Positive harmonic function

Herglotz-Riesz 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

${\displaystyle f(z)={\int }_0^{2\pi }\frac{1+e^{-i\theta }z}{1-e^{-i\theta }z}d\mu (\theta ).}$

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

Carathéodory's positivity criterion for holomorphic functions

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

${\displaystyle \sum _m\sum _n{a}_{m-n}{\lambda }_m\overline{{\lambda }_n}\ge 0}$

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

${\displaystyle \sum _m\sum _n{a}_{m-n}{\lambda }_m\overline{{\lambda }_n}={\int }_0^{2\pi }{\left|\sum _n{\lambda }_n{e}^{-in\theta }\right|}^{2}d\mu (\theta )\ge 0.}$

Conversely, setting λ_n = zⁿ,

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