TITLE: Complex analysis, normal families
QUESTION [0 upvotes]: I'm trying to determine whether or not the following families of functions in $H(\mathbb{D})$ are normal or not.
The first family is 
$$\mathcal{F}=\left\{f\in H(\mathbb{D}); f(0)=0,\ \sup_{z\in\mathbb{D}}\ |f''(z)|\cdot(1-|z|^2)<+\infty\right\}$$
The second family is
$$\mathcal{G}=\left\{ g\in H(\mathbb{D}); \sup_{z\in\mathbb{D}}\left|g(z)-\dfrac{z^3}{(1-z^2)} \right|<+\infty\right\}$$
I imagine the idea is trying to prove these are locally uniformly bounded in $\mathbb{D}$ and then apply Montel's theorem, but I can't make heads or tails of how to bound these. Also, I'm supposed to provide additional conditions for normality in case these families aren't normal. Any hints?
Thank you.

REPLY [1 votes]: First family: 

Let $\mathcal{F}_1\subset \mathcal{F}$ be the family of linear functions such that $f(0)=0$, i.e. $\mathcal{F}_1=\{az; a\in \mathbb{C}\}$. This family is not normal, as it is not locally uniformly bounded ($\underset{ n\in \mathbb{N}}{\forall}\underset{z\in \mathbb{D}-\{0\}}{\forall} \exists f\in \mathcal{F}:|f(z)|=n$, namely $f(w):=\frac{n}{z}w$), and so neither is $\mathcal{F}$.
However, if one imposes the condition $|f'(0)|<k$ and a stricter condition like $\sup_{\mathbb{D}}|f''(z)|(1-|z|^2)<h$ the result follows. In fact, given a compact set $\Gamma \subset \mathbb{D}$ of diameter $\rho<1$, we have
 $$|f'(z)|\le |f'(0)|+\sup_{\Gamma}|f''(z)||z|\le |f'(0)|+|z|\frac{h}{1-\rho^2}\le k+\frac{\rho h}{1-\rho^2}\\
|f(z)|\le |f(0)|+\sup_{\Gamma}|f'(z)||z|\le \left(k+\frac{\rho h}{1-\rho^2}\right)|z|\le \left(k+\frac{\rho h}{1-\rho^2}\right)\rho$$

Second family

Note that $\mathcal{G}$ contains the family $\mathcal{G}_1:=\left\{f(z)=a+\frac{z^3}{1-z^2}; a\in \mathbb{C}\right\}$, which is not locally bounded (as it is not bounded in $0$). However, one condition which is sufficient to ensure normality is to modify the definition of $\mathcal{G}$ as follows:
$$\mathcal{G}=\left\{f\in H(\mathbb{D});\sup_{z\in \mathbb{D}}\left|f(z)-\frac{z^3}{1-z^2}\right|<k\right\}$$
The normality of these redefined $\mathcal{G}$ follows from the fact that, given a compact set $\Gamma\subset \mathbb{D}$ of diameter $\rho<1$, we have
$$|f(z)|\le \left|\frac{z^3}{1-z^2}\right|+k\le \frac{\rho^3}{1-\rho^2}+k$$
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