Reference Book: Joel L.Schiff- Normal Families

Some Classical Theorems

Weierstrass Theorem Let $\{ f_{n}\}$ be a sequence of analytic functions on a domain $\Omega$ which converges uniformly on compact subsets of $\Omega$ to a function $f$ . Then $f$ is analytic in $\Omega$ , and the sequence of derivatives $\{ f_{n}^{(k)}\}$ converges uniformly on compact subsets to $f^{(k)}, k=1,2,3...$ .

Hurwitz Theorem Let $\{ f_{n}\}$ be a sequence of analytic functions on a domain $\Omega$ which converges uniformly on compact subsets of $\Omega$ to a non-constant analytic function $f(z)$ . If $f(z_{0})=0$ for some $z_{0}\in\Omega$ , then for each $r>0$ sufficiently small, there exists an $N=N(r)$ , such that for all $n>N$ , $f_{n}(z)$ has the same number of zeros in $D(z_{0},r)$ as does $f(z)$ . (The zeros are counted according to multiplicity).

The Maximum Principle If $f(z)$ is analytic and non-constant in a region $\Omega$ , then its absolute value $|f(z)|$ has no maximum in $\Omega$ .

The Maximum Principle’ If $f(z)$ is defined and continuous on a closed bounded set $E$ and analytic on the interior of $E$ , then the maximum of $|f(z)|$ on $E$ is assumed on the boundary of $E$ .

Corollary 1.4.1 If $\{ f_{n}\}$ is a sequence of univalent analytic functions in a domain $\Omega$ which converge uniformly on compact subsets of $\Omega$ to a non-constant analytic function $f$ , then $f$ is univalent in $\Omega$ .

Definition 1.5.1 A family of functions $\mathcal{F}$ is locally bounded on a domain $\Omega$ if, for each $z_{0}\in \Omega$ , there is a positive number $M=M(z_{0})$ and a neighbourhood $D(z_{0},r)\subset \Omega$ such that $|f(z)|\leq M$ for all $z\in D(z_{0}, r)$ and all $f\in \mathcal{F}$ .

Theorem 1.5.2 If $\mathcal{F}$ is a family of locally bounded analytic functions on a domain $\Omega$ , then the family of derivatives $\mathcal{F}^{'}=\{ f^{'}: f\in \mathcal{F}\}$ form a locally bounded family in $\Omega$ .

The converse of Theorem 1.5.2 is false, since $\mathcal{F}=\{n: n=1,2,3...\}$ . However, the following partial converse does hold.

Theorem 1.5.3 Let $\mathcal{F}$ be a family of analytic functions on $\Omega$ such that the family of derivatives $\mathcal{F}^{'}$ is locally bounded and suppose that there is some $z_{0}\in \Omega$ with $|f(z_{0})|\leq M<\infty$ for all $f\in \mathcal{F}$ . Then $\mathcal{F}$ is locally bounded. (Hint: find a path connecting $z_{0}$ and $z$ .)

Definition 1.6.1 A family $\mathcal{F}$ of functions defined on a domain $\Omega$ is said to be equicontinuous (spherically continuous) at a point $z^{'}\in \Omega$ if, for each $\epsilon>0$ , there is a $\delta=\delta(\epsilon,z^{'})>0$ such that $|f(z)-f(z^{'})|<\epsilon$ , $(\chi(f(z),f(z^{'}))<\epsilon)$ whenever $|z-z^{'}|<\delta$ , for every $f\in \mathcal{F}$ . Moreover, $\mathcal{F}$ is equicontinuous (spherical continuous) on a subset $E\subset \Omega$ if it is continuous (spherically continuous) at each point of $E$ .

Normal Families of Analytic Functions

Definition 2.1.1 A familiy $\mathcal{F}$ of analytic functions on a domain $\Omega\subset \mathbb{C}$ is normal in $\Omega$ if every sequence of functions $\{f_{n}\}\subset \mathcal{F}$ contains either a subsequence which converges to a limit function $f\not\equiv \infty$ uniformly on each compact subset of $\Omega$ , or a subsequence which converges uniformly to $\infty$ on each compact subset.

The family $\mathcal{F}$ is said to be normal at a point $z_{0}\in\Omega$ if it is normal in some neighbourhood of $z_{0}$ .

Theorem 2.1.2 A family of analytic functions $\mathcal{F}$ is normal in a domain $\Omega$ if and only if $\mathcal{F}$ is normal at each point in $\Omega$ .

Theorem 2.2.1 Arzela-Ascoli Theorem. If a sequence $\{f_{n}\}$ of continuous functions converges uniformly on a compact set $K$ to a limit function $f\not\equiv \infty$ , then $\{f_{n}\}$ is equicontinuous on $K$ , and $f$ is continuous. Conversely, if $\{f_{n}\}$ is equicontinuous and locally bounded on $\Omega$ , then a subsequence can be extracted from $\{f_{n}\}$ which converges locally uniformly in $\Omega$ to a (continuous) limit function $f$ .

Montel’s Theorem If $\mathcal{F}$ is a locally bounded family of analytic functions on a domain $\Omega$ , then $\mathcal{F}$ is a normal family in $\Omega$ .

Koebe Distortion Theorem Let $f(z)$ be analytic univalent in a domain $\Omega$ and $K$ a compact subset of $\Omega$ . Then there exists a constant $c=c(\Omega, K)$ such that for any $z,w\in K$ , $c^{-1}\leq |f^{'}(z)| / |f^{'}(w)| \leq c$ .

Vitali-Porter Theorem Let $\{f_{n}\}$ be a locally bounded sequence of analytic functions in a domain $\Omega$ such that $\lim_{n\rightarrow \infty}f_{n}(z)$ exists for each $z$ belonging to a set $E\subset \Omega$ which has an accumulation point in $\Omega$ . Then $\{ f_{n}\}$ converges uniformly on compact subsets of $\Omega$ to an analytic function.

Proof. From Montel’s Theorem, $\{ f_{n}\}$ is normal, extract a subsequence $\{ f_{n_{k}}\}$ which converges normally to an analytic function $f$ . Then $\lim_{k\rightarrow \infty} f_{n_{k}}(z)=f(z)$ for each $z\in E$ . Suppose, however, that $\{ f_{n}\}$ does not converge uniformly on compact subsets of $\Omega$ to $f$ . Then there exists some $\epsilon>0$ , a compact subset $K\subset \Omega$ , as well as a subsequence $\{f_{m_{j}}\}$ and points $z_{j}\in K$ satisfying $|f_{m_{j}}(z_{j})- f(z_{j})| \geq \epsilon,$ $j=1,2,3,...$ . Now $\{ f_{m_{j}}\}$ itself has a subsequence which converges uniformly on compact subsets to an analytic function $g$ , and $g\not\equiv f$ from above. However, since $f$ and $g$ must agree at all points of $E$ , the Identity Theorem for analytic functions implies $f\equiv g$ on $\Omega$ , a contradiction which establishes the theorem.

Fundamental Normality Test Let $\mathcal{F}$ be the family of analytic functions on a domain $\Omega$ which omit two fixed values $a$ and $b$ in $\mathbb{C}$ . Then $\mathcal{F}$ is normal in $\Omega$ .

Generalized Normality Test Suppose that $\mathcal{F}$ is a family of analytic functions in a domain $\Omega$ which omit a value $a\in \mathbb{C}$ and such that no function of $\mathcal{F}$ assumes the value $b\in \mathbb{C}$ at more that $p$ points. Then $\mathcal{F}$ is normal in $\Omega$ .

2.3 Examples:

Assume $U$ is the unit disk in the complex plane, $\Omega$ is a region (connected open set) in $\mathbb{C}$ .

1. $\mathcal{F}=\{ f_{n}(z)=z^{n}: n=1,2,3...\}$ in $U$ . Then $\mathcal{F}$ is normal in $U$ , but not compact since $0 \notin \mathcal{F}$ . In the domain $U^{'}: |z|>1$ , $\mathcal{F}$ is normal.

2. $\mathcal{F}=\{ f_{n}(z)=\frac{z}{n}: n=1,2,3...\}$ is a normal family in $\mathcal{C}$ but not compact.

3. $\mathcal{F}=\{ f: f$ analytic in $\Omega$ and $|f|\leq M \}$ . Then $\mathcal{F}$ is normal in $\Omega$ and compact.

4. $\mathcal{F}=\{ f: f$ analytic in $\Omega$ and $\Re f>0\}$ . Then $\mathcal{F}$ is normal but not compact. Hint: $\mathcal{G}=\{g=e^{-f}:f\in \mathcal{F}\}$ is a uniformly bounded family.

5. $\mathcal{S}=\{ f: f$ analytic, univalent in $U$ , $f(0)=0, f^{'}(0)=1 \}$ . These are the normalised “Schlicht” functions in $U$ . $\mathcal{S}$ is normal and compact.

Normal Families of Meromorphic Functions

Assume a function $f(z)$ is analytic in a neighbourhood of $a$ , except perhaps at $a$ itself. In other words, $f(z)$ shall be analytic in a region $0<|z-a|<\delta$ . The point $a$ is called an isolated singularity of $f(z)$ . There are three cases about an isolated singularity. The first one is a removable singularity, the second one is a pole, the third one is an essential singularity. A function $f(z)$ which is analytic in a region $\Omega$ , except for poles, is said to be meromorphic in $\Omega$ .

The chordal distance $\chi(z_{1}, z_{2})$ between $z_{1}$ and $z_{2}$ is

$\chi(z_{1}, z_{2}) = \frac{|z_{1}-z_{2}|}{\sqrt{1+|z_{1}|^{2}}\sqrt{1+|z_{2}|^{2}}}$ if $z_{1}$ and $z_{2}$ are in the finite plane, and

$\chi(z_{1}, \infty) = \frac{1}{\sqrt{1+|z_{1}|^{2}}},$ if $z_{2}=\infty$ . Clearly, $\chi(z_{1}, z_{2})\leq 1$ , and $\chi(z_{1}^{-1}, z_{2}^{-1}) = \chi(z_{1}, z_{2})$ . The chordal metric and spherical metric are uniformly equivalent and generate the same open sets on the Riemann sphere.

Definition 1.2.1 A sequence of functions $\{ f_{n}\}$ converges spherically uniformly to $f$ on a set $E\subset \mathbb{C}$ if, for any $\epsilon>0$ , there is a number $n_{0}$ such that $n\geq n_{0}$ implies $\chi(f(z), f_{n}(z))<\epsilon$ , for all $z\in E$ .

Definition 3.1.1 A family $\mathcal{F}$ of meromorphic functions in a domain $\Omega$ is normal in $\Omega$ if every sequence $\{ f_{n} \} \subset \mathcal{F}$ contains a subsequence which converges spherically uniformly on compact subsets of $\Omega$ .

Theorem 3.1.3 Let $\{ f_{n}\}$ be a sequence of meromorphic functions on a domain $\Omega$ . Then $\{ f_{n}\}$ converges spherically uniformly on compact subsets of $\Omega$ to $f$ if and only if about each point $z_{0}\in \Omega$ there is a closed disk $K(z_{0},r)$ in which $|f_{n}-f|\rightarrow 0$ or $|1/f_{n} - 1/f| \rightarrow 0$ uniformly as $n\rightarrow \infty$ .

Corollary 3.1.4 Let $\{ f_{n}\}$ be a sequence of meromorphic functions on $\Omega$ which converges spherically uniformly on compact subsets to $f$ . Then $f$ is either a meromorphic function on $\Omega$ or identically equal to $\infty$ .

Corollary 3.1.5 Let $\{ f_{n}\}$ be a sequence of analytic functions on a domain $\Omega$ which converge spherically uniformly on compact subsets of $\Omega$ to $f$ . Then $f$ is either analytic on $\Omega$ or identically equal to $\infty$ .

Theorem 3.2.1 A family $\mathcal{F}$ of meromorphic functions in a domain $\Omega$ is normal if and only if $\mathcal{F}$ is spherically equicontinuous in $\Omega$ .

Fundamental Normality Test Let $\mathcal{F}$ be a family of meromorphic functions on a domain $\Omega$ which omit three distinct values $a, b, c \in \mathbb{C}$ . Then $\mathcal{F}$ is normal in $\Omega$ .

Vitali-Porter Theorem Let $\{f_{n}\}$ be a sequence belonging to a spherically equicontinuous family of meromorphic functions such that $\{ f_{n}(z)\}$ converges spherically on a point set $E$ having an accumulation point in $\Omega$ . Then $\{ f_{n}\}$ converges spherically uniformly on compact subsets of $\Omega$ .

Let $f(z)$ be meromorphic on a domain $\Omega$ . If $z\in \Omega$ is not a pole, the derivative in the spherical metric, called the spherical derivative, is given by $f^{\#}(z) =\lim_{z^{'}\rightarrow z}\frac{\chi(f(z),f(z^{'}))}{|z-z^{'}|} =\frac{|f^{'}(z) |}{1+|f(z)|^{2}}$ . If $\zeta$ is a pole of $f(z)$ , define $f^{\#}(\zeta) = \lim_{z\rightarrow \zeta} \frac{|f^{'}(z)|}{1+|f(z)|^{2}}$ .

Marty’s Theorem A family $\mathcal{F}$ of meromorphic functions on a domain $\Omega$ is normal if and only if for each compact subset $K\subset \Omega$ , there exists a constant $C=C(K)$ such that spherical derivative $f^{\#}(z) =\frac{|f^{'}(z) |}{1+|f(z)|^{2}}\leq C, z\in K, f\in \mathcal{F},$ that is, $f^{\#}$ is locally bounded.

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ZHANG RONG

Daily Archives: October 20, 2014

Normal Families

Some Classical Theorems

Normal Families of Analytic Functions

Normal Families of Meromorphic Functions

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