Questions tagged [normal-families]

For questions about normal families and their properties. A normal family is a precompact family of continuous functions.

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Family of harmonic functions is normal if and only if family of harmonic conjugates is normal

(a) Let $u$ be a harmonic function in a simply connected region and $v$ be its conjugate. Is it true that $u$ is bounded if and only if $v$ is bounded. (b) Assume $\mathcal{U} = \{u_n\}$ is a family ...
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Is this family normal?

Suppose that $R$ is a region, $a \in R$ and $\mathcal{F}=\{f \in H(R) : \mbox{$|f(a)|<1$and$0,1 \notin f(R)$}\}$. Is this family normal? I guess it is, but I fail to find the reasoning behind.
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the family of analytic functions with positive real part is normal. [duplicate]

I'm reviewing Complex Analysis and I don't quite understand the concept of normal family. There is an exercise in Ahlfors' Complex Analysis: Prove that in any region the family of analytic functions ...
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Let $\mathcal{F}$ denote all analytic functions such that $|f(z)| \leq \frac{1}{(1-|z|)^{2015}}$ for all $z\in \mathbb{D}$.

Let $\mathcal{F}$ denote all analytic functions $f:\mathbb{D} \to \mathbb{C}$ satisfying the inequality $|f(z)| \leq \frac{1}{(1-|z|)^{2015}}$ for all $z\in \mathbb{D}$. Show that $\mathcal{F}$ is ...
1 vote
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Why don't Montel's great theorem and Montel's theorem contradict each other?

Montel's Great theorem states Let $\mathcal{F}$ be a collection of analytic functions on a region $\Omega$ such that all of the $f\in \mathcal{F}$ omit the same two values $a,b$. Then the family is ...
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Normal family with $\Im f(z)>0$

I have to decide if this family is normal or not $\mathscr{F}=\{f\in\mathscr{H}(D(0,1)): f(0)=2i\ \&\ \Im f(z)>0\ \forall z\in D(0,1)\}$ where $\Im f(z)$ is the imaginary part of $f(z)$. I ...
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What am I missing (Fundamental Normality Test)?

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 ...
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$\{g_{n_k}\}_{k \in \Bbb N}$ normally convergent $\implies \{g^2_{n_k}\}_{k \in \Bbb N}$ normally convergent (as meromorphic functions)?
Let $\mathscr F$ be a family of one-to-one holomorphic functions on a simply-connected domain $D \subset \Bbb C$ such that $\mathscr F$ omits 0. Show that $\mathscr F$ is a normal family (when ...