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):

$\mu((\lambda_1,\lambda_2]) = \lim_{\delta\rightarrow0} \lim_{\varepsilon\rightarrow 0} \frac{1}{\pi} \int_{\lambda_1+\delta}^{\lambda_2+\delta} \mathrm{Im}(N(\lambda+i\varepsilon))d\lambda.$

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 $a.$)

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

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

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 (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: $\frac{iz+i-2}{z+1+i}$

• $1+i+z$ and $i+e^{iz}$ 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.