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# Harnack’s principle

If the functions $u_{1}(z)$, $u_{2}(z)$, … are harmonic in the domain $G\subseteq\mathbb{C}$ and

$u_{1}(z)\leq u_{2}(z)\leq\cdots$ |

in every point of $G$, then $\lim_{{n\to\infty}}u_{n}(z)$ either is infinite in every point of the domain or it is finite in every point of the domain, in both cases uniformly in each closed subdomain of $G$. In the latter case, the function $u(z)=\lim_{{n\to\infty}}u_{n}(z)$ is harmonic in the domain $G$ (cf. limit function of sequence).

Major Section:

Reference

Type of Math Object:

Theorem

Parent:

## Mathematics Subject Classification

30F15*no label found*31A05

*no label found*

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