8.4 A Spherical Generalization of Normal Families
Picard’s Great Theorem requires a more profound concept by generalizing normal families in the one-point compactification of .
Definition 8.4.1.
Let be a (not necessarily analytic) complex function sequence on a connected set . If compact, , such that , , , then locally uniformly spherically on .
When the “locally uniform limit” is taken to be , the condition of -closeness is instead replaced by the requirement that the values eventually leave every fixed compact subset of (the given definition is equivalent to: compact, compact, such that , , ). In this way, convergence to infinity is treated symmetrically with convergence to finite values by working in the Riemann sphere , where is simply another accumulation point.
By equipping the extended complex plane with the spherical metric instead of the Euclidean metric, convergence to can be treated like convergence to any finite point. In this setting, is simply another accumulation point, so there is no need to handle it differently from other values.
Let be a sequence. Then we say spherically iff , such that , , where is the spherical distance.
Definition 8.4.2.
A family of meromorphic functions on some is said to be spherically normal iff every sequence has a locally uniformly spherically convergent subsequence on .
Montel’s Theorem for holomorphically normal families in Theorem 5.2.3 can be generalized via the spherical metric by the statement of Marty’s Criterion (Theorem 8.4.1).
Definition 8.4.3 (Spherical Derivative).
Let be an open region or domain. Suppose is meromorphic. Then the spherical derivative of is given by
for and
otherwise.
Proposition 8.4.1.
Any linear fractional transformation is spherically uniformly continuous on .
Proof.
Let , where . Then,
The spherical distance between two points is given by
where joins and . The spherical distance is bounded by the integral over the Euclidean straight line joining and :
Since as and , it is bounded by some constant on . Hence, we have
Hence, , ,
Proposition 8.4.2.
Let be a sequence of holomorphic functions on a domain . If locally uniformly spherically, then is either holomorphic on or identically .
Proof.
A result analogous to Theorem 2.3.5 can be used to show that is spherically continuous. Let be arbitrary.
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If , then by spherical continuity, such that ,
Similarly, such that ,
Hence, we have
By the reverse triangle inequality, we have
By Weierstrass (Theorem 4.1.1), is holomorphic on .
-
Consider . Assume, for the sake of contradiction, is an isolated pole of . Hence, such that is holomorphic on .
Because each is holomorphic on , by the Maximum Modulus Principle (Theorem 3.4.1), ,
By letting , we have
contradicting the assumption that is an isolated pole. Hence, must be an accumulation of values evaluating to . By spherical continuity, such that
Similarly, such that ,
Hence, we have
By the reverse triangle inequality, we have
Hence each is holomorphic on and converges locally uniformly spherically to on . By Weierstrass (Theorem 4.1.1), is holomorphic on and has zeros that accumulate at . By the Identity Theorem, on .
Let be the set of all such that is finite. By the argument above, is open. The complement then consists of all points where . By the argument above, is also open. Since is connected, by Theorem 3.2.9, either or . In the former case, on , and in the latter case, is holomorphic on .
Theorem 8.4.1 (Marty's Criterion).
A family of meromorphic functions on some is spherically normal iff
or the family of spherical derivatives, is locally uniformly bounded in .
Proof.
The condition is equivalent to that of
for all compact , , where depends only on . Under the assumption that this holds, then
where joins and where . Hence, , such that , , and hence is uniformly spherically equicontinuous. Since for any two points by geometry of , is also uniformly spherically bounded (the compactness of ). Then the Arzelà–Ascoli Theorem (Theorem 5.2.2) under the spherical metric gives that is a normal family.
Conversely, assume for the sake of contradiction that is a normal family such that conclusion is not satisfied. Then, compact and a sequence such that the sequence
tends to (specifically, suppose that , ). By normality, we may extract a locally uniformly spherically convergent subsequence . By Theorem 2.3.5 under the spherical metric, the uniform spherical limit of , , is spherically continuous on . For every point , there are two possibilities:
-
If , then by continuity, such that ,
Similarly, such that ,
Hence, we have
By the reverse triangle inequality, we have
Hence, the meromorphy of each is actually holomorphy. By continuity, is locally uniformly bounded on . Hence, locally uniformly converges on . By a result of Weierstrass (Theorem 4.1.1), is holomorphic on and the sequence locally uniformly converges to on .
By holomorphy of on , such that . Uniform convergence of gives the existence of some such that ,
Therefore, is uniformly bounded by
on this compact disk. Hence, ,
-
, then by continuity, such that ,
Similarly, such that ,
Hence, we have
By the reverse triangle inequality, we have
Hence, each is holomorphic on . By continuity, is locally uniformly bounded on . It can also be realized that locally uniformly converges on . By a result of Weierstrass (Theorem 4.1.1), is holomorphic on and the sequence locally uniformly converges to on .
By holomorphy of on , such that . Uniform convergence of gives the existence of some such that ,
Therefore, is uniformly bounded by
on this compact disk. Hence, ,
In essence, for any point , there exists an open disk centered at on which the spherical derivatives are bounded by some constant for . By Heine–Borel (Theorem 1.1.3), there exists a finite collection of disks that cover . Thus, is uniformly bounded on by , where , contradicting the assumption that for all .
Theorem 8.4.2 (Fundamental Normality Test).
Let be a region and suppose that is a family of holomorphic functions on . If there exist two different points such that , then must be a spherically normal family.
Proof.
Map and to by a linear function . Then the family of holomorphic functions
omits and for all .
By Proposition 8.3.1, such that for
as in (8.4.5),
Therefore, if we let , then
Let be arbitrary and let satisfy . By Corollary 8.3.1, the pullback of from to satisfies
. Since ,
Hence, there exist open neighborhoods of respectively on which . Since , by Theorem 1.2.13, such that on . Let , and
Hence, , we have by virtue of (8.4.1),
for any . Now restricting to , we have
For any compact , the collection of open disks
forms an open cover of . Hence, by Heine–Borel (Theorem 1.1.3), it admits a finite subcover
for some . Then is uniformly bounded on by
and is thus locally uniformly bounded on . Marty’s Criterion (Theorem 8.4.1) gives the normality of ; since is linear, it follows that is also normal on .
Corollary 8.4.1 (Montel–Carathéodory).
Let be a region and suppose that is a family of meromorphic functions on . If there exist three different points such that , then must be a spherically normal family.
Proof.
Let be a Möbius transformation mapping to , respectively. Hence, the family of meromorphic functions
omits , , and (and hence each function is holomorphic). By the Fundamental Holomorphic Normality Test (Theorem 8.4.2), is normal.
By Proposition 8.4.1, , such that in ,
Let be any function sequence in and let be locally uniformly convergent to on a compact set . Then such that ,
Therefore, , , we have
Hence, every sequence has a locally uniformly spherically convergent subsequence, and the normality of follows.