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

Every Nevanlinna function N admits a representation

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

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

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

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

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

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):

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

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

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

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

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

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

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

• ${\displaystyle \tan(z)}$ (an example that is surjective but not injective).

• A Möbius transformation

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

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

• ${\displaystyle 1+i+z}$ and ${\displaystyle i+\operatorname {e} ^{iz}}$ 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 ${\displaystyle f}$ is an arbitrary vector, then the function

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

is a Nevanlinna function.

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

Nevanlinna functions appear in the study of Operator monotone functions.