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Consider $\Omega=\{z\in\mathbb{C}:|z|<1\}$ and let $\mathcal{F}=\{f(z)=z+n:n\in\mathbb{N}\}$. We have that $\mathcal{F}$ is a family of holomorphic functions defined over $\Omega$ whose range omits infinite points of $\mathbb{C}$; hence by the Fundamental Normality Test we can say that $\mathcal{F}$ is a normal family, but this family is not uniformly bounded under compact sets of $\Omega$.

I am sure it has to be something obviuos but I have been thinking for a while and I do not find any clue of why my reasoning is wrong.

What am I missing here? Any help would be appreciated.

G. Acd
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1 Answers1

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This family is certainly not normal with respect to $(\Omega,\mathbb{C})$, as it is not locally uniformly bounded. Indeed, take any compact subset $K$ of $\Omega$ that contains the origin, and suppose the family $\{f_n\}$ were uniformly bounded by some $M>0$. Consider the image of $0$ under the family. If we choose $N$ so that $N>M,$ then $|f_N(0)|=N>M.$

Read your version of the fundamental normality test again carefully and make sure that it applies exactly. Does it apply to $(\Omega,\mathbb{C})$ or to $(\Omega,{\hat{\mathbb{C}}})?$ Is the question about normality with respect to $(\Omega,\mathbb{C})$ or to $(\Omega,{\hat{\mathbb{C}}})?$

Recall that normality can be defined on the Riemann sphere, in which case the above work is not sufficient. In this scenario, we either need that a subsequence converges uniformly on each compact subset as a sequence into $\mathbb{C}$ (we can rule this out) or that it converges to $\infty$ uniformly on compact subsets.

This part is true. Fix any compact $K\subset\Omega.$ Note that if $z\in K,$ then $$|f_n(z)|=|n+z|\geq n-|z|\geq n-1\rightarrow\infty$$ uniformly in $n$ as $n\rightarrow\infty$. This is because the FNT that you've seen was likely applied to $\hat{\mathbb{C}}.$

cmk
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  • My original statement to prove was: Let $\Omega\subset\mathbb{C}$ be a domain with $0\in\Omega$. Let $\mathcal{F}\subset\mathcal{H}(\Omega)$ with sup${|f(0)|:f\in\mathcal{F}}<\infty$ and all the functions in $\mathcal{F}$ omit a ball. Prove that $\mathcal{F}$ is normal. Searching for some result that could help me I read about the Fundamental Normality Test in here: https://www.encyclopediaofmath.org/index.php/Normal_family, which a priori solved my problem but then the example I wrote originally in the post seemed to contradict this Fundamental Normality Test. – G. Acd Jun 15 '19 at 12:24
  • Still don't understand exactly what I am missing, but the truth is that I have only worked with normality referring to $\mathbb{C}$ and not $\hat{\mathbb{C}}$. Anyway, thanks for your help. – G. Acd Jun 15 '19 at 12:30
  • I am only able to take a brief glance of the page you linked at the moment, but I don't see the FNT in there. You might want to reference Schiff's Normal Families book, but the version that they have is just like the one you'll find on wikipedia. You might like this link for your problem in your first comment: https://math.stackexchange.com/questions/5016/normal-families – cmk Jun 15 '19 at 12:34
  • @G.Acd In Schiff, their version of normality is defined on $\hat{\mathbb{C}}$, and so their version of FNT is applied to $(\Omega,\hat{\mathbb{C}}).$ I suspect that the version of the theorem you've seen does, as well. The family that you listed is normal here. – cmk Jun 15 '19 at 13:10
  • I've edited my post accordingly. – cmk Jun 15 '19 at 13:15