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 {\displaystyle z-a} 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
A real number is not considered to be in the upper half-plane.
See for example Section 4, "Poisson representation", of Louis de Branges (1968). Hilbert spaces of entire functions. Prentice-Hall. ASIN B0006BUXNM.. 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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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