On the poisson integral representation of harmonic functions

Abstract

On the Poisson Integral Representation of Harmonic Functions A positive harmonic function h in the unit disk has the Poisson integral representation h = $P[d\mu]$, where $\mu$ is a positive finite Borel measure on the unit circle. If $\phi(z)$ is a holomorphic self mapping of the unit disk into itself, h 0 $\phi$ can also be represented by h $\phi=P[d\mu\phi]$ where $\mu\,\phi$ is a positive finite Borel measure on the unit circle. For the special cases $\phi(z)=\frac{Z-a}{1-aZ}$ and $\phi(Z)=Z^n$, we consider the relations between $d\mu$ and $d\mu\phi$. We apply these results to holomorphic functions with positive real parts and also extend to positive M-harmonic functions in the units ball of $C^n$.