Nevanlinna function

Short description: A complex analysis 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 ${\mathscr{H}}$ 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 N admits a representation

$N(z)=C+Dz+\underset{\mathbb{ℝ}}{\int }(\frac{1}{\lambda -z}-\frac{\lambda }{1+{\lambda }^2})\mathrm{d}\mu (\lambda ),z\in {\mathscr{H}},$

where C is a real constant, D is a non-negative constant, ${\mathscr{H}}$ is the upper half-plane, and μ is a Borel measure on ℝ satisfying the growth condition

$\underset{\mathbb{ℝ}}{\int }\frac{\mathrm{d}\mu (\lambda )}{1+{\lambda }^2}<\mathrm{\infty }.$

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

$C=\mathrm{\Re }(N(i))\text{ and }D=\underset{y\to \mathrm{\infty }}{\lim }\frac{N(iy)}{iy}$

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):

$\mu (({\lambda }_1,{\lambda }_2])=\underset{\delta \to 0}{\lim }\underset{\epsilon \to 0}{\lim }\frac{1}{\pi }\underset{{\lambda }_1+\delta }{\overset{{\lambda }_2+\delta }{\int }}\mathrm{\Im }(N(\lambda +i\epsilon ))\mathrm{d}\lambda .$

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) $\bar{a}d-b\bar{c}$ is a positive real number and $\mathrm{\Im }(\bar{b}d)=\mathrm{\Im }(\bar{a}c)=0$. 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: $\frac{iz+i-2}{z+1+i}$

• $1+i+z$ and $i+{\mathrm{e}}^{iz}$ 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

$⟨(S-z{)}^{-1}f,f⟩$

is a Nevanlinna function.

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

Importance in operator theory

Nevanlinna functions appear in the study of Operator monotone functions.

References

General

• Vadim Adamyan, ed (2009). Modern analysis and applications. p. 27. ISBN 3-7643-9918-X. 
• Naum Ilyich Akhiezer and I. M. Glazman (1993). Theory of linear operators in Hilbert space. ISBN 0-486-67748-6. 
• Marvin Rosenblum and James Rovnyak (1994). Topics in Hardy Classes and Univalent Functions. ISBN 3-7643-5111-X.