Montel's theorem

(Compactness principle, P. Montel, 1910). Prove that a set $M$ in the space $\mathcal{A}(D)$ is precompact if and only if it is bounded in the space $\mathcal{A}(D)$ (i.e., $\sup \left\{ \left|f(z)\right|: f \in M, z \in Q\right\} < \infty$ for every compact set $Q \subset D$).

Is the set of analytic functions precompact in the space $C^{1}[0,1]$

where the series converges in a disk $D \subset \mathbb {C}$ containing $[0,1]$?

Which of the following families are normal, and which is compact? Justify your answers.

Suppose that $1<p<\infty$ and $c>0$ is a real number. Let $\mathcal F$ be the set of all analytic functions $f$ on $\left\{ \left|z\right|<1\right\}$ such that

Show that $\mathcal F$ is a normal family.

State whether the following statement is True or False, and prove your assertion.

For each positive integer $n$ there exists an entire function $f_n$ such that

Let $G = D \setminus (-1,0]$, where $D = \left\{ z:\left|z\right|<1\right\}$.

Let $\mathcal F$ denote the family of functions $f(z) = \sum_{n=0}^{\infty } a_n z^n$ which are holomorphic in the unit disk $D$ and satisfy $\left|a_n\right| \leq n$ for $n=0,1,2,\ldots$. Prove that any sequence in $\mathcal F$ has a subsequence which converges uniformly on compact subsets of $D$.

Let $\mathscr {F}$ be the class of all $f \in H(U)$ such that $\mathrm{Re}\left(f\right) > 0$ and $f(0) = 1$. Show that $\mathscr {F}$ is a normal family. Can the condition " $f(0) = 1$ " be omitted? Can it be replaced by " $\left|f(0)\right| \leq 1$ "?

Let $\mathscr {F}$ be the class of all $f \in H(U)$ for which

Is this a normal family?

Suppose $\Omega$ is a bounded region, $a \in \Omega , f \in H(\Omega ), f(\Omega ) \subset \Omega$, and $f(a) = a$.

Suppose $\Omega$ is a region, $p \in \Omega , R < \infty$. Let $\mathscr {F}$ be the class of all $f \in H(\Omega )$ such that $\left|f(p)\right| \leq R$ and $f(\Omega )$ contains neither 0 nor 1. Prove that $\mathscr {F}$ is a normal family.

Let the functions $f_{n}(z), \quad n = 1,2, \ldots$, be regular in the disk $\left|z\right| < 1$ and satisfy the conditions

where the numbers $M$ and $m$ do not depend on $n$. Prove that from the sequence $\left\{ f_{n}(z)\right\}$ one can select a subsequence $\left\{ f_{n_{k}}(z)\right\}$ converging uniformly on every compact subset of the disk $\left|z\right| < 1$.

Let the boundary $\partial D$ of a bounded domain $D$ consist of a finite number of simple piecewise-smooth curves, and let functions $\varphi_{n}(\zeta )$ be defined and continuous on $\partial D$. Prove that if the sequence $\left\{ \varphi_{n}(\zeta )\right\}$ is uniformly bounded on $\partial D$, then from the sequence of functions $\left\{ f_{n}(z)\right\}$,

($m$ a fixed integer) one can extract a subsequence $\left\{ f_{n_{k}}(z)\right\}$ converging uniformly on every compact subset of the domain $D$.

Let the functions $f_{n}(z), \quad n = 1,2, \ldots$, be regular in a domain $D$ and satisfy the conditions $\mathrm{Re}\left(f_{n}(z)\right) \geqslant 0 \quad (z \in D, n = 1,2, \ldots )$. Prove that from the sequence $\left\{ f_{n}(z)\right\}$ one can extract a subsequence $\left\{ f_{n_{k}}(z)\right\}$ converging uniformly on every compact subset of the domain $D$ (possibly to infinity).

Let the functions $u_{n}(x, y), n = 1,2, \ldots$, be harmonic in a domain $D$ and satisfy the conditions

Prove that from the sequence $\left\{ u_{n}(x, y)\right\}$ one can extract a subsequence $\left\{ u_{n_{k}}(x, y)\right\}$ converging uniformly on every compact subset of the domain $D$.

Let the functions $f_{n}(z), \quad n = 1,2, \ldots$, be regular in a domain $D$, not vanish in this domain, and satisfy the inequalities

(the constant $M$ does not depend on $n$). Prove that from the sequence $\left\{ \sqrt[n]{\left|f_{n}(z)\right|}\right\}$ one can extract a subsequence converging uniformly on every compact subset of the domain $D$ to a function $\left|g(z)\right|$, where $g(z)$ is a function regular in the domain $D$.