Normal sequences and Montel's Theorem

$\begingroup$

I am currently stuck on an exam question involving normal sequences and Montel's theorem:

1. Give two examples of non-constant normal sequences one in the $(a)$ unit disk $\mathbb{D}$ and one in $(b)$ $\mathbb{C}$.

2. Let $(a_n)$ be a sequence of complex numbers satisfying $|a_n|\lt 1$ and an $a_n\to 0$ and consider the sequence of holomorphic functions $(f_n)$ in $\mathbb{D}$ defined by:

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\large f_n(z)=\frac{z-a_n}{1-\overline{a_n}z}$

State and justify if $(f_n)$ is normal and whether it satisfies the condition of Montel’s theorem? Explain if $(f_n)$ converges uniformly in $\mathbb{D}$.

Firstly I will state Montel's theorem and then I will write what I have tried so far.

Montel's theorem: A family $\mathbb{F}\subset \mathbb{H}(U)$ is normal if and only if it is locally bounded. (Where $U\subset\mathbb{C}$, open, and $\mathbb{H}(U)$ is the space of holomorphic functions on $U$.

My examples for 1. are:

$(a)$ $g_n:\mathbb{D}\to\mathbb{C}, g_n(z)=\large e^{\frac{iz}{n}}$

$(b)$ $h_n:\mathbb{C}\to\mathbb{C}, h_n(z)=\large \frac{z}{n^2}$

Now for $2$. I have chosen to show that $f_n$ has a subsequence that converges uniformly on compact subsets of $\mathbb{D}$, so let $K\subset \mathbb{D}$ be compact, thus if $z\in K\Rightarrow |z|\le 1$.

I know that since $a_n\to 0$, $f_n(z)\to z$ pointwise.

Now: $\large|f_n(z)-z|=|\frac{z-a_n}{1-\overline{a_n}z}-z|=|\frac{z-a_n-z+\overline{a_n}z^2}{1-\overline{a_n}z}|$

$\large=|\frac{\overline{a_n}z^2-a_n}{1-\overline{a_n}z}|\le \frac{|\overline{a_n}z^2-a_n|}{1-|a_n|}\le \frac{|a_n|(|z|^2+1)}{1-|a_n|}\le \frac{2|a_n|}{1-|a_n|}$

Now we can construct a subsequence:

since $a_n\to 0$ it follows that $\exists N\in \mathbb{N}:\forall j\ge N\,:|a_j|\lt\frac{1}{N}$

So we consider the subsequence $\{f_j\,|j\ge N\}$

Here $|f_j(z)-z|\le\frac{2|a_j|}{1-|a_j|}\lt \frac{\frac{2}{N}}{1-\frac{1}{N}}=\frac{2}{N-1}\to 0$ as $N\to\infty$.

So the subsequence converges uniformly in $\mathbb{D}$, thus $(f_n)$ is normal, and thus by Montel's theorem it is locally bounded, so it satisfies the condition of montels theorem.

Now explain if $(f_n)$ is uniformly convergent on $\mathbb{D}$, this is the part that I am stuck on, and would really appreciate help with.

I would also like clarification if what I have done for the previous parts is at all correct?

Thanks!

There is a third part to the question:

1. Let $(f_n)$ be a sequence of holomorphic functions from $U$ to the upper half-plane. Prove that $(f_n)$ has a subsequence converging uniformly on all compact subsets of $U$ to either a holomorphic function $f$ or to $\infty$

My approach is to consider the sequence:

$g_n(z)=\large\frac{\frac{f_n(z)}{|f_n(z)|}-a_n}{1-\overline{a_n}\frac{f_n(z)}{|f_n(z)|}}$, here $g_n$ is of the form of part 2. and takes values in the unit disk.

$\endgroup$ 36 Sorted by: Reset to default

Know someone who can answer? Share a link to this question via email, Twitter, or Facebook.

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy