Reference Book: Joel L.Schiff- Normal Families
Some Classical Theorems
Weierstrass Theorem Let be a sequence of analytic functions on a domain which converges uniformly on compact subsets of to a function . Then is analytic in , and the sequence of derivatives converges uniformly on compact subsets to .
Hurwitz Theorem Let be a sequence of analytic functions on a domain which converges uniformly on compact subsets of to a non-constant analytic function . If for some , then for each sufficiently small, there exists an , such that for all , has the same number of zeros in as does . (The zeros are counted according to multiplicity).
The Maximum Principle If is analytic and non-constant in a region , then its absolute value has no maximum in .
The Maximum Principle’ If is defined and continuous on a closed bounded set and analytic on the interior of , then the maximum of on is assumed on the boundary of .
Corollary 1.4.1 If is a sequence of univalent analytic functions in a domain which converge uniformly on compact subsets of to a non-constant analytic function , then is univalent in .
Definition 1.5.1 A family of functions is locally bounded on a domain if, for each , there is a positive number and a neighbourhood such that for all and all .
Theorem 1.5.2 If is a family of locally bounded analytic functions on a domain , then the family of derivatives form a locally bounded family in .
The converse of Theorem 1.5.2 is false, since . However, the following partial converse does hold.
Theorem 1.5.3 Let be a family of analytic functions on such that the family of derivatives is locally bounded and suppose that there is some with for all . Then is locally bounded. (Hint: find a path connecting and .)
Definition 1.6.1 A family of functions defined on a domain is said to be equicontinuous (spherically continuous) at a point if, for each , there is a such that , whenever , for every . Moreover, is equicontinuous (spherical continuous) on a subset if it is continuous (spherically continuous) at each point of .
Normal Families of Analytic Functions
Definition 2.1.1 A familiy of analytic functions on a domain is normal in if every sequence of functions contains either a subsequence which converges to a limit function uniformly on each compact subset of , or a subsequence which converges uniformly to on each compact subset.
The family is said to be normal at a point if it is normal in some neighbourhood of .
Theorem 2.1.2 A family of analytic functions is normal in a domain if and only if is normal at each point in .
Theorem 2.2.1 Arzela-Ascoli Theorem. If a sequence of continuous functions converges uniformly on a compact set to a limit function , then is equicontinuous on , and is continuous. Conversely, if is equicontinuous and locally bounded on , then a subsequence can be extracted from which converges locally uniformly in to a (continuous) limit function .
Montel’s Theorem If is a locally bounded family of analytic functions on a domain , then is a normal family in .
Koebe Distortion Theorem Let be analytic univalent in a domain and a compact subset of . Then there exists a constant such that for any , .
Vitali-Porter Theorem Let be a locally bounded sequence of analytic functions in a domain such that exists for each belonging to a set which has an accumulation point in . Then converges uniformly on compact subsets of to an analytic function.
Proof. From Montel’s Theorem, is normal, extract a subsequence which converges normally to an analytic function . Then for each . Suppose, however, that does not converge uniformly on compact subsets of to . Then there exists some , a compact subset , as well as a subsequence and points satisfying . Now itself has a subsequence which converges uniformly on compact subsets to an analytic function , and from above. However, since and must agree at all points of , the Identity Theorem for analytic functions implies on , a contradiction which establishes the theorem.
Fundamental Normality Test Let be the family of analytic functions on a domain which omit two fixed values and in . Then is normal in .
Generalized Normality Test Suppose that is a family of analytic functions in a domain which omit a value and such that no function of assumes the value at more that points. Then is normal in .
Assume is the unit disk in the complex plane, is a region (connected open set) in .
1. in . Then is normal in , but not compact since . In the domain , is normal.
2. is a normal family in but not compact.
3. analytic in and . Then is normal in and compact.
4. analytic in and . Then is normal but not compact. Hint: is a uniformly bounded family.
5. analytic, univalent in , . These are the normalised “Schlicht” functions in . is normal and compact.
Normal Families of Meromorphic Functions
Assume a function is analytic in a neighbourhood of , except perhaps at itself. In other words, shall be analytic in a region . The point is called an isolated singularity of . 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 which is analytic in a region , except for poles, is said to be meromorphic in .
The chordal distance between and is
if and are in the finite plane, and
if . Clearly, , and . 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 converges spherically uniformly to on a set if, for any , there is a number such that implies , for all .
Definition 3.1.1 A family of meromorphic functions in a domain is normal in if every sequence contains a subsequence which converges spherically uniformly on compact subsets of .
Theorem 3.1.3 Let be a sequence of meromorphic functions on a domain . Then converges spherically uniformly on compact subsets of to if and only if about each point there is a closed disk in which or uniformly as .
Corollary 3.1.4 Let be a sequence of meromorphic functions on which converges spherically uniformly on compact subsets to . Then is either a meromorphic function on or identically equal to .
Corollary 3.1.5 Let be a sequence of analytic functions on a domain which converge spherically uniformly on compact subsets of to . Then is either analytic on or identically equal to .
Theorem 3.2.1 A family of meromorphic functions in a domain is normal if and only if is spherically equicontinuous in .
Fundamental Normality Test Let be a family of meromorphic functions on a domain which omit three distinct values . Then is normal in .
Vitali-Porter Theorem Let be a sequence belonging to a spherically equicontinuous family of meromorphic functions such that converges spherically on a point set having an accumulation point in . Then converges spherically uniformly on compact subsets of .
Let be meromorphic on a domain . If is not a pole, the derivative in the spherical metric, called the spherical derivative, is given by . If is a pole of , define .
Marty’s Theorem A family of meromorphic functions on a domain is normal if and only if for each compact subset , there exists a constant such that spherical derivative that is, is locally bounded.