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Fall 2013 - Problem 5

normal families

Let $\set{f_n}_n$ be a sequence of holomorphic functions on $\D$ and suppose that

\int_\D \abs{f_n\p{z}} \,\diff\lambda\p{z} \leq 1

for all $n \in \N$ , where $\diff\lambda$ denotes integration with respect to Lebesgue measure $\lambda$ on $\C$ . Show that then there exists a subsequence $\set{f_{n_k}}_k$ that converges uniformly on all compact subsets of $\D$ .

Solution.

Suppose $\cl{B\p{z, r}} \subseteq \D$ . Then by the mean value property

\begin{aligned} \abs{f_n\p{z}} &\leq \frac{1}{\pi r^2} \int_{B\p{z, r}} \abs{f_n\p{\zeta}} \,\diff\lambda\p{\zeta} \\ &\leq \frac{1}{\pi r^2}. \end{aligned}

Thus, if $K = \cl{B\p{0, R}}$ for $0 < R < 1$ , then let $\delta = \frac{1}{2}d\p{K, \D^\comp} > 0$ so that $\cl{B\p{z, \delta}} \subseteq \D$ for any $z \in K$ . Hence,

\abs{f_n\p{z}} \leq \frac{1}{\pi \delta^2},

so $\set{f_n}_n$ form a normal family. Thus, there exists a subsequence which converges locally uniformly on $\D$ , which completes the proof.