# Harnack's Principle question

1. Jan 26, 2009

### redrzewski

Ahlfors version of this theorem says that a sequence of harmonic functions {Un} tends UNIFORMLY to infinity on compact subsets, or tends to a harmonic limit function uniformly on compact sets.

Can someone please clarify what tending uniformly to infinity means?

In particular, it seems like a set of harmonic {Un} where Uk = k (such that each function is constant) tends non-uniformly to infinity.

So I must be missing something somewhere.

thanks

2. Jan 30, 2009

### jostpuur

$f_n:K\to\mathbb{C}$ ($n=1,2,3\ldots$) converges towards infinity uniformly, if for all $R>0$ there exists $N\in\mathbb{N}$ such that $|f_n(z)|>R$ for all $z\in K$ and $n\geq N$.

Simply extend $[0,\infty[$ to $[0,\infty]$ (with topology homeomorphic with $[0,1]$), and threat $\infty$ as a constant so that a sequence of functions can converge uniformly towards the corresponding constant function.

In point-wise convergence to infinity would mean that for each $z\in K$ and $R>0$ there exists $N\in\mathbb{N}$ such that $|f_n(z)|>R$ for all $n\geq N$.

Constant functions surely converge uniformly if they converge somehow.

3. Jan 30, 2009

### redrzewski

Thanks. That definition clears things up greatly.