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 H and has non-negative imaginary part. A Nevanlinna function maps the upper half-plane to itself or to a real constant,^[1] 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

$N(z)=C+Dz+\int _{{{\mathbb {R}}}}\left({\frac {1}{\lambda -z}}-{\frac {\lambda }{1+\lambda ^{2}}}\right)d\mu (\lambda ),\quad z\in {\mathbb {H}},$

where C is a real constant, D is a non-negative constant and μ is a Borel measure on R satisfying the growth condition

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

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

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

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):

$\mu ((\lambda _{1},\lambda _{2}])=\lim _{{\delta \rightarrow 0}}\lim _{{\varepsilon \rightarrow 0}}{\frac {1}{\pi }}\int _{{\lambda _{1}+\delta }}^{{\lambda _{2}+\delta }}{\mathrm {Im}}(N(\lambda +i\varepsilon ))d\lambda .$

A very similar representation of functions is also called the Poisson representation.^[2]

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 some real number $a.$)

$z^{p}{\text{ with }}0\leq p\leq 1$

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

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 $i(z/i)^{p}{\text{ with }}-1\leq p\leq 1.$

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

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

• A Möbius transformation

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

is a Nevanlinna function if (but not only if) $a^{\ast }d-bc^{\ast }$ is a positive real number and ${\mathrm {Im}}(b^{\ast }d)={\mathrm {Im}}(a^{\ast }c)=0.$ 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: ${\frac {iz+i-2}{z+1+i}}$

• ${\displaystyle 1+i+z}$ and ${\displaystyle i+e^{iz}}$ 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

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

is a Nevanlinna function.

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

References

1. A real number is not considered to be in the upper half-plane.
2. See for example Section 4, "Poisson representation", of Louis de Branges. Hilbert spaces of entire functions. Prentice-Hall.. De Branges gives a form for functions whose real part is non-negative in the upper half-plane.

• 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.