Positive harmonic function

In mathematics, a positive harmonic function on the unit disc in the complex numbers is characterized as the Poisson integral of a positive measure on the circle. This result, the Herglotz representation theorem, was proved by Gustav Herglotz in 1911. It can be used to give a related formula and characterization for any holomorphic function on the unit disc with positive real part. Such functions had already been characterized in 1907 by Constantin Carathéodory in terms of the positive definiteness of their Taylor coefficients.

Herglotz representation theorem for harmonic functions

A positive function f on the unit disk with f(0) = 1 is harmonic if and only if there is a probability measure μ on the unit circle such that

$f(re^{i\theta })=\int _{0}^{2\pi }{1-r^{2} \over 1-2r\cos(\theta -\varphi )+r^{2}}\,d\mu (\varphi ).$

Herglotz representation theorem for holomorphic functions

A holomorphic function f on the unit disk with f(0) = 1 has positive real part if and only if there is a probability measure μ on the unit circle such that

$f(z)={1 \over 2\pi }\int _{0}^{2\pi }{1+e^{-i\theta }z \over 1-e^{-i\theta }z}\,d\mu (\theta ).$

This follows from the previous theorem because:

• the Poisson kernel is the real part of the integrand above
• the real part of a holomorphic function is harmonic and determined by the holomorphic function up to a scalar
• the above formula defines a holomorphic function, the real part of which is given by the previous theorem

Carathéodory's positivity criterion for holomorphic functions

Let

$f(z)=1+a_{1}z+a_{2}z^{2}+\cdots$

be a holomorphic function on the unit disk. Then f(z) has positive real part on the disk if and only if

$\sum _{m}\sum _{n}a_{m-n}\lambda _{m}{\overline {\lambda _{n}}}\geq 0$

for any complex numbers λ₀, λ₁, ..., λ_N, where

$a_{0}=2,\,\,\,a_{-m}={\overline {a_{m}}}$

for m > 0.

References

• Duren, P. L. (1983), Univalent functions, Grundlehren der Mathematischen Wissenschaften, vol. 259, Springer-Verlag, ISBN 0-387-90795-5 {{citation}}: Text "Duren, Peter L." ignored (help)
• Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht