Normal Families

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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