper half-plane to itself or 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 $N$ admits a representation


 * $$ N(z) = C + D z + \int_{\mathbb{R}} \bigg(\frac{1}{\lambda - z} - \frac{\lambda}{1 + \lambda^2} \bigg) \operatorname{d} \mu(\lambda), \quad z \in \mathcal{H},$$

where $C$ is a real constant, $D$ is a non-negative constant, $$\mathcal{H}$$ is the upper half-plane, and $μ$ is a Borel measure on $ℝ$ satisfying the growth condition
 * $$ \int_{\mathbb{R}} \frac{\operatorname{d} \mu(\lambda)}{1 + \lambda^2} < \infty.$$

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


 * $$ C = \Re \big( N(i) \big) \qquad \text{ and } \qquad D = \lim_{y \rightarrow \infty} \frac{N(i y)}{i y} $$

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


 * $$ \mu \big( (\lambda_1, \lambda_2 ] \big) =

\lim_{\delta\rightarrow 0} \lim_{\varepsilon\rightarrow 0} \frac{1}{\pi} \int_{\lambda_1+\delta}^{\lambda_2+\delta} \Im \big( N(\lambda + i \varepsilon) \big) \operatorname{d} \lambda.$$

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). ($$z$$ can be replaced by $$z - a$$ for any real number $$a$$.)


 * $$z^p\text{ with } 0 \le p \le 1$$


 * $$-z^p\text{ with } -1 \le p \le 0$$


 * These are injective but when $p$ does not equal 1 or −1 they are not surjective and can be rotated to some extent around the origin, such as $$i(z/i)^p ~\text{ with }~-1\le p\le 1$$.


 * 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


 * $$z \mapsto \frac{az+b}{cz+d} $$


 * is a Nevanlinna function if (sufficient but not necessary) $$\overline{a} d - b \overline{c}$$ is a positive real number and $$\Im (\overline{b} d ) = \Im (\overline{a} c) = 0$$. 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: $$\frac{i z + i - 2}{z + 1 + i}$$


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


 * $$ \langle (S-z)^{-1} f, f \rangle $$


 * 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.