Positive harmonic function

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

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

The formula clearly defines a positive harmonic function with f(0) = 1.

Conversely if f is positive and harmonic and r_n increases to 1, define

${\displaystyle f_{n}(z)=f(r_{n}z).\,}$

Then

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

where

${\displaystyle d\mu _{n}(\varphi )={1 \over 2\pi }f(r_{n}e^{i\varphi })\,d\varphi }$

is a probability measure.

By a compactness argument (or equivalently in this case Helly's selection theorem for Stieltjes integrals), a subsequence of these probability measures has a weak limit which is also a probability measure μ.

Since r_n increases to 1, so that f_n(z) tends to f(z), the Herglotz formula follows.

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

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

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 determines the holomorphic function up to addition of a scalar
• the above formula defines a holomorphic function, the real part of which is given by the previous theorem

Let

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

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

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

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

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

for m > 0.

In fact from the Herglotz representation for n > 0

${\displaystyle a_{n}=2\int _{0}^{2\pi }e^{-in\theta }\,d\mu (\theta ).}$

Hence

${\displaystyle \sum _{m}\sum _{n}a_{m-n}\lambda _{m}{\overline {\lambda _{n}}}=\int _{0}^{2\pi }\left|\sum _{n}\lambda _{n}e^{-in\theta }\right|^{2}\,d\mu (\theta )\geq 0.}$

Conversely, setting λ_n = zⁿ,

${\displaystyle \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }a_{m-n}\lambda _{m}{\overline {\lambda _{n}}}=2(1-|z|^{2})\,\Re \,f(z).}$

• Bochner's theorem

• Carathéodory, C. (1907), "Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen" (PDF), Math. Ann., 64: 95–115, doi:10.1007/bf01449883
• Duren, P. L. (1983), Univalent functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, ISBN 0-387-90795-5
• Herglotz, G. (1911), "Über Potenzreihen mit positivem, reellen Teil im Einheitskreis", Ber. Verh. Sachs. Akad. Wiss. Leipzig, 63: 501–511
• Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, 15, Vandenhoeck & Ruprecht
• Riesz, F. (1911), "Sur certains systèmes singuliers d'équations intégrale", Ann. Sci. Éc. Norm. Supér., 28: 33–62, doi:10.24033/asens.633