APPLICATIONS OF ZALCMAN’S
LEMMA TO Qm -NORMAL FAMILIES
Shahar Nevo
Abstract. A family F of meromorphic functions on a plane domain D is called
quasi-normal on D if each sequence S of functions in F has a subsequence which
converges locally χ-uniformly on D \ E, where E = E(S) is a subset of D having no
accumulation points in D. The notion of quasi-normality generalizes the concept of
normal family, which corresponds to E = ∅.
Chi-Tai Chuang extended the notion of quasi-normal family further in an inductive
fashion. According to Chuang, a family F of meromorphic functions on a plane
domain D is Qm -normal (m = 0, 1, 2, . . . ) if each sequence S of functions in F has
a subsequence which converges locally χ-uniformly on the domain D \ E, where
(m)
(m)
E = E(S) ⊂ D satisfies ED = ∅. (Here ED is the m-th derived set of E in D.)
In particular, a Q0 -normal family is a normal family, and a Q1 -normal family is a
quasi-normal family.
This paper gives generalizations of Zalcman’s Lemma to Qm -normal families,
together with applications of these generalizations – specifically, determining the
degree of normality (m) of families of meromorphic functions obtained as linear
combinations of functions taken from given families.
1. Introduction.
A family F of meromorphic functions on a plane domain D is called quasi-normal
on D if each sequence S of functions in F has a subsequence which converges locally
χ-uniformly on D \ E, where E = E(S) is a subset of D having no accumulation
points in D. The notion of quasi-normality was introduced by Montel in [6]; it
generalizes the concept of normal family, which corresponds to E = ∅.
Chi-Tai Chuang extended the notion of quasi-normal family further in an inductive fashion. According to Chuang, a family F of meromorphic functions on a
plane domain D is Qm -normal (m = 0, 1, 2, . . . ) if each sequence S of functions in
F has a subsequence which converges locally χ-uniformly on the domain D \ E,
(m)
where E = E(S) ⊂ D satisfies ED
(m)
= ∅. (Here ED
is the m-th derived set of
Typeset by AMS-TEX
1
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SHAHAR NEVO
E in D.) In particular, a Q0 -normal family is a normal family, and a Q1 -normal
family is a quasi-normal family. A Qm -normal family is said to be of order at most
(m−1)
ν if E can be taken to satisfy |ED
| ≤ ν. The theory of quasi-normal families, of
Qm -normal families, and of Qm -normal families of finite order is developed in great
detail in Chuang’s monograph [2].
In 1975, Zalcman derived the following characterization of normality [10].
Zalcman’s Lemma. A family F of functions meromorphic (resp., analytic) on
the unit disk ∆ is not normal if and only if there exist
(a) a number 0 < r < 1;
(b) points zn , |zn | < r;
(c) functions fn ∈ F; and
(d) numbers ρn → 0+ ,
such that
fn (zn + ρn ζ) → g(ζ)
locally χ-uniformly (resp., locally uniformly) on C, where g is a nonconstant meromorphic (entire) function on C.
Moreover, g(ζ) can be taken to satisfy the normalization g # (ζ) ≤ g # (0) = 1,
ζ ∈ C.
This lemma has proved to be a powerful tool in complex function theory [1], [11].
In this paper we develop generalizations of Zalcman’s Lemma. First we prove
a local version (Lemma 4.1) of the Lemma for normal families. We then prove
successive extensions of the Lemma to quasi-normal families of finite order (Theorem 4.3), general quasi-normal families (Theorem 4.4), Qm -families of finite order
(Theorem 4.5), and general Qm -families (Theorem 4.6).
These four theorems are very close in their ideas and formulations. We state, for
example,
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
3
Theorem 4.6. Let F be a family of meromorphic functions in a domain D, m ≥ 1
an integer. In order that F not be a Qm -normal family in D, it is necessary and
sufficient that there exist
(m)
(A) a set E ⊂ D satisfying ED
6= ∅, to each point z ∈ E of which corresponds
(B) a sequence of points {ωn,z }∞
n=1 belonging to D such that ωn,z → z;
(C) a sequence ρn,z → 0+ ;
(D) a nonconstant function gz , meromorphic on C; and
(E) a sequence of functions of F, S = {fn }∞
n=1 such that
(F) fn (ωn,z + ρn,z ζ) → gz (ζ) locally χ-uniformly on C (z ∈ E).
All of this is done in Section 4. Before that we introduce some background. In
Section 2 we establish notations and discuss basic notions concerning the geometrical structure underlying the notion of Qm -normal family. In Section 3, we collect
the basic definitions and results which will be required in the sequel. Section 5
deals with applications of the results of Section 4. Given ` families F1 , . . . , F` of
meromorphic functions on a domain D, each of which has its degree of normality,
½
¾
P̀
determine the degree of normality of the family
ci fi : ci ∈ C, fi ∈ Fi .
i=1
Preliminaries.
2.1) Let z0 ∈ C, r > 0.
∆(z0 , r) = {|z − z0 | < r}. ∆ denotes the unit disk, unless otherwise stated.
∆(z0 , r) = {z − z0 | ≤ r}
∆0 (z0 , r) = {0 < |z − z0 | < r.
2.2) Let A ⊂ C, z ∈ C. Then z · A = {z · a : a ∈ A}.
2.3) Let D ⊂ C be a domain and A ⊂ D. We say that A is compactly contained
in D if A ⊂ D.
2.4) Let A1 , A2 , . . . be sets in C. These sets are said to be strongly disjoint if
Ai ∩ Aj = ∅ whenever i 6= j. This definition is also valid for a finite number
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SHAHAR NEVO
of sets.
2.5) Let {gn } be a sequence of meromorphic functions on a domain D ⊂ C.
If {gn } converges uniformly on compact subsets of D to g (where g is a
meromorphic function on D or the constant ∞) with respect to the spherical
metric χ on Ĉ, we say that {gn } converges to g locally χ-uniformly on D
χ
and write gn ⇒ g on D.
In case the functions gn are holomorphic in D, then either the convergence
is locally uniform with respect to the Euclidean metric, in which case the limit
function g is holomorphic on D, or {gn } diverges uniformly to ∞ on compacta. In
this case, write gn ⇒ g on D or gn ⇒ ∞ on D, respectively.
Likewise, if A ⊂ D and {gn } converges uniformly with respect to χ to g (g can
χ
be ∞) on A, we shall write gn → g on A for meromorphic functions, gn , n ≥ 1; and
gn → g on A in the case of holomorphic functions.
When S is a sequence of points in C, we shall sometimes treat S as a point set,
in the context of intersection of sets, union, inclusion and derived sets. In the same
manner, a sequence of meromorphic functions will sometimes be treated as a family
of functions.
Let E be a point set in a domain D ⊂ C. We denote by E 0 the derived set of
E, i.e., the set of all accumulation points of E (see [5], p.120). Set E 00 = (E 0 )0 ,
E 000 = (E 00 )0 , etc. In this way, we define inductively the successive derived sets
of E. Writing for the sake of uniformity E 0 = E (1) , E 00 = E (2) , etc., we have
E (n+1) = (E (n) )0 .
In what follows we define the notion of derived set with respect to a domain D
and to record certain properties of and assertions about these sets.
Definition 2.6. Let D be a domain in the complex plane and E ⊂ D a point
set. The derived set of the first order (order 1) of E with respect to D, denoted
(1)
(1)
by ED , is the set of all accumulation points of E in D, that is, ED = E 0 ∩ D.
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
5
For every m ≥ 2 define the derived set of order m of E with respect to D by
(m)
ED
(m−1) (1)
)D .
= (ED
(0)
(1)
Set ED = E. More generally, we write EA = E 0 ∩ A where
(m)
A is an arbitrary subset of C and define EA
in a similar fashion.
Properties. (cf. [5], p.120)
(m)
= E (m) ∩ D.
(m)
⊂ ED
2.7. For each m ≥ 0, ED
2.8. For each m ≥ 2, ED
(m−1)
.
(m)
2.9. If E1 , E2 ⊂ D and m ≥ 0 is an integer, then (E1 ∪ E2 )D
(m)
=
(m)
(E1 )D ∪ (E2 )D .
2.10. Let {Et }t∈A be a collection of sets, each contained in D, and let m ≥ 0 be
an integer. Then
[
Ã
(m)
(Et )D
t∈A
⊂
[
!(m)
Et
t∈A
.
D
(m)
Lemma 2.11. Let m ≥ 0 be an integer E ⊂ D. If z0 ∈ ED
and ∆0 is a neigh-
(m)
(m)
borhood of z0 such that ∆0 ⊂ D, then (∆0 ∩ E)∆0 6= ∅ and z0 ∈ (∆0 ∩ E)∆0 .
Lemma 2.12. Let D, D0 be domains in C, ϕ : D → D0 a homeomorphism, and
(m)
(m)
m ≥ 0 an integer. Then (ϕ(E))D0 = ϕ((E)D ), for any E ⊂ D.
This lemma follows easily by mathematical induction.
(m)
Lemma 2.13. Let D be a domain E ⊂ D. If |E| = ℵ, then |ED | = ℵ for each
m ≥ 1.
Proof. It is sufficient to prove the lemma for the case m = 1. Define two sets
contained in E :
E1 = {z ∈ E : z is an accumulation point of E},
E2 = E \ E1 .
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SHAHAR NEVO
Each point z ∈ E2 has a positive distance dz from E2 \ {z}. Consider the disk
∆(z, dz /3). The collection {∆(z, dz /3)}z∈E2 is made up of strongly disjoint disks;
(1)
hence E2 is at most enumerable. Therefore, |E1 | = ℵ. By Definition 2.6, E1 ⊂ ED
so the assertion follows.
We mention that we also have the following assertions.
¯ ∞
¯
¯ T (m) ¯
¯
Lemma 2.14. If |E| = ℵ, then ¯
ED ¯¯ = ℵ.
m=0
Lemma 2.15. If E 6= φ, E =
(1)
ED ,
then |E| = ℵ.
3. Background on Qm -normal families of meromorphic functions.
All references in brackets are to [2], from which much of the discussion is taken,
more or less verbatim.
Definition 3.1. [Definition 1.4] Let S = {fn } be a sequence of meromorphic
functions in a domain D. A point z0 of D is called a C0 -point of S, if there is a
disk ∆ = {|z − z0 | < r} contained in D such that the sequence S is uniformly
convergent in ∆ with respect to the spherical distance. The sequence S is said to
be a C0 -sequence in D if each point of D is a C0 -point of S.
Suppose that S is a C0 -sequence in D and let z0 be a point of D. Then, by
hypothesis, the sequence S is uniformly convergent in some disk ∆ = {|z = z0 | < r}
with respect to the spherical distance. Consequently, there exists a function F
defined in ∆ such that fn converges uniformly to F in ∆ with respect to the
spherical distance. The collection of local limit functions of {fn } defines a global
function on D. Thus we have the following theorem.
Theorem 3.2. [Theorem 1.2] Let S = {fn } be a C0 -sequence of meromorphic
functions in a domain D. Then there exists a function F defined in D such that
χ
fn ⇒ F on D.
The nature of the limit function is specified in the following theorem.
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
7
Theorem 3.3. [Theorem 1.3] Let S = {fn } be a C0 -sequence of meromorphic
functions in a domain D. Then the limit function F of S is either a meromorphic
function in D or the constant ∞.
Definition 3.4. [Definition 1.5] Let F be a family of meromorphic functions in
a domain D. We say that the family F is normal in D if from every sequence of
∞
functions {fn }∞
1 of the family F, we can extract a subsequence {fnk }k=1 which is
a C0 -sequence in D.
For details on the highly developed theory of normal families, the reader is
referred to [7] and [8].
We turn now to the theory of quasi-normal families of meromorphic functions.
Definition 3.5. [Definition 5.1] Let S = {fn } be a sequence of meromorphic
functions in a domain D. A point z0 of D is called a C1 -point of S if there is
a disk ∆ = {|z − z0 | < r} contained in D such that each point of the domain
∆0 = {0 < |z − z0 | < r} is a C0 -point of S. S is called a C1 -sequence in D if each
point of D is a C1 -point of S.
According to this definition, if a point z0 of D is a C0 -point of S, then z0 is also
a C1 -point of S. Consequently, if S is a C0 -sequence in D, then S is also a C1 sequence in D. Now assume that S is a C1 -sequence in D, but not a C0 -sequence
in D. Then there exists at least one point z0 of D which is not a C0 -point of S.
We call such a point z0 a non C0 -point of S. Denote by E the set of non C0 -points
of S in D. E has no accumulation point in D. In the domain D1 = D \ E, S is a
χ
C0 -sequence. Hence, by Theorems 3.2 and 3.3, fn ⇒ F on D1 , where F (z) is a
n→∞
meromorphic function on D or the constant ∞.
In particular, if {fn } is a C1 -sequence of holomorphic functions in a domain D,
χ
but is not a C0 -sequence in D, then fn ⇒ F on D1 = D \ E, where F is either
a holomorphic function or the constant ∞. However, the first alternative cannot
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SHAHAR NEVO
occur, because otherwise, by the maximum modulus principle, the points of E
would be C0 -points of the sequence {fn }. Thus, the second alternative must hold.
Definition 3.6. [Definition 5.2] Let F be a family of meromorphic functions in a
domain D. We say that F is quasi-normal in D, if from every sequence of functions
{fn } of F we can extract a subsequence {fn } which is a C1 -sequence in D. F is
said to be quasi-normal at a point z0 ∈ D if there is a disk ∆(z0 , r) belonging to D
such that F is quasi-normal in ∆(z0 , r).
Theorem 3.7. [Theorem 5.1] Let F be a family of meromorphic functions in a
domain D. If F is quasi-normal at each point of D, then F is quasi-normal in D.
Note that if a family F of meromorphic functions in a domain D is normal in
D, then it is also quasi-normal in D.
Definition 3.8. [Definition 5.3] Let F be a family of meromorphic functions in a
domain D and ν ≥ 0 an integer. We say that F is quasi-normal of order at most
ν in D, if from every sequence of functions {fn } belonging to F, we can extract
a subsequence {fnk } which is a C1 -sequence in D, and the number of whose non
C0 -points does not exceed ν. In particular, when ν = 0, the family F is normal in
D. If ν ≥ 1 and F is quasi-normal of order at most ν in D, but is not quasi-normal
of order at most ν − 1, we say that F is quasi-normal of exact order ν in D.
Theorem 3.9. [Theorem 5.4] Let D be a domain, 0 < δ < 1 a number, and pj ≥ 0
(j = 1, 2, 3) three integers such that p1 ≤ p2 ≤ p3 . Let F be the family of functions
satisfying the conditions
(1) f is meromorphic in D; and
(2) there exist three values aj (f ) ∈ Ĉ (j = 1, 2, 3) such that
χ(aj (f ), aj 0 (f )) > δ
(j 6= j 0 )
and for each 1 ≤ j ≤ 3, the equation f (z) = aj (f ) has at most pj distinct
roots in D.
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
9
Then the family F is quasi-normal of order at most p2 in D.
Definition 3.10. [Definition 5.6] Let S = {fn } be a C1 -sequence of meromorphic
functions in a domain D and z0 a non C0 -point of S in D. Then two cases are
possible:
(1) We can extract from the sequence S a subsequence S 0 of which z0 is a
C0 -point. In this case, we say that S is reducible with respect to the point
z0 .
(2) z0 is a non C0 -point of every subsequence of the sequence S. In this case, we
say that S is irreducible with respect to the point z0 . The sequence is said
to be an irreducible sequence in D if S has non C0 -points in D with respect
to which S is irreducible.
We shall use the notions of reducible or irreducible sequence with respect to a
given point even in the case in which the sequence is not necessarily a C1 -sequence
in D.
Lemma 3.11. [Lemma 5.13] Let S = {fn } be a C0 -sequence of meromorphic functions in a domain D whose limit function f is a meromorphic function in D. Let E
be a compact subset of D. Assume that f is finite on E. Then there exists a positive
integer N such that fn (z) is finite on E for n ≥ N, and fn → f on E.
n→∞
Lemma 3.12. [Lemma 5.14] Let S = {fn } be a sequence of meromorphic functions
in the disk ∆ = {|z − z0 | < r} satisfying the following conditions.
(1) S is a C0 -sequence in the domain D = {0 < |z − z0 | < r}.
(2) The limit function f of S in D is a non-constant meromorphic function.
(3) There exists a value w0 ∈ Ĉ and positive integers p and N such that, when
n ≥ N, the equation fn (z) = w0 has at most p roots in ∆ (with due count
of order of multiplicity).
Then the function f is meromorphic in ∆, provided that f (z0 ) is suitably defined.
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SHAHAR NEVO
Lemma 3.13. [Lemma 6.1] Let f1 and f2 be two meromorphic functions in a
domain D. Then the function ϕ(z) = χ(f1 (z), f2 (z)) is continuous in D.
Lemma 3.14. [Lemma 7.1] Let {fn } be a C0 -sequence of meromorphic functions
in the domain D and {an } a sequence of holomorphic functions in D, converging
locally uniformly to a holomorphic function in D. Then the sequence of functions
gn = fn + an (n = 1, 2, . . . ) is a C0 -sequence in D.
We have defined the notions of C0 -point, C0 -sequence and normal family of
meromorphic functions (Definitions 3.1 and 3.4) and also the notions of C1 -point,
C1 -sequence, and quasi-normal family of meromorphic functions in a domain D
(Definitions 3.5 and 3.6).
Denote by E the set of non C0 -points of S in D. Then E = ∅ if S is a C0 -sequence
(1)
in D, and ED = ∅ if S is a C1 -sequence in D (cf. Definition 2.6).
It is natural to go still further and introduce notions corresponding, respectively,
(2)
(3)
to the conditions ED = ∅, ED = ∅, . . . . In what follows we shall define, for each
integer m ≥ 0, the notions of Cm -point, Cm -sequence and Qm -normal family of
(m)
meromorphic functions, corresponding to the condition ED
= ∅. In particular,
a Q0 -normal family of meromorphic functions is a normal family of meromorphic
functions and a Q1 -normal family of meromorphic functions is a quasi-normal family
of meromorphic functions.
Definition 3.15. [Definition 8.1] (Cm -point, m ≥ 2) Let m ≥ 2 be an integer. Let
S = {fn } a sequence of meromorphic functions in the domain D and z0 a point of
D. We say that z0 is a Cm -point of S if there is a disk {|z − z0 | < r} contained in
D such that each point of the domain {0 < |z − z0 | < r} is a Cm−1 -point of S.
Combining this definition with Definitions 3.1 and 3.5 completes the definition
of Cm -point for each m ≥ 0.
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
11
Lemma 3.16. [Lemma 8.1] If for some integer m ≥ 0, z0 is a Cm -point of S,
then the following assertions hold:
(1) z0 is a Cm+1 -point of S;
(2) there is a disk {|z − z0 | < r} contained in D, each point of which is a
Cm -point of S; and
(3) z0 is a Cm -point of every subsequence of S.
This is easily proved by induction.
Definition 3.17. [Definition 8.2] Let m ≥ 0 be an integer. A sequence S of
meromorphic functions in a domain D is a Cm -sequence in D, if each point of D is
a Cm -point of S.
Lemma 3.18. [Lemma 8.2] If, for some integer m ≥ 0, S is a Cm -sequence in D,
then S is a Cm+1 -sequence in D, and every subsequence of S is a Cm -sequence in
D.
This follows immediately from Lemma 3.16.
Definition 3.19. [Definition 8.3] Let S be a sequence of meromorphic functions
in a domain D and m ≥ 0 an integer. A point z0 of D is said to be a non Cm -point
of S, if z0 is not a Cm -point of S.
Theorem 3.20. [Theorem 8.1] Let S be a sequence of meromorphic functions in
the domain D. If, for some integer m ≥ 0, S is a Cm -sequence in D, then the set
E of non C0 -points of S in D is at most enumerable.
Definition 3.21. [Definition 8.4] Let E be a set of points in a domain D. If,
(j)
for some integer m ≥ 0, the sets ED j = 0, 1, . . . , m (see Definition 2.6) are all
nonempty, we say that the set E has property Wm with respect to D.
(m)
Note that it is enough to demand ED
follows.
6= ∅. Theorem 3.20 can be sharpened as
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SHAHAR NEVO
Theorem 3.22. [Theorem 8.2] Let S be a sequence of meromorphic functions in
a domain D, and m ≥ 0 as integer. In order for S to be a Cm -sequence in D, it is
necessary and sufficient that the set of non C0 -points of S in D not have property
Wm with respect to D.
We can also characterize non Cm -points by the following remark.
Remark 3.23. Let S = {fn } be a sequence of meromorphic functions in a domain
D, z0 ∈ D, and let m ≥ 0 be an integer. Then z0 is a non Cm -point of S if and
(m)
only if z0 ∈ ED , where E is the set of the non C0 -points of S in D.
The proof of this remark follows at once by mathematical induction.
Definition 3.24. [Definition 8.5] Let F be a family of meromorphic functions in
a domain D and m ≥ 0 an integer. We say that the family F is Qm -normal in D,
if from every sequence of functions of the family F, we can extract a subsequence
which is a Cm -sequence in D. F is said to be Qm -normal at a point z0 of D if
there is a disk ∆(z0 , r) contained in D such that F is Qm -normal in ∆(z0 , r). In
particular, a Q0 -normal family is a normal family and a Q1 -normal family is a
quasi-normal family, respectively, according to Definitions 3.4 and 3.6.
Evidently, according to Definition 3.24, if F is Qm -normal in D, then F is Qm normal at each point of D. Conversely, we have the following theorem.
Theorem 3.25. [Theorem 8.3] Let F be a family of meromorphic functions in a
domain D and m ≥ 0 an integer. If F is a Qm -normal at each point of D, then F
is Qm -normal in D.
Lemma 3.26. [Lemma 8.3] If, for an integer m ≥ 0, a family F of meromorphic
functions in a domain D is Qm -normal in D, then F is Qm+1 -normal in D.
This is an immediate consequence of Lemma 3.18.
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
13
Necessary and sufficient conditions for Qm -normality.
Definition 3.27. [Definition 8.6] Let S = {fn } be a sequence of meromorphic
functions in a domain D and z0 a point of D. We say that z0 is a µ1 -point of S, if
for each closed disk
∆ = {|z −z0 | ≤ r} contained in D, we have lim max fn# (z) =
n→∞ z∈∆
+∞. We say that z0 is a µ2 -point of S, if for each disk ∆ = {|z − z0 | < r} contained
in D, S has a µ1 -point z 0 in the domain {0 < |z −z0 | < r}. In general, if m ≥ 2 is an
integer, z0 is called a µm -point of S, if for each disk ∆ = {|z − z0 | < r} contained in
D, S has a µm−1 -point in the domain {0 < |z − z0 | < r}. Let S 0 be a subsequence
of S. Obviously, if z0 is a µ1 -point of S, then it is a µ1 -point of S 0 . By induction,
we see that, in general, if z0 is a µm -point of S, then it is a µm -point of S 0 .
Theorem 3.28. [Theorem 8.6] Let F be a family of meromorphic functions in the
domain D. For F to be Q0 -normal (or normal) in D, it is necessary and sufficient
that each sequence S = {fn } of functions of F have no µ1 -point in D.
The notion of µm -point is closely connected to the notion of Qm -normality of
finite order.
Definition 3.29. [Definition 8.7] Let m ≥ 1 be an integer. Let F be a family
of meromorphic functions in a domain D and ν ≥ 0 an integer. We say that F
is Qm -normal of order at most ν in D if from every sequence of functions of the
family F we can extract a subsequence which is a Cm -sequence in D and has at
most ν non Cm−1 -points in D. In particular, when ν = 0, F is Qm−1 -normal in D.
If F is a Qm -normal family of order at most ν ≥ 1 in D but not a Qm -normal
family of order at most ν − 1 in D, we say that F is a Qm -normal family of exact
order ν on D. If F is a Qm -normal family but not a Qm -normal family of order at
most ν for any ν ≥ 1, we say that F is a Qm -normal family of infinite order in D.
Theorem 3.30. [Theorem 8.9] Let m ≥ 1 be an integer. Let F be a family of
meromorphic functions in the domain D and ν ≥ 0 an integer. For F to be Qm -
14
SHAHAR NEVO
normal of order at most ν in D, it is necessary and sufficient that each sequence
S = {fn } of functions of F have at most ν µm -points in D.
Theorem 3.31. [Theorem 8.10] Let m ≥ 1 be an integer. Let F be a family
of meromorphic functions in the domain D. For F to be Qm -normal in D, it is
necessary and sufficient that each sequence S = {fn } of functions of F have no
µm+1 -points in D.
The µm -points can also be characterized by the following remark.
Remark 3.32. Let S = {fn } be a sequence of meromorphic functions in the
domain D and m ≥ 0 an integer. Then z0 ∈ D is a µm+1 -point of S if and only if
z0 is a non Cm -point of any subsequence of S. (That is, S is irreducible with respect
to z0 ; cf. Definition 3.33).
Proof of Remark 3.32.
For m = 0, Marty’s Theorem and the definition of µ1 -point easily yield the result.
Assuming the correctness of the claim for k = m, let us prove it now for k = m + 1.
Suppose z0 is a µm+2 -point of S. If z0 is a Cm+1 -point of some subsequence of S,
0
S 0 = {fnk }∞
k=1 , then by Definition 3.15 there exists r > 0 such that ∆ (z0 , r) ⊂ D
consists only of Cm -points of S 0 . Hence, according to the induction assumption for
k = m, S has no µm+1 -points in ∆0 (z0 , r); and, by Definition 3.27, z0 is not a
µm+2 -point of S, a contradiction.
Conversely, suppose that z0 is a non Cm+1 -point of any subsequence of S. Considering S as a family of functions, we deduce from Definition 3.24 that S is not a
Qm+1 -normal family in any neighborhood of z0 . Hence, taking a sequence rk → 0+ ,
we get by Theorem 3.31 that for each k, ∆(z0 , rk ) contains µm+2 -points of S,
so z0 is either an accumulation point of µm+2 -points of S (in which case, z0 is
a µm+3 -point of S) or a µm+2 -point. In either case, z0 is a µm+2 -point of S, as
required.
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
15
The following definition will be used in building Qm -normal families of certain
types.
Definition 3.33. [Definition 8.8] Consider a domain D. Let σ(1, n) = {zj : 1 ≤
j ≤ n} be a system of points of D. We say that σ(1, n) satisfies condition (Γ1,n )
with respect to D if either n = 1 or n > 1 and the n points zj (1 ≤ j ≤ n) are
distinct. Let σ(1) = {zj : j ≥ 1} be a sequence of points of D. We say that σ(1)
satisfies condition (Γ01 ) with respect to D if the points zj (j ≥ 1) are distinct and
the limit lim zj = λ exists, with λ = λσ(1) ∈ D.
j→∞
Let σ(2, n) = {zij : 1 ≤ i ≤ n, j ≥ 1} be a double sequence of points of D. We
say that σ(2, n) satisfies condition (Γ2,n ) with respect to D if for each i (1 ≤ i ≤ n),
the sequence σi (1) = {zij : j ≥ 1} satisfies condition (Γ01 ) with respect to D and
the system σ(1, n) = {λi = λ(σi (1)) : 1 ≤ i ≤ n} satisfies condition (Γ1,n ) with
respect to D. Let σ(2) = {zij : 1 ≥ 1, j ≥ 1} be a double sequence of points of D.
We say that σ(2) satisfies condition (Γ02 ) with respect to D if for each i (i ≥ 1), the
sequence σi (1) = {zij : j ≥ 1} satisfies condition (Γ01 ) with respect to D and the
sequence σ(1) = {λi = λ(σi (1)) : i ≥ 1} satisfies condition (Γ01 ) with respect to D.
In general, let σ(m, n) = {zj1 j2 ,...,jm : 1 ≤ j1 ≤ n, j2 ≥ 1, . . . , jm ≥ 1} be an
m-ple (m ≥ 3) sequence of points of D. We say that σ(m, n) satisfies condition
(Γm,n ) with respect to D, if for each choice of indices j1 , j2 , . . . , jm−1 (1 ≤ j1 ≤ n,
j2 ≥ 1, . . . , jm−1 ≥ 1), the sequence σj1 ,j2 ,...,jm−1 (1) = {zj1 ,j2 ,...,jm−1 ,jm : jm ≥ 1}
satisfies condition Γ01 with respect to D, and the (m−1)-ple sequence λj1 ,j2 ,...,jm−1 =
λ(σj1 ,j2 ,...,jm−1 (1)) (1 ≤ j1 ≤ n, j2 ≥ 1, . . . , jm−1 ≥ 1) satisfies condition (Γm−1,n )
with respect to D. Let σ(m) = {zj1 ,j2 ,...,jm : j1 ≥ 1, j2 ≥ 1, . . . , jm ≥ 1} be
an m-ple (m ≥ 3) sequence of points of D. We say that σ(m) satisfies condition
(Γ0m ) with respect to D, if for each choice of indices j1 , j2 , . . . , jm−1
(j1 ≥ 1,
j2 ≥ 1, . . . , jm−1 ≥ 1), the sequence {σj1 ,j2 ,...,jm−1 ,jm (1) : jm ≥ 1} satisfies condition (Γ01 ) with respect to D, and the (m − 1)-ple sequence
16
SHAHAR NEVO
λj1 ,j2 ,...,jm−1 = λ(σj1 ,j2 ,...,jm−1 (1)) (j1 ≥ 1, j2 ≥ 1, . . . , jm−1 ≥ 1) satisfies condition (Γ0m−1 ) with respect to D.
Definition 3.34. [Definition 8.14] Let m ≥ 1 be an integer. Let S = {fn } be a
Cm -sequence of meromorphic functions in a domain D. Assume that S has a non
Cm−1 -point z0 in D. Then two cases are possible:
(1) We can extract from the sequence S a subsequence S 0 of which z0 is a
Cm−1 -point. In this case, we say that S is reducible with respect to the point
z0 .
(2) z0 is a non Cm−1 -point of every subsequence of S. In this case, we say that
S is irreducible with respect to the point z0 . The sequence S is said to be an
irreducible Cm -sequence in D if S has non Cm−1 -points in D, with respect
to which S is irreducible.
Lemma 3.35. [Lemma 8.7] Let m ≥ 1 be an integer. Let S = {fn } be a Cm sequence of meromorphic functions in a domain D. Assume that S has non Cm−1 points in D. Then we can extract from S a subsequence S 0 which is either a Cm−1 sequence in D or an irreducible Cm -sequence in D.
Other criteria of Qm -normality.
Lemma 3.36. (see [2], pp.341-342) Let m ≥ 1 be an integer and let E be a
nonempty set of a given domain D. Assume that E does not have property Wm
(1)
with respect to D. Then D \ E is a domain if and only if ED ⊂ E.
Definition 3.37. [Definition 9.1] Let E be a set of points in the domain D, and
m ≥ 0, ν > 0 integers. We say that E has property Wm,ν with respect to D, if the
(m)
set ED
(m)
consists of at least ν points. If |ED | = ν, we say that E has property
Wm,ν exactly with respect to D.
We shall also need the following.
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
17
Definition 3.38. [Definition 9.3] Let En (n = 1, 2, . . . ) and E be sets of points in
the complex plane C. We say that E is a limit set of the sequence En (n = 1, 2, . . . )
if for any point z0 ∈ E, any positive number ε, and any positive integer N, we can
find an integer n ≥ N such that the set En ∩ {|z − z0 | < ε} is nonempty.
In view of Definition 3.38, we introduce the following modification to P.2.9.
Remark 3.39. Let D be a domain, and S = {An }∞
n=1 a sequence of sets contained
in D such that S has no nonempty limit set in D.
µ∞
¶(m)
∞
S
S
(m)
Then
An
=
(An )D .
n=1
n=1
D
For the proof we need the following lemma.
Lemma 3.40. Let {An }∞
n=1 be a sequence of sets belonging to a domain D. Then
{An } has a nonempty limit set E, if and only if there exists a sequence of points
{ak }∞
k=1 , ak ∈ Ank (n1 < n2 < . . . ), such that ak → a0 for some a0 ∈ D.
k→∞
The proof of this lemma is obvious.
Proof of Remark 3.39. By the inductive character of Definition 2.6, it is enough to
∞
S
prove the remark for m = 1. Let E =
An . We have to show that
(A)
∞
S
n=1
n=1
(1)
(1)
(An )D ⊆ ED .
(1)
(B) ED ⊆
∞
S
(1)
(An )D .
n=1
(1)
(A) is obvious. To prove (B) let us suppose that z0 ∈ ED . If the assertion is
false, then for every δ > 0 and every n ≥ 1 there exists at most a finite number
(1)
of points of An in ∆(z0 , δ). Therefore, since z0 ∈ ED , there exists a sequence of
1
distinct points in E, {ank }∞
k=1 , n1 < n2 < . . . , ank ∈ Ank , satisfying ank ∈ ∆(z0 , k ).
So ank → z0 , and by the lemma we deduce that S has a nonempty limit set in
k→∞
D. We have reached a contradiction.
18
SHAHAR NEVO
Lemma 3.41. [Lemma 9.18] Given integers m ≥ 1 and n ≥ 1, there exists an
m-ple sequence of points
σ(m) = {xj1 ,j2 ,...,jm : 1 ≤ j1 ≤ n, j2 ≥ 1, . . . , jm ≥ 1}
in the interval I : 0 < x < 1, satisfying condition (Γm,n ) with respect to I, and an
m-ple sequence of points
σ1 (m) = {xj1 ,j2 ,...,jm : j1 ≥ 1, j2 ≥ 1, . . . , jm ≥ 1}
satisfying condition (Γ0m ) with respect to I (see Definition 3.33).
4. Extensions of Zalcman’s Lemma.
In this section, we extend Zalcman’s Lemma to quasi-normal familes of finite
order (Theorem 4.3), general quasi-normal families (Theorem 4.4), Qm -families of
finite order (Theorem 4.5) and general Qm -families (Theorem 4.6).
We begin with a local version of Zalcman’s Lemma [9] for normal families. Our
proof takes the result of the original lemma as given.
Lemma 4.1. A family F of meromorphic (holomorphic) functions defined on a
domain containing z0 ∈ C is not normal at z0 if and only if there exist
(a) points zn , zn → z0 ;
n→∞
(b) functions fn ∈ F;
(c) numbers ρn → 0+ ; and
(d) a nonconstant meromorphic (entire) function g on C such that
(4.1)
χ
fn (zn + ρn ζ) ⇒ g(ζ)
on
C.
The function g may be taken to satisfy the normalization
g # (ζ) ≤ g # (0) = 1,
ζ ∈ C.
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
19
Proof. Without loss of generality, we may assume that z0 = 0. Assume that the
conditions of the theorem are satisfied and that F is normal at z0 . Then by Marty’s
theorem,
f # (z) ≤ M
(4.2)
in {|z| ≤ ρ} for some ρ > 0, for each f ∈ F. From (4.1) it follows that
ρn fn# (zn + ρn ζ) → g # (ζ) and by (4.2) g # (ζ) = 0 for each ζ ∈ C, so g is constant,
n→∞
a contradiction.
Conversely, suppose that F is not normal at 0. The members of F are defined
in {|z| ≤ ρ} for some ρ > 0. Let k0 ∈ N be such that
√1
k0
< ρ; then for each k ≥ k0
there is fk ∈ F with
½
(4.3)
sup
fk# (z)
¶¾
µ
1
≥ k.
: z ∈ ∆ 0, √
2 k
³ ´
For k ≥ k0 , set gk (z) = fk √zk . Each gk is defined on ∆ and satisfies gk# (z) =
³ ´
n
√
¡ 1 ¢o
#
#
√1 f
√z
there.
By
(4.3)
we
have
sup
g
(z)
:
z
∈
∆
0,
≥
k; so, again
k
k
2
k
k
by Marty’s theorem, {gk }∞
k=1 regarded as a family is not normal in ∆. Now, by
the original version of Zalcman’s Lemma there exist: (a) a number 0 < r < 1;
(b) points z`∗ , |z`∗ | < r; (c) numbers ρ∗` → 0+ ; and (d) functions gke ∈ {gk } such
χ
that gk` (z`∗ + ρ∗` ζ) ⇒ g(ζ) on C, where g # (ζ) ≤ g # (0) = 1, ζ ∈ C. But this means
´ χ
³ ∗
ρ∗
z∗
ρ∗
z
fk` √k` + √k` ζ ⇒ g(ζ) on C. Setting z` = √k` , ρ` = √k` completes the proof.
`
`
`
`
Before turning to extensions of Zalcman’s Lemma for (non) Qm -normal families,
we introduce the following notation.
Definition 4.2. Let S = {fn } be a sequence of meromorphic functions on a domain
D and let E ⊂ D. E and S are said to satisfy Zalcman’s condition with respect to
D if for any ζ ∈ E there exist zn,ζ → ζ, zn,ζ ∈ D; ρn,ζ → 0+ such that
χ
fn (zn,ζ + ρn,ζ z) ⇒ gζ (z)
on
where g is a nonconstant meromorphic function on C.
C,
20
SHAHAR NEVO
Theorem 4.3. Let F be a family of meromorphic functions in a domain D and
let p ≥ 0 be an integer. In order that F not be a Q1 -normal family of order at
most p in D (see Definition 3.8), it is necessary and sufficient that there exist a
set E = {b1 , . . . , bp+1 } of p + 1 distinct points b1 , b2 , . . . , bp+1 belonging to D, and a
sequence S of D, such that E and S satisfy Zalcman’s condition with respect to D.
Proof. Assume first that such E and S do exist. By Definition 3.8, it is enough to
show that there exists a sequence of functions of F, every subsequence of which has
at least p + 1 non C0 -points. We take the sequence to be S. According to Definition
3.10, it will be enough to show that for each 1 ≤ j ≤ p + 1, S is irreducible with
respect to the point bj . But this is obvious in view of Lemma 4.1, regarding S as a
family of functions.
Conversely, suppose that F is not a Q1 -normal family of order at most p in D.
Then there is a sequence S0 = {hn }∞
1 ⊂ F, such that every subsequence of S has
at least p + 1 non C0 -points in D.
Step 1. Evidently, as a family, S0 is not normal in D. Because normality is a local
property, there exist
(A1 ) b1 ∈ D such that S0 is not normal at b1 . By Lemma 4.1 there exist
(B1 ) a sequence wn,1 → b1 ;
n→∞
(C1 ) a sequence δn,1 → 0+ ;
n→∞
(D1 ) a nonconstant meromorophic function g1 on C (which can be taken to
satisfy g1# (z) ≤ g1# (0) = 1, z ∈ C);
(E1 ) a subsequence S1 = {hn,1 }∞
n=1 of S0 such that
χ
(F1 ) hn,1 (wn,1 + δn,1 ζ) ⇒ g1 (ζ) on C.
n→∞
Step 2. S1 , as a family, is not Q1 -normal of order at most p − 1 in the domain
D1 = D \ {b1 }. For otherwise, S1 would have a subsequence S ∗ with at most
p − 1 non C0 -points in D; and then S ∗ would have at most p non C0 -points in D,
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
21
contradicting the property of S0 .
As in the first step, we conclude the existence of
(A2 ) b2 ∈ D1 such that S1 is not normal at b2 and
(B2 ) wn,2 → b2 ;
(C2 ) δn,2 → 0+ ;
(D2 ) a nonconstant function g2 , meromorphic on C; and
(E2 ) a subsequence S2 = {hn,2 }∞
n=1 of S1 such that
χ
(F2 ) hn,2 (wn,2 + δn,2 ζ) ⇒ g2 (ζ) on C.
n→∞
Continuing this way, we get for each 1 ≤ i ≤ p :
(Ai ) bi ∈ Di−1 = D \ {b1 , . . . , bi−1 } (D0 = D) such that Si−1 is not normal at bi
and
(Bi ) wn,i → bi ;
n→∞
(Ci ) δn,i → 0+ ;
n→∞
(Di ) a nonconstant function gi , meromorphic on C;
(Ei ) a subsequence Si = {hn,i }∞
n=1 of Si−1 such that
χ
(Fi ) hn,i (wn,i + δn,i ζ) ⇒ gi (ζ) on C.
n→∞
Now, Sp as a family is not normal (or Q1 -normal of order 0) in the domain
Dp = D \ {b1 , . . . , bp }, since otherwise Sp would have a subsequence S 0 which is
a C0 -sequence in Dp and hence S 0 would have at most p non C0 -points in D,
contradicting the property of S0 . Hence, as in the previous steps, there exist:
(Ap+1 ) bp+1 ∈ Dp such that Sp is not normal at bp+1 and
(Bp+1 ) wn,p+1 → bp+1 ;
n→∞
+
(Cp+1 ) δn,p+1 → 0 ;
(Dp+1 ) a nonconstant function gp+1 , meromorphic on C; and
(Ep+1 ) a subsequence Sp+1 = {hn,p+1 }∞
n=1 of Sp , such that
χ
(Fp+1 ) hn,p+1 (wn,p+1 + δn,p+1 ζ) ⇒ gp+1 (ζ) on C.
n→∞
22
SHAHAR NEVO
For each 1 ≤ j ≤ p + 1, Sp+1 is a subsequence of Sj . It then follows that
E = {b1 , . . . , bp+1 } and S = Sp+1 satisfy Zalcman’s condition with respect to D,
and the proof is completed.
Theorem 4.4. Let F be a family of meromorphic functions in a domain D. In
order that F not be a Q1 -normal family in D, it is necessary and sufficient that
there exist a set E ⊂ D having property W1 with respect to D (see Definition 3.21)
and a sequence S of F, such that E and S satisfy Zalcman’s condition with respect
to D.
(1)
Proof. We first prove the sufficiency of Zalcman’s condition. Indeed, let b0 ∈ ED .
There is a sequence {bj } of E, bj → b0 . By Lemma 4.1 and Definition 4.2, each bj
is a non C0 -point of any subsequence of S (see Definition 3.5), but this makes b0 a
non C0 -point of any subsequence of S, so F cannot be a Qm -normal family on D.
Conversely, suppose that F is not a Q1 -normal family in D. Then, by Theorem
3.31, there exists a sequence S0 = {hn }∞
n=1 ⊂ F which has a µ2 -point, b0 ∈ D
(see Definition 3.27). As mentioned in Definition 3.27, b0 is a µ2 -point of any
subsequence of S0 . We shall construct the sequence S inductively.
Step 1. There exists
(A1 ) b1 ∈ D, |b1 − b0 | < 1, a µ1 -point of S0 . Now S0 is not normal (as a family)
at b1 ; therefore, by Lemma 4.1, there exist
(B1 ) a sequence {wn,1 }∞
n=1 , wn,1 → b1 ;
(C1 ) a sequence of numbers
n→∞
{δn,1 }∞
n=1
δn,1 → 0+ ;
(D1 ) a nonconstant function g1 , meromorphic on C; and
(E1 ) a subsequence S1 of S0 , S1 = {hn,1 }∞
n=1 , such that
χ
(F1 ) hn,1 (wn,1 + δn,1 ζ) ⇒ g1 (ζ) on C.
n→∞
Step 2. Because b0 is also a µ2 -point of S1 , there exists
(A2 ) b2 ∈ D, a µ1 -point of S1 , b2 6= b1 , |b2 − b0 | ≤ 21 . Now S1 is not normal (as a
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
23
family) at b2 ; therefore, by Lemma 4.1, there exist
(B2 ) wn,2 → b2 ;
n→∞
(C2 ) δn,2 → 0+ ;
n→∞
(D2 ) a nonconstant function g2 , meromorphic on C; and
(E2 ) a subsequence S2 of S1 , S2 = {hn,2 }∞
n=1 , such that
χ
(F2 ) hn,2 (wn,2 + δn,2 ζ) ⇒ g2 (ζ) on C.
n→∞
Continuing in this way, let us suppose that the j-th step has been executed.
Step j + 1. Since b0 is also a µ2 -point of Sj , there exists
(Aj+1 ) bj+1 ∈ D, a µ1 -point of Sj , bj+1 ∈
/ {b1 , . . . , bj }, |bj+1 − b0 | <
1
j+1 .
Now Sj
is not normal (as a family) at bj+1 ; therefore, by Lemma 4.1, there exist
(Bj+1 ) wn,j+1 → bj+1 ;
n→∞
+
(Cj+1 ) δn,j+1 → 0 ;
(Dj+1 ) a nonconstant function gj+1 , meromorphic on C; and
(Ej+1 ) a subsequence Sj+1 of Sj , Sj+1 = {hn,j+1 }∞
n=1 , such that
χ
(Fj+1 ) hn,j+1 (wn,j+1 + δn,j+1 ζ) ⇒ gj+1 (ζ) on C.
n→∞
∞
∞
In this way, we create sequences {bj }∞
j=1 , bj → b0 ; {wn,j }n=1 , j ≥ 1; {δn,j }n=1 ,
j→∞
j ≥ 1; {gj }; and
{hn,j }∞
n=1 ,
j ≥ 1. Now let fn = hn,n , and set S = {fn }∞
n=1 . Then
for each j ≥ 1, Ŝj = {fn }∞
n=j is a subsequence of Sj . If we now set E = {Sj ; j ≥ 1}.
(Evidently, E has property W1 with respect to D), then it is obvious that E and
S satisfying Zalcman’s condition with respect to D, and the theorem is proven.
More generally, we have the following two generalizations of Theorem 4.3 and
Theorem 4.4.
Theorem 4.5. Let F be a family of meromorphic functions in a domain D. In
order that F not be a Qm -normal family of order at most ν in D (see Definition
3.29), it is necessary and sufficient that there exist a set E ⊂ D having property
24
SHAHAR NEVO
Wm−1,ν+1 with respect to D (see Definition 3.35) and a sequence S of F such that
E and S satisfy Zalcman’s condition with respect to D.
The case m = 1 gives Theorem 4.3.
Theorem 4.6. Let F be a family of meromorphic functions in a domain D, m ≥ 1
an integer. In order that F not be a Qm -normal family in D, (see Definition 3.24),
it is necessary and sufficient that there exist a set E ⊂ D having property Wm with
respect to D (see Definition 3.21) and a sequence S of F, such that E and S satisfy
Zalcman’s condition with respect to D.
The case m = 1 gives Theorem 4.4.
Proof of Theorem 4.5. For the purpose of the proof, we shall assume that Theorem
4.6 holds (and prove it later). Suppose first that such E and S do exist. Let
(m−1)
b1 , b2 , . . . , bν+1 be distinct points of ED
. We assert that for each 1 ≤ j ≤ ν + 1,
bj is a non Cm−1 -point of any subsequence of S. (That is, S is irreducible with
respect to bj ; see Definition 3.33). This will prove that F is not a Qm -normal
family of order at most ν in D.
Let S ∗ = {fn` }∞
`=1 be a subsequence of S. Define for each 1 ≤ j ≤ ν + 1
a disk ∆(bj , rj ) compactly contained in D. By Lemma 2.11, E ∩ ∆(bj , rj ) has
property Wm−1 with respect to ∆(bj , rj ), and hence also with respect to D. By the
∞
assumptions, for any z ∈ E ∩ ∆(bj , rj ), there exist {ωn` ,z }∞
`=1 , {ρn` ,z }`=1 , gz as in
χ
Definition 4.2, such that fn` (ωn` ,z + ρn` ,z ζ) ⇒ gz (ζ) on C. Then by Theorem
n→∞
4.6 for m − 1, {fn` } as a family is not Qm−1 -normal in ∆(bj , rj ). Since rj can be
arbitrarily small, it follows that bj is a non Cm−1 -point of S ∗ , as claimed.
Conversely, assume that F is not Qm -normal of order at most ν in D. Then, by
Theorem 3.30, there is a sequence of functions from F, S0 = {hn }∞
n=1 , that has
at least ν + 1 µm -points in D. Let b1 , . . . , bν+1 be ν + 1 distinct point of this set.
As mentioned in Definition 3.27, each bj , 1 ≤ j ≤ ν + 1, is a µm -point of every
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
25
subsequence of S0 .
Define now ν + 1 strongly disjoint disks ∆j = {|z − bj | < rj } 1 ≤ j ≤ ν + 1
compactly contained in D. Since bj is a µm -point, it follows from Theorem 3.31
that S0 as a family is not Qm−1 -normal in ∆j , nor is any subsequence of S0 .
Step 1. S0 is not Qm−1 -normal in ∆1 , by Theorem 4.6 for m − 1, there exist
(A1 ) E1 ⊂ ∆1 having property Wm−1 with respect to ∆ such that for any z ∈ E1
there exist
(B1 ) wn,z → z;
n→∞
(C1 ) δn,z → 0+ ;
n→∞
(D1 ) a nonconstant function gz , meromorphic on C; and
(E1 ) a subsequence of S0 , S1 = {hn,1 }∞
n=1 , such that
χ
(F1 ) hn,1 (wn,z + δn,z ζ) ⇒ gz (ζ) on C, z ∈ E1 .
As a subsequence of S0 , S1 is not Qm−1 -normal in ∆2 . Hence there is a set E2 ⊂
∆2 having property Wm−1 with respect to ∆2 such that to every z ∈ E2 correspond
sequences {wn,z } and {δn,z }, a global nonconstant meromorphic function gz , and
a subsequence S2 of S1 , S2 = {hn,2 }∞
n=1 , which together satisfy the appropriate
conditions (A2) - (F2). This is the second step.
Continuing as in the proof of Theorem 4.3, after ν + 1 steps, we have for each
1≤j ≤ν+1:
(Aj ) A set Ej ⊂ ∆j with property Wm−1 with respect to ∆j , such that for any
z ∈ Ej there exist
(Bj ) wn,z → z;
n→∞
+
(Cj ) δn,z → 0 ;
(Dj ) a nonconstant function gz , meromorphic on C, and
(Ej ) a subsequence Sj of Sj−1 , Sj = {hn,j }∞
n=1 , such that
χ
(Fj ) hn,j (wn,z + δn,z ζ) ⇒ gz (ζ) on C, z ∈ Ej .
˙ n+1 . Since {∆1 , ∆2 , . . . , ∆ν+1 } are strongly disjoint, it follows
Set E = E1 ∪˙ . . . ∪E
26
SHAHAR NEVO
(m−1)
from P.2.9 that |ED
| ≥ ν + 1, so E has property Wm−1,ν+1 with respect to D.
Choose fn = hn,ν+1 for n ≥ 1 and set S = {fn }∞
n=1 . As in the proof of Theorem
4.3, it follows that E and S satisfy Zalcman’s condition with respect to D. THe
proof is completed.
Proof of Theorem 4.6. Theorem 4.4 gives the proof for m = 1, so suppose that
m ≥ 2 and the theorem holds for m − 1. Assume first the existence of E and S. By
(m)
Definition 3.21, ED
(m)
6= ∅. Let b0 be a point of ED . We shall show that b0 is a
non Cm -point of any subsequence of S. Then by Definition 3.24, it follows that F
is not Qm -normal in D.
Indeed, let S ∗ = {fnk } be subsequence of S and let D0 = {0 < |z − b0 | < r} be
(m−1)
a punctured disk contained in D. As b0 is an accumulation point of ED
(m−1)
exists some b∗ ∈ ED
, there
∩ D0 . Let ∆∗ be an open disk centered at b∗ , such that
∆∗ ⊂ D0 . By Lemma 2.11, E ∩ ∆∗ has property Wm−1 with respect to ∆∗ . Note
that for each z ∈ E ∩ ∆∗
χ
fnk (ωnk ,z + ρnk ,z ζ) ⇒ gz (ζ)
on
C,
where ωnk ,z → z, ρnk ,z → 0+ .
k→∞
k→∞
The conditions of the theorem are now fulfilled with m̂ = m − 1, D̂ = ∆∗ ,
∗
Ê = E ∩ ∆∗ , and F̂ = Ŝ = {fnk }∞
k=1 . Then by the induction assumption, S as a
family is not Qm−1 -normal in ∆∗ . In particular, S ∗ must have (as a sequence) a
non Cm−1 -point b0 ∈ ∆∗ ⊂ D0 , as required.
Conversely, assume that F is not a Qm -normal family in D. By Theorem 3.31,
there exists a sequence of functions of F, S0 = {hn }∞
n=1 , having a µm+1 -point
b0 ∈ D. Now b0 is a µm+1 -point of any subsequence of S0 and is also an accumulation
point of µm -points of S0 (each of which is a µm -point of any subsequence of S0 ).
We turn now to our construction.
Step 1. Take a µm -point of S0 , b1 ∈ D, 0 < |b1 − b0 | < 1. By Definition 3.24
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
27
and Theorem 3.31, S0 as a family, is not Qm−1 -normal at b1 . Enclose b1 in a disk
∆1 = ∆(b1 , r1 ) such that ∆1 ⊂ D, b0 ∈
/ ∆1 . Then S0 is not a Qm−1 -normal family
in ∆1 ; therefore, by the induction assumption, for m − 1, there exist
(A1 ) E1 ⊂ ∆1 having property Wm−1 with respect to ∆1 , such that for any
z ∈ E1 there exist
(B1 ) wn,z → z;
(C1 ) δn,z → 0+ ;
(D1 ) a nonconstant function gz , meromorphic on C; and
(E1 ) a subsequence S1 = {hn,1 }∞
n=1 of S0 , such that
χ
(F1 ) hn,1 (wn,z + δn,z ζ) ⇒ gz (ζ) on
C, z ∈ E1 .
Step j. (j ≥ 2) Let bj ∈ D be a µm -point of Sj−1 , Sj−1 = {hn,j−1 }∞
n=1 , such that
(4.4)
0 < |bj − b0 | <
1
j
and
˙ j−1 .
bj ∈
/ ∆1 ∪˙ . . . ∪∆
Enclose bj in a disk ∆j = ∆(bj , rj ), such that ∆j ⊂ D, b0 ∈
/ ∆j ; and
(4.5)
∆i ∩ ∆j = ∅
for
1 ≤ i ≤ j − 1.
Sj−1 as a family is not a Qm−1 -normal in ∆j ; hence, according to the induction
assumption for m − 1, there exist
(Aj ) Ej ⊂ ∆j having property Wm−1 with respect to ∆j , such that for any
z ∈ Ej there exist
(Bj ) wn,z → z;
(Cj ) δn,z → 0+ ;
(Dj ) a nonconstant function gz , meromorphic on C; and
(Ej ) a subsequence Sj = {hn,j }∞
n=1 of Sj−1 ,such that
(Fj ) hn,j (wn,z + δn,z ζ) ⇒ gz (ζ) on C, z ∈ Ej .
∞
S
(m−1)
Set now E = ˙ Ej . For each j ≥ 1, there exist cj ∈ E∆j
(cj may be taken
j=1
28
SHAHAR NEVO
to be bj ). By (4.5), ci 6= cj , i 6= j, and by (4.4) cj → b0 , cj 6= b0 . By P.2.10,
j→∞
(m−1)
ED
⊃
∞
[
(m−1)
(Ej )D
⊃
j=1
∞
[
(m−1)
(Ej )∆j
⊃ {cj }∞
j=1 .
j=1
(m−1)
Hence b0 is an accumulation point of ED
(m−1) (1)
)D
in D, i.e., b0 ∈ (ED
(m)
= ED
(see Definition 2.6), and E has property Wm with respect to D.
As in the proof of Theorem 4.4, we set fn = hn,n and S = {fn }∞
n=1 . Similarly, it
follows that E and S satisfy Zalcman’s condition with respect to D, and the proof
is completed.
5. The degree of normality of families of linear combinations.
In this section, we discuss families which are generated by taking linear combinations of a finite number of Qm -normal families, with or without limitations on
their order. We shall determine the degree and order of normality for these families.
Examples which establish the sharpness of these results are also provided.
Definition. Let F1 , F2 , . . . , F` be ` families of meromorphic functions on a domain
D. Define
G(F1 , . . . , F` ) = {c1 f1 + · · · + c` f` : ci ∈ C, fi ∈ Fi ; 1 ≤ i ≤ `}.
Let F be a Qm -normal family on a domain D. In what follows we shall discuss the
nature of the domain on which the limit functions of locally χ-uniformly convergent
sequences of F are defined. We then extend our argument to ` Qm -normal families
F1 , F2 , . . . , F` , all defined on D, and discuss the nature of the common domain of
definition of any ` limit functions, f1 , f2 , . . . , f` , where fi is a limit function of a
locally χ-uniformly convergent sequence of Fi , 1 ≤ i ≤ `. This discussion is needed
for the definition of certain families of linear combinations corresponding to these
families; see Definitions 5.2 and 5.3.
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
29
Indeed, let F be a Qm -normal family of meromorphic functions on a domain D.
If S = {fn } is a sequence of F, then according to Definition 3.24 and Theorem
3.22, S has a subsequence (which, by renumbering, we may take to be S itself)
which is a Cm -sequence. In view of Remark 3.23, the set E of non C0 -points of S
(m)
satisfies ED
= ∅. Since E is relatively closed in D, by Lemma 3.34, D \ E is a
χ
domain. Thus, fn ⇒ f on D \ E, where f is a meromorphic function on D \ E or
the constant ∞. If, in addition, F is Qm -normal of order at most ν in D, then by
(m−1)
Remark 3.23 one may assume that |ED
| ≤ ν.
Now let F1 , . . . F` be Qm -normal families of meromorphic functions on a domain
D; by what has been mentioned above, we may assume, without loss of generality,
χ
that for each 1 ≤ i ≤ `, Si = {fn,i }∞
n=1 is a sequence of Fi with fn,i ⇒ fi on D \ Ei .
Here fi is a meromorphic function or fi ≡ ∞, Ei is the set of non C0 -points of Si ,
(m)
(Ei )D
(m−1)
= ∅ (or |(Ei )D
| ≤ Ni if Fi is a Qm -normal family of order at most Ni in
D), and D \Ei is a domain. By P.2.7, E1 ∪· · ·∪E` does not have property Wm with
(1)
(1)
(1)
respect to D; moreover, (E1 ∪ · · · ∪ E` )D = (E1 )D ∪ · · · ∪ (E` )D ⊂ E1 ∪ · · · ∪ E` .
This implies by Lemma 3.36 that D \ (E1 ∪ · · · ∪ E` ) is a domain.
Starting from another point of departure, we get according to Definition 3.24
and what has been mentioned above that if F is a Qm -normal family on D and f is
a limit function (or identically ∞) of some locally χ-uniformly convergent sequence
of F on an open subset of D, then it may be assumed that f is extended to be
(m)
defined on some domain D \ Ef , where (Ef )D
(m−1)
= ∅ (or |(Ef )D
| ≤ ν, if F is of
χ
order at most ν in D) and there exists some sequence {fn } of F such that fn ⇒ f
on D \ Ef .
In view of this entire discussion, it is convenient and useful to present the following two definitions.
Definition 5.2. Let F be a Qm -normal family of meromorphic functoins on a
domain D. Define L(F) to be the collection of all limit functions, (including, if
30
SHAHAR NEVO
necessary, f ≡ ∞) of all Cm -sequences of F, each of which is defined on some
(m)
domain D \ Ef , where (Ef )D
(m−1)
= ∅ (or |(Ef )D
| ≤ ν if F is Qm -normal of order
at most ν).
Let L∗ (F) = L(F) \ {f ≡ ∞}.
Definition 5.3. Let F1 , . . . , F` be ` families of meromorphic functions on a domain
D. Denote by H(F1 , . . . , F` ) the collection of all nontrivial linear combinations,
f = c1 f1 + · · · + c` f` , where fi ∈ L∗ (Fi ), 1 ≤ i ≤ `.
We are ready for our first theorem.
Theorem 5.4. Let F1 , . . . , F` be ` normal families of meromorphic functions on
a domain D. Suppose that
(5.1)
{f ≡ ∞} ∈
/ L(Fi )
and that
(5.2)
{f ≡ 0} ∈
/ H(F1 , . . . , F` ).
Then G(F1 , . . . , F` ) is a Q1 -normal family on D.
Proof. Induction on the number of families, `.
`=1:
Let S = {ck,1 fk,1 }∞
k=1 be a sequence of G(F1 ), ck,1 ∈ C, k ≥ 1. By passing to a
χ
subsequence, we may assume that fk,1 ⇒ f1 on D, and ck,1 → c1 . By (5.1) and
k→∞
(5.2)
(5.3)
f1 (z) 6≡ 0, ∞.
χ
If c1 6= 0, ∞ then ck,1 fk,1 ⇒ c1 f1 on D.
If c1 = ∞, then in some neighborhood of each z0 ∈ D such that f1 (z0 ) 6= 0, one
χ
has ck,1 fk,1 ⇒ ∞; and so z0 is a C0 -point of S. If, on the other hand, f1 (z0 ) = 0,
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
31
take a closed disk ∆(z0 , r) in D on which f1 is holomorphic. Then, by Lemma 3.11,
for k large enough, fk,1 is holomorphic on ∆(z0 , r). By (5.3), z0 is an isolated zero
of f1 ; therefore, r can be taken so small that f1 (z) 6= 0 for z ∈ ∆(z0 , r) − {z0 }.
Thus, for large enough k, |fk,1 (z) − f1 (z)| < |f1 (z)|, z ∈ {|z − z0 | = r}. Thus,
by Rouché’s Theorem, fk,1 vanishes at some point of ∆(z0 , r) and so does ck,1 fk,1 .
Since ck,1 fk,1 ⇒ ∞ on ∆0 (z0 , r), z0 must be a non C0 -point of S, and S is irreducible
with respect to z0 . On the other hand, according to Definition 3.5, z0 is a C1 -point
of S.
The case c1 = 0 is settled similarly. Indeed, since {fk,1 } ({ck,1 fk,1 }) converges
n
o ³n
o´
1
1
1
χ-uniformly if and only if fk,1
does (and ck,1
→ ∞), and since the
ck,1 fk,1
poles of f1 are the zeros of
1
f1 ,
we get that if f1 (z0 ) 6= ∞, then z0 is a C0 -point
of S. Likewise, each pole of f1 is a non C0 -point of S with respect to which S is
irreducible and is also a C1 -point of S.
To summarize, in the cases c1 = ∞ or c1 = 0, each z0 ∈ D is a C1 -point of S; so
by Definitions 3.5 and 3.6, F is Q1 -normal in D.
`≥2:
Suppose that G(F1 , . . . , F` ) is not a Q1 -normal family in D. According to Theorem 4.3, there exist
(A) bn → b0 , b0 ∈ D;
h→∞
and, for each n ≥ 1, sequences
(B) ωk,n → bn ;
k→∞
+
(C) ρk,n → 0 ;
(D) nonconstant meromorphic functions gn on C, as well as
(E) ` sequences of complex numbers {ck,i }∞
k=1 , 1 ≤ i ≤ `, corresponding to `
sequences of functions, Si = {fk,i }∞
k=1 , 1 ≤ i ≤ `, where fk,i ∈ Fi , such that
χ
(F) ck,1 fk,1 (ωk,n + ρk,n z) + · · · + ck,` fk,` (ωk,n + ρk,n z) ⇒ gn (z) on C, n ≥ 1.
32
SHAHAR NEVO
By passing to subsequences, we may assume that
χ
(5.4)
fk,i ⇒ fi
on D,
1 ≤ i ≤ `,
where by (5.2) f1 , . . . , f` are linearly independent on D and that ck,i → ci , 1 ≤ i ≤ `.
In addition, because bn → b0 ∈ D, we get by (5.1) and (5.2) that fi (bn ) 6= 0, ∞
when n is large enough, for 1 ≤ i ≤ `. Therefore, by (F) and Lemma 3.11, it can
be assumed that each gn is an entire function; and by (5.4),
(5.5)
fk,i (ωk,n + ρk,n z) ⇒ fi (bn ) on C,
1 ≤ i ≤ `.
Now, if for example c1 6= ∞, then from (5.5) and (F) we get for each n ≥ 1
ck,2 fk,2 (ωk,n + ρk,n z) + · · · + ck,` fk,` (ωk,n + ρk,n z) ⇒ gn (z) − c1 f1 (bn ).
k→∞
Thus, by Theorem 4.4, G(F2 , . . . , F` ) is not a Q1 -normal family on D, contradicting
the induction assumption for ` − 1. Hence we must have that
(5.6)
ci = ∞,
1 ≤ i ≤ `.
Passing to subsequences and renumbering, if necessary, we may assume also that
|ck,1 | ≥ |ck,2 | ≥ · · · ≥ |ck,` |, k ≥ 1
and that
ck,i
→
ck,1 k→∞
αi , 1 ≤ i ≤ ` with 0 ≤ |α` | ≤ · · · ≤ |α2 | ≤ α1 = 1. Now for
n ≥ 1 divide (F) by ck,1 , and let k tend to infinity. We get by (5.6)
fk,1 (ωk,n + ρk,n z) +
ck,2
ck,`
fk,2 (ωk,n + ρk,n z) + · · · +
fk,1 (ωk,n + ρk,n z) ⇒ 0,
k→∞
ck,1
ck,1
and passing to the limit gives by (5.5):
f1 (bn ) + α2 f2 (bn ) + · · · + α` f` (bn ) = 0
for any n ≥ 1. It follows from (A) that
f1 (z) + α2 f2 (z) + · · · + α` f` (z) ≡ 0,
which contradicts (5.2). The proof of Theorem 5.4 is completed.
We now show that neither of the conditions (5.1) and (5.2) can be omitted. We
do this for the case ` = 1.
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
33
Example 5.5.
The importance of condition (5.1)
Let 0 < r < 1. Consider a sequence of holomorphic functions gn (z) =
n ≥ 1, on ∆(1, r) and set F1 = {gn : n ≥ 1}. Then gn
zn
(1−r)n ,
⇒ ∞ on ∆(1, r); in
n→∞
n
particular, F1 is a normal family there. Define hn (z) = (1 − r) gn (z) = z n , n ≥ 1.
Then hn ∈ G(F1 ), n ≥ 1, and {hn } satisfies hn ⇒ 0 on ∆ ∩ ∆(1, r) and hn ⇒ ∞ on
c
∆ ∩ ∆(1, r). Hence, any point of ∂∆ ∩ ∆(1, r) is a non C0 -point of {hn }, and {hn }
is irreducible with respect to it. Since this set is of power ℵ, we conclude by Lemma
2.13 and Theorem 3.22 that G(F1 ) is not a Qm -normal family for any m ≥ 0.
Example 5.6.
The importance of condition (5.2)
Under the same conditions of Example 5.5, define wn (z) =
zn
(1+r)n ,
and F1∗ =
{wn : n ≥ 1}. Then wn ⇒ 0 on ∆(1, r), so condition (5.2) does not hold. It is
obvious that G(F1∗ ) = G(F1 ).
As a consequence of Theorem 5.4, we have
Corollary 5.7. Let f1 , . . . , f` be meromorphic functions in a domain D. Then the
collection of all linear combinations of these functions is a Q1 -normal family in D.
From Corollary 5.7, it is easy to construct examples that show that the order of
the family G(F1 , . . . , F` ) (as a Q1 -normal family) is not bounded in general. The
following two examples demonstrate this for a bounded domain D.
Example 5.8. Let f1 be a holomorphic function in ∆, f1 6≡ 0, which has infinitely
many zeros in ∆. Then G({f1 }), which is the family of all scalar multiples of f1 , is
not a Q1 -normal family of any finite order, as the sequence {kf1 }∞
k=1 shows.
Example 5.9. Let ϕ be a Möbius transformation mapping ∆ onto the upper half
plane H = {Im w > 0}. Set f1 (z) = eϕ(z) , f2 (z) = e2ϕ(z) , and
34
SHAHAR NEVO
A = ϕ−1 {2πin : n ≥ 1}. Each z ∈ A satisfies
1 · f1 (z) + (−1)f2 (z) = 0.
Now, taking ck,1 = k, ck,2 = −k, k ≥ 1 easily implies that each z ∈ A is a non
C0 -point of the sequence S = {ck,1 f1 + ck,2 f2 }∞
k=1 and S is irreducible with respect
to each z ∈ A. It follows that G({f1 }, {f2 }) is not a Q1 -normal family of any finite
order.
Despite these last two examples, in some cases we can control the order of normality of G({f1 }, . . . , {f` }).
Theorem 5.10. Let y1 , . . . , y` be meromorphic functions in a domain D. Assume
that any linear combination of these functions that is not the function f ≡ 0 has at
most N distinct zeros in D. Let P be the number of distinct poles of the collection
of functions {y1 , . . . , y` }. (This means that each z0 ∈ D which is a pole of some
yi , 1 ≤ i ≤ `, is counted precisely once, without considering its multiplicity). Then
G = G({y1 }, . . . , {y` }) (see Corollary 5.7) is a Q1 -normal family of order at most
max{N, P } in D.
Before proving this theorem in the general case, we prove it in the case that
y1 , . . . , y` are all rational functions.
Proof for the rational case. Take w0 6= 0, ∞. Because all the functions are rational,
there exists some n0 ≥ 1, such that the equation f (z) = w0 has no more than n0
roots in D, for any f ∈ G.
Define δ = min{χ(0, w0 ), χ(∞, w0 )} > 0, and set p1 = N, p2 = n0 , p3 = P. For
each f ∈ G, set a1 (f ) = 0, a2 (f ) = w0 , a3 (f ) = ∞. Let q be the median of N, P, n0 .
By the definitions of N, P, n0 and Theorem 3.9, it follows that G is a Q1 -normal
family of order q at most in D. But for any arrangement of N, P, n, q satisfies
q ≤ max{N, P }.
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
35
Proof of Theorem 5.10. Since G is a family of linear combinations of y1 , . . . , y` , we
may assume without loss of generality that
(5.7)
y1 , . . . , y` are linearly independent on D.
For each 1 ≤ i ≤ `, let pi be the number of the distinct poles of yi , none of which
is a pole of yj for j 6= i, 1 ≤ j ≤ `.
Let S = {fk } be a sequence of functions from G :
fk = ck,1 y1 + · · · + ck,` y` , k ≥ 1.
By passing to a subsequence and renumbering if necessary, we may assume that
ck,` 6= 0, k ≥ 1, and also |ck,1 | ≤ |ck,2 | ≤ · · · ≤ |ck,` | for k ≥ 1. As before, we can
also assume that ck,i → ci and
k→∞
ck,i
→ αi , 1 ≤ i ≤ `,
ck,`
where 0 ≤ |α1 | ≤ · · · ≤ |α`−1 | ≤ α` = 1.
Let i = i1 , 1 ≤ i1 ≤ ` be the smallest integer such that αi 6= 0. Consider the
linear combination
F = αi1 yi1 + αi1 +1 yi1 +1 + · · · + y` .
By (5.7)
(5.8)
F 6≡ 0 in D.
We consider three cases.
Case (A). c` = ∞.
It is clear that for a point z0 ∈ D which is not a pole of any yi , 1 ≤ i ≤ ` and
is such that F (z0 ) 6= 0, there exists a neighborhood of z0 in D, on which fk ⇒ ∞;
thus z0 is a C0 -point of S. Now let ζ1 , . . . , ζP be the distinct poles of y1 , . . . , y` in
36
SHAHAR NEVO
D, and let z1 , . . . , zt be the distinct zeros of F in D, (t ≤ N ), which are not poles
of any yi , 1 ≤ i ≤ `.
Construct P + t strongly disjoint disks, compactly contained in ∆,
(5.9)
∆(ζ1 , r1 ), . . . , ∆(ζp , rP ); ∆(z1 , R1 ), . . . , ∆(zt , Rt ).
By (5.8), this can be done in such a way that for each 1 ≤ i ≤ P
(5.10)
F (z) 6= 0
z ∈ ∆(ζi , ri ) \ {ζi },
and for each 1 ≤ j ≤ t
(5.11)
F (z) 6= 0,
z ∈ ∆(zj , Rj ) \ {zj }.
Denote γi = ∂∆(ζ, ri ), 1 ≤ i ≤ P, and Γj = ∂(zj , Rj ), 1 ≤ j ≤ t. By Lemma
3.11,
(5.12)
fk ∆
= hk ⇒ F on D \ {ζ1 , . . . , ζP }.
ck,`
By (5.11) and (5.12), there exists k ∈ N such that
|hk (z) − F (z)| < |F (z)|, z ∈ Γj
for each k ≥ k1 , 1 ≤ j ≤ t. By Rouché’s Theorem, fk vanishes in ∆(zj , Rj ) for
1 ≤ j ≤ t. Since fk is a linear combination of y1 , . . . , y` , by assumption
(5.13)
it has at most N − t distinct zeros in the domain D \
t
[
∆(zj , Rj ).
j=1
Now, if P ≤ N − t, then S has at most N non C0 -points in D; and we are finished.
Otherwise, by (5.13) and the construction in (5.9), there exists a subsequence
S 0 = {fkj }∞
j=1 such that for each j ≥ 1
(5.14) fkj (z) 6= 0 in ∆(ζi , ri ) for some P − (N − t) specific values of i, 1 ≤ i ≤ P.
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
37
Without loss of generality, we may assume that this is true for 1 ≤ i ≤ P − N + t.
By (5.12), hkj → F on γi ; and since by (5.10) F (z) 6= 0, we get by Lemma 3.11
that
1
hkj
→
1
F
on γi , 1 ≤ i ≤ P − N + t. Thus
1
→
fkj j→∞
0 there. Then (5.14) and
the maximum principle yield that fkj → ∞ on ∆(ζi , ri ), 1 ≤ i ≤ P − N + t; and it
follows that S 0 has at most t + (N − t) = N non C0 -points in D.
Case (B). c` = 0.
In this case, if z0 is not a pole of yi for some 1 ≤ i ≤ `, then fk → 0 in a
sufficiently small neighborhood of z0 . Hence there are at most P non C0 -points of
S0.
Case (C). c` 6= 0, ∞.
If yi (z0 ) 6= ∞, 1 ≤ i ≤ `, then z0 is a C0 -point of S. If yi (z0 ) = ∞ for a single
1 ≤ i ≤ `, say i = 1, then evidently z0 is a C0 -point of the sequence {fk,1 − ck,1 y1 }.
The functions fk,1 − ck,1 y1 (k ≥ 1) are holomorphic in some neighborhood of z0 .
Moreover, since c` 6= ∞,
{fk,1 −ck,1 y1 } converges uniformly to an analytic function
on some neighborhood of z0 . Now fk,1 = fk,1 − ck,1 y1 + ck,1 y1 ; so Lemma 3.14 then
implies that z0 is a C0 -point of S. Hence, only a point that is a common pole of at
least two functions, yi , yj , 1 ≤ i 6= j ≤ `, can be a non C0 -point of S. That is, there
P̀
are at most P −
pi non C0 -points of S in D. The results in cases (A), (B) and
i=1
(C) yield that in any case S has some subsequence with at most max{N, P } non
C0 -points in D, and the theorem is proved.
Remark 5.11. It is obvious that Theorem 5.10 is true also in the case
max{N, P } = ∞.
Corollary 5.12. Theorem 5.10 is sharp. Specifically, for any y1 , . . . , y` as in Theorem 5.10, which have a “maximal” linear combination with exactly N distinct zeros
in D. The corresponding family G({y1 }, . . . , {y` }) is a Q1 -normal family of order
exactly max{N, P } in D.
38
SHAHAR NEVO
Proof. It is enough to show the existence of two sequences: one with exactly N non
C0 -points with respect to each of which this sequence is irreducible and the second
with exactly P such points.
I. A sequence with N non C0 -points
Suppose that the linear combination H = α1 y1 + · · · + α` y` has exactly N zeros,
P̀
kαi yi ; and set S = {fk }∞
z1 , . . . , zN in D. Define, for any k ≥ 1, fk =
k=1 .
i=1
Evidently, S satisfies the required property.
II. A sequence with P non C0 -points
Define, for each k ≥ 1,
(5.15)
fk =
X̀ 1
yi ;
ki
i=1
and set S = {fk }. Let z1 , . . . , zP be the P distinct poles of the collection of functions
{y1 , y2 , . . . , y` } (cf. the formulation of Theorem 5.10). Clearly, any z0 ∈ D, such
that z0 ∈
/ {z1 , . . . , zP } has a neighborhood on which fk ⇒ 0. So such a point is
a C0 -point of S. Now, for each 1 ≤ q ≤ P and 1 ≤ i ≤ `, consider the Laurent
expansion of yi around zq . Writing fk as a sum of ` such expansions, we get in some
small neighborhood ∆(zq , rq ) of zq
(5.16)
fk (z) =
X̀ 1
w (z) + gk,q (z),
i i,q
k
i=1
where

(q)
O(i,q)
Ai,j
P




j


 i=1 (z − zq )
wi,q (z) =







if zq is a pole of yi (z) of order O(i, q)
(q)
(and Ai,O(i,q) 6= 0)
0
otherwise
Thus, wi,q is the singular part of fk (z) in ∆(zq , rq ) and gk,q (z) is holomorphic on
∆(zq , rq ). By (5.15), (5.16), and the properties of Laurent expansions, it follows
that gk,q ⇒ 0 on D.
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
39
It is possible that the maximal order for some pole zq , is attained by more than
one function, yi . But since
1/kj
→
1/ki k→∞
0 for 1 ≤ i < j ≤ `, we get according to
(5.15) that each zq , 1 ≤ q ≤ P, is a non C0 -point of S with respect to which S is
irreducible, as required.
There exists a connection between N and the pi ’s from Theorem 5.10, as the
following corollary shows.
Corollary 5.13. Under the conditions of Theorem 5.10, given that ` ≥ 2 and
y1 , . . . , y` are linearly independent, there exists for each 1 ≤ i ≤ ` a nontrivial
linear combination of y1 , . . . , y` having at least pi distinct zeros in D. Thus, N ≥
max{p1 , . . . , p` }.
Remark 5.14. Corollary 5.13 is false for linearly dependent y1 , . . . , y` .
Remark 5.15. Corollary 5.13 is sharp. For example, setting D = C, y1 (z) =
1
(z−1)(z−2)(z−3)
and y2 (z) =
1
(z−2)(z−3)(z−4) ,
we have P = 4, p1 = p2 = 1; and it is
easy to verify that N = 1.
Proof of Corollary 5.13. Write y1 (z) =
A1 (z)
B1 (z) , . . . , y` (z)
=
A` (z)
B` (z) ,
where for each
1 ≤ i ≤ `, Ai (z) and Bi (z) are holomorphic functions on D without common
zeros. We shall prove that N ≥ p1 .
A linear combination of y1 , . . . , y` is
(5.17)
H = c1 y1 + · · · + c` y`
=
c1 (A1 · B2 · · · Bm ) + c2 (B1 · A2 · B3 · · · Bm ) + · · · + cm (B1 · · · Bm−1 · Am )
B1 · · · Bm
Fix c2 , . . . , c` such that ci0 6= 0 for some 2 ≤ i0 ≤ `, and define
f (z) = c2 (B1 · A2 · B3 · · · Bm )(z) + · · · + cm (B1 · · · Bm−1 · Am )(z)
g(z) = f (z) + c1 (A1 · B2 · · · Bm )(z).
40
SHAHAR NEVO
Now, suppose p1 > 0 (otherwise there is nothing to prove), and let z1 , . . . , zp1 be
the p1 zeros of B1 (z) alone, so that Bi (zj ) 6= 0, 2 ≤ i ≤ `, 1 ≤ j ≤ p1 .
Construct p1 strongly disjoint disks, ∆(zj , rj ), 1 ≤ j ≤ p1 , compactly contained
in D. These disks can be taken to satisfy
(5.18)
Bi (z) 6= 0 for z ∈ ∆(zj , rj ), 1 ≤ j ≤ p1 , 2 ≤ i ≤ `;
moreover,
B1 (z) 6= 0 on ∆0 (zj , rj ), 1 ≤ j ≤ p1 .
(5.19)
Since y2 , . . . , y` are linearly independent on D, we can also choose r1 , . . . , rp1 such
that
(5.20)
f (z) 6= 0, z ∈ γj , 1 ≤ j ≤ p1 ,
where γj = ∂(∆j , rj ).
For small enough values of c1 , it then follows by (5.20) that
(5.21)
|f (z) − g(z)| < |f (z)|, z ∈ γj , for 1 ≤ j ≤ p1 .
By Rouché’s Theorem, g vanishes at some ζj ∈ ∆(zj , rj ); but by the definitions of
zj and g, g(zj ) 6= 0, so ζj 6= zj . According to (5.18) and (5.19), ζj is not a zero of
the denominator of H in (5.17). Since g is actually the numerator of H in (5.17),
it follows that H has ζ1 , . . . , ζp1 as distinct roots in D, as desired.
Remark 5.16. We have dealt above with the case p1 < ∞. If p1 = ∞, then it
cannot be guaranteed that (5.21) holds for every j ≥ 1; but, in the same way, we
can create a linear combination of y1 , . . . , y` which has as large a number of zeros
in D as we desire. Thus, in this case also, N = ∞ = max{p1 , . . . , p` }.
Theorem 5.10 implies the following result about linear combinations of holomorphic functions.
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
41
Corollary 5.17. Let y1 , . . . , y` be holomorphic functions on a domain D any linear
combination of which that is not identically zero has at most N distinct roots in D.
Then G({y1 }, . . . , {y` }) is a Q1 -normal family of order at most N in D.
Remark 5.18. According to the definition of the family G = G({y1 }, . . . , {y` }),
it follows that for each w ∈ C, w 6= 0, the least upper bound for the number of
distinct points at which some f ∈ G takes on the value w does not depend on w;
denote this number by T (T might be ∞). On the other hand, since a holomorphic
function is an open map, T is greater or equal to the least upper bound of times
that some f ∈ G attains the value 0 in D.
Thus Corollary 5.17 is also true when it is given that each linear combination
takes on some specific value w0 6= 0 not more than N times. Moreover, in this
case, the family G({1}, {y1 }, . . . , {y` }) of linear combinations of 1, y1 , . . . , y` is also
a Q1 -normal family of order at most N in D.
By similar considerations, the same holds with regard to Theorem 5.10. Specifically, the conclusions of that theorem hold if N is the least upper bound for the
number of times some value w0 6= 0, ∞ is attained by any of the functions in
G({y1 }, . . . , {y` }).
Remark 5.19. Let
(5.22)
y (n) + a1 y (n−1) + · · · + an y = 0
be a linear differential equation of order n in the disk ∆(z0 , R) (0 < R ≤ ∞), where
a1 (z), . . . , an (z) are bounded and analytic in ∆(z0 , R). The equation (5.22) is said
to be disconjugate in ∆(z0 , R) if no nontrivial solution of (5.22) has more than n−1
zeros (where the zeros are counted with their multiplicity). For details see [4] and
[3]. Corollary 5.17, together with the fundamental theorem of linear differential
equations, implies that the collection of solutions of (5.22) is a Q1 -normal family
of order at most n − 1 in ∆(z0 , R).
42
SHAHAR NEVO
We now present the generalization of Theorem 5.4 for any m ≥ 1.
Theorem 5.20. Let m, ` ≥ 1 be integers and F1 , . . . , F` be Qm -normal families of
meromorphic functions on a domain D. Suppose that
(5.23)
{f ≡ ∞} ∈
/ L(Fi ),
1 ≤ i ≤ `,
and that
(5.24)
{f ≡ 0} ∈
/ H(F1 , . . . , F` }.
Then G(F1 , . . . , F` ) is a Qm+1 -normal family on D.
Remark 5.21. For families of holomorphic functions which are Qm -normal on D
and yet not normal on D (m ≥ 1), Theorem 5.20 has no value, since its conditions
χ
cannot be satisfied. Indeed, in the notation of Theorem 5.20, assume that yk,i ⇒ yi
on D \ Ei , 1 ≤ i ≤ `. Let z0 ∈ Ei be a non C0 -point of Si . By Theorem 3.20, Ei can
be assumed to be at most enumerable; hence there exists r > 0 such that ∂∆(z0 , r)
is contained in D, and consists of C0 -points of Si only. Hence, if yi is a holomorphic
function, it follows that Si is a uniformly convergent Cauchy sequence on ∂∆(z0 , r),
i.e.,
max |yn,i (z) − ym,i (z)|
|z−z0 |=r
→
m,n→∞
0; and by the maximum principle this is also
true for ∆(z0 , r). Hence z0 is a C0 -point of Si and Ei = ∅ actually. Thus, yi must
be the constant ∞, and condition (5.23) cannot be satisfied (cf. [2], p.131).
Proof of Theorem 5.20. We shall prove the theorem by induction on the number of
families, `.
` = 1.
Let S = {ck,1 yk,1 } be a sequence of functions of G(F1 ). We have to prove that
S has subsequence which is a Cm+1 -sequence in D. As usual, by passing to a
subsequence, we may assume that ck,1 → c1 and that yk,1 ⇒ y1 on the domain
D∗ = D \ E1 , where
(5.25)
(m)
(E1 )D
=∅
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
43
(recall the discussion at the beginning of this chapter). By (5.23) and (5.24),
χ
y1 6≡ 0, ∞. Now, if c1 6= 0, ∞, then ck,1 yk,1 ⇒ c1 y1 on D∗ ; and by Theorem 3.22, S
is a Cm -sequence on D.
If c1 = ∞, then for any z0 ∈ D∗ with y1 (z0 ) 6= 0, there exists a neighborhood
∆(z0 , r) such that ck,1 yk,1 ⇒ ∞ in ∆(z0 , r); and z0 is a C0 -point of S. Hence, the
set of non C0 -points of S in D∗ (“irreducible” in this case) is A = {z : y1 (z) = 0};
(1)
and, by (5.24), A is isolated in D∗ . So AD ⊂ E1 ; and E, the set of non C0 -points
of S in D, is contained in A ∪ E1 . Then by P.2.9 and (5.25),
(m+1)
ED
(m+1)
⊂ (A ∪ E1 )D
(m+1)
= AD
(m+1)
∪ (E1 )D
(1) (m)
(m)
= (AD )D ∪ ∅ ⊂ (E1 )D
= ∅.
By Theorem 3.22, S is a Cm+1 -sequence in D. The case c1 = 0 is handled similarly,
the set A being replaced by the set A∗ = {z : y1 (z) = ∞} ⊂ D∗ .
` ≥ 2.
Suppose, to the contrary, that G(F1 , . . . , F` ) is not a Qm+1 -normal family in D.
Then, by Theorem 4.6, there exist
(A) A set E ⊂ D having property Wm+1 with respect to D and for each z ∈ E
sequences,
(B) ωk,z → z,
k→∞
+
(C) ρk,z → 0 , and
(D) a nonconstant meromorphic function gz on C, with the property that there
exists a sequence S ⊂ G(F1 , . . . , F` )
(E) S : fk = ck,1 yk,1 + · · · + ck,` yk,` , k ≥ 1, where yk,i ∈ Fi , 1 ≤ i ≤ `, such
that for every z ∈ E
χ
(F) fk (ωk,z + ρk,z ζ) ⇒ gz (ζ) on C.
χ
As usual, we assume that for each 1 ≤ i ≤ `, Si : yk,i ⇒ yi on Di , where yi is
defined on the domain Di = D \ Ei and by (5.23) and (5.24), yi 6≡ 0, ∞. Here Ei
(m)
is the set of C0 -points of Si in D, (Ei )D
= ∅. In addition, y1 , . . . , y` are linearly
independent on the domain D∗ = D \ (E1 ∪ · · · ∪ E` ) by (5.24).
44
SHAHAR NEVO
For each 1 ≤ i ≤ `, set
Bi = {z ∈ Di : yi (z) = 0 or yi (z) = ∞}.
(1)
Evidently, (Bi )Di = ∅; and thus
(1)
(5.26)
(Bi )D∗ = ∅ for 1 ≤ i ≤ `.
(1)
Set B = B1 ∪ B2 ∪ · · · ∪ B` . By P.2.9 and (5.26), BD∗ = ∅; hence the accumulation
(m+1)
points of B in D are contained in E1 ∪ · · · ∪ E` . We obtain BD
(m)
(E1 ∪ · · · ∪ E` )D
(5.27)
(1) (m)
= (BD )D
⊂
= ∅. Again by (A) and P.2.9,
E ∗ = E \ (B ∪ E1 ∪ · · · ∪ E` ) has property Wm+1 with respect to D.
Substitute E ∗ for E in (A). Then (B) - (F) hold for any z ∈ E ∗ , and by the
definition of Bi (1 ≤ i ≤ `), yi (z) 6= 0, ∞, z ∈ E ∗ . Take z ∈ E ∗ ; then z ∈ D \ Ei for
each 1 ≤ i ≤ ` and has a neighborhood in D∗ in which y1 , . . . , y` are holomorphic.
Hence, by Lemma 3.11
yk,i (ωk,z + ρk,z ζ) ⇒ yi (z) on C, z ∈ E ∗ , 1 ≤ i ≤ `;
(5.28)
and so in (F), gz is an entire function for each z ∈ E ∗ .
Now, as in the proof of Theorem 5.4, we may assume that ck,i → ∞ for 1 ≤ i ≤ `.
(Otherwise, we get a contradiction to the induction assumption for ` − 1.) It can
also be assumed that for each k ≥ 1, |ck,1 | ≥ |ck,2 | ≥ · · · ≥ |ck,` |; and
ck,i
→ αi , 1 ≤ i ≤ `
ck,1 k→∞
with
0 ≤ |α` | ≤ · · · ≤ |α2 | ≤ α1 = 1.
Divide (F) by ck,1 to get
yk,1 (ωk,z + ρk,z ζ)
(5.29)
+
ck,`
ck,2
yk,2 (ωk,z + ρk,z ζ) + · · · +
yk,` (ωk,z + ρk,z ζ) ⇒ 0 on C,
ck,1
ck,1
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
45
for each z ∈ E ∗ .
Passing to the limit in (5.29) yields by (5.28) that
y1 (z) + α2 y2 (z) + · · · + α` y` (z) = 0, z ∈ E ∗ .
(5.30)
Now, if E ∗ has no accumulation point in D∗ , then
(m+1)
(E ∗ )D
(1) (m)
= ((E ∗ )D )D
(m)
⊂ (E1 ∪ · · · ∪ E` )D
= ∅,
which violates (5.27). Thus, by (5.30), we obtain a contradiction to (5.24), and the
proof of Theorem 5.20 is complete.
We now turn to the analogous result concerning Qm -normal families of finite
order, based on the discussion of those families at the beginning of this section.
Theorem 5.22. Let `, m ≥ 1 be integers and for i = 1, 2, . . . , ` let Fi be a Qm normal family of order at most Ni in a domain D, 1 ≤ i ≤ `. Suppose further
that
(5.31)
{f ≡ ∞} ∈
/ L(Fi ), 1 ≤ i ≤ `
and
(5.32)
{f ≡ 0} ∈
/ H(F1 , . . . , F` }.
Then G(F1 , . . . , F` ) is a Qm+1 -normal family of order at most N0 =
P̀
i=1
Ni in D.
Proof. Suppose, to the contrary, that G(F1 , . . . , F` ) is not a Qm+1 -normal family
of order at most N0 in D. Then by Theorem 4.5, there exist
(A) A set E ⊂ D having property Wm,N0 +1 with respect to D (see Definition
3.37), such that for each z ∈ E there exist sequences
(B) ωk,z → z
(C) ρk,z → 0+ , and
46
SHAHAR NEVO
(D) a nonconstant meromorphic function gz on C with the property that there
exists a sequence S ⊂ G(F1 , . . . , F` ),
(E) S : fk = ck,1 yk,1 + · · · + ck,` yk,` , k ≥ 1, where yk,i ∈ Fi , 1 ≤ i ≤ `, such
that for every z ∈ E
χ
(F) fk (ωk,z + ρk,z ζ) ⇒ gz (ζ) on C.
χ
As before, we may assume that Si : yk,i ⇒ yi on D \ Ei , where Ei is the set of non
C0 -points of Si in D, satisfying
(m−1)
(5.33)
|(Ei )D
| ≤ Ni for each 1 ≤ i ≤ `
(recall the preceding discussion). Set E ∗ = E \ (E1 ∪ · · · ∪ E` ).
Now, by (5.31), yi 6≡ ∞, 1 ≤ i ≤ `; and by (5.32) y1 , . . . , y` are linearly independent on the domain D∗
(m)
(E1 ∪ · · · ∪ E` )D
=
D \ (E1 ∪ · · · ∪ E` ). By (5.33) and P.2.9,
(m)
= ∅ so that ED
(m)
⊂ (E ∗ )D
(m)
= (E ∗ )D .
(m)
contains at
∪ (E1 ∪ · · · ∪ E` )D
(m)
Again by (5.33) and P.2.9, |(E1 ∪ · · · ∪ E` )D | ≤ N0 . Hence (E ∗ )D
(m−1)
least one point, say z0 , that does not belong to (Ei )D
(m)
, for each 1 ≤ i ≤ `; and
by Remark 3.23, z0 is a Cm−1 -point of Si for every 1 ≤ i ≤ `.
Definitions 3.15 and 3.17 and Lemma 3.16 yield that there exists r > 0 such
that for each 1 ≤ i ≤ `, Si is a Cm−1 -sequence on ∆(z0 , r) (and considered as
a family is a Qm−1 -normal family there). It is easy to see that S1 , . . . , S` satisfy
the assumptions of Theorem 5.20 on ∆(z0 , r). Thus G(S1 , . . . , S` ) is a Qm -normal
family in ∆(z0 , r). On the other hand, by Lemma 2.11, E ∗ ∩ ∆(z0 , r) has property
Wm with respect to ∆(z0 , r). Moreover, for each z ∈ E ∗ ∩ ∆(z0 , r), (B) - (F) hold.
But then by Theorem 4.6, G(S1 , . . . , S` ) is not a Qm -normal family in ∆(z0 , r), a
contradiction. The proof is completed.
In certain situations, the degree of normality of the family of linear combinations
G(F1 , . . . , F` ) is equal to the largest degree of normality of the generating families
F1 , . . . , F` .
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
47
Theorem 5.23. Let m ≥ 1 be an integer and let F1 , . . . , F` be Qm -normal families on a domain D. Suppose {f ≡ ∞} ∈
/ L(Fi ), 1 ≤ i ≤ `, and {f ≡ 0} ∈
/
H(F1 , . . . , F` ). Suppose, in addition, that there exist (not necessarily distinct)
w1 , . . . , w` ∈ Ĉ and non-negative integers p1 , . . . , p` such that for each 1 ≤ i ≤ `
and each f ∈ Fi , the equation f (z) = wi has at most pi roots in D, counted with
their multiplicities. Then G(F1 , . . . , F` ) is a Qm -normal family on D.
Note. In the case m = 0, Theorem 5.23 fails. For example, take D = ∆, ` = 1,
F1 = {f (z) = z}, w1 = ∞.
Before proving the theorem, we present the following result, which is a slight
modification of Lemma 3.12.
Lemma 5.24. Let m ≥ 1 be an integer and S = {fn } a Cm -sequence of meromorphic functions on a domain D. Let E be the set of non C0 -points of S in D.
Suppose there exist w0 ∈ Ĉ and integers P, N ≥ 0 such that for n > N, the equation
fn (z) = w0 has at most P roots in D, counted with multiplicities. Assume also that
χ
fn ⇒ f on D \ E, where f is a nonconstant meromorphic function there. Then f
(1)
can be extended to a meromorphic function on the domain D∗ = D \ ED .
Proof. Suppose E 6= ∅, since otherwise there is nothing to prove. Because E is
(1)
(1)
(m)
closed in D, ED ⊂ E; and E \ ED 6= ∅, since otherwise ED
(m+1)
= ED
for each
m ≥ 0, contradicting the fact that E is enumerable (see Lemma 2.15 and Theorem
3.20).
χ
(1)
Let z0 ∈ E \ ED . Then there exists r > 0 such that ∆(z0 , r) ⊂ D and fn ⇒ f on
∆0 (z0 , r). By assumption, f is a nonconstant meromorphic function in ∆0 (z0 , r); so
by Lemma 3.12, f has an extension to a meromorphic function on ∆(z0 , r). Since
(1)
this can be done for any z0 ∈ E \ ED , we are finished.
(1)
Note. D \ ED is a domain by Property P.2.8 and Lemma 3.34.
Proof of Theorem 5.23. Induction on the number of families, `.
48
SHAHAR NEVO
` = 1.
Let S1 = {ck,1 yk,1 }∞
k=1 be a sequence of G(F1 ). We have to find a subsequence
of S1 which is a Cm -sequence on D. As before, we may assume that
ck,1 → c1
and
χ
yk,1 ⇒ y1 on D \ E1 ,
k→∞
where E1 is the set of non C0 -points of {yk,1 }, satisfying
(m)
(5.34)
(E1 )D
= ∅.
By our assumption, y1 6≡ 0, ∞. There are two possibilties:
1) y1 ≡ w for some w 6= 0, ∞, in which case y1 evidently has a meromorphic
extension to D.
2) y1 is a nonconstant meromorphic function on D \ E1 . Then by Lemma 5.24
(1)
there is a meromorphic extension of y1 to the domain D \ (E1 )D .
χ
Now, if c1 6= 0, ∞, then ck,1 yk,1 ⇒ c1 y1 on D \ E1 ; so by (5.34) S1 is a Cm sequence on D.
If c1 = ∞, then, as usual, any z0 ∈ D \ E1 with y1 (z0 ) 6= 0 is a C0 -point of
S1 . Hence the set of non C0 -points of S1 in D \ E1 is the set N = y1−1 ({0}), and
the set E of non C0 -points of S1 in D is contained in N ∪ E1 . The set N has
(1)
no accumulation points in the domain D \ (E1 )D , else by Lemma 5.24, y1 ≡ 0.
Therefore,
(m)
ED
(m)
⊂ (N ∪ E1 )D
(m)
= ND
(m)
∪ (E1 )D
(m)
= (E1 )D
(1) (m−1)
= (ND )D
(1) (m−1)
∪ ∅ ⊂ ((E1 )D )D
= ∅,
and S1 is a Cm -sequence in D, as expected. The case c1 = 0 is settled in a similar
fashion.
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
49
`≥2
Suppose, to the contrary, that G(F1 , . . . , F` ) is not a Qm -normal family on D.
Then by Theorem 4.6 there exist
(A) A set E ⊂ D having the property Wm with respect to D, and
(B)-(F) as in the proof of Theorem 5.22.
χ
As usual, we assume that for each 1 ≤ i ≤ `, Si : yk,i ⇒ yi on the domain D \ Ei ,
(m)
where Ei is the set of non C0 -points of Si in D, (Ei )D
= ∅. Likewise, yi 6≡ ∞; and
y1 , . . . , y` are linearly independent on the domain
(1)
(1)
D∗ = D \ ((E1 )D ∪ · · · ∪ (E` )D ).
(5.35)
By Lemma 5.24, D∗ is a common domain of definition for y1 , . . . , y` . Denote for
1≤i≤`
(1)
Bi = {z ∈ D \ (Ei )D : yi (z) = 0 or yi (z) = ∞}.
(1)
Clearly, Bi is isolated in the domain D\(Ei )D ; hence it has no accumulation points
in D∗ . Set B = B1 ∪ · · · ∪ B` . By P.2.9, B has no accumulation point in D∗ . Thus
(1)
(1)
the accumulation points of B in D belong to (E1 )D ∪ · · · ∪ (E` )D , so that
(m)
BD
(1) (m−1)
= (BD )D
(1)
(1) (m−1)
⊂ ((E1 )D ∪ · · · ∪ (E` )D )D
= ∅.
Setting E ∗ = E \ (B ∪ E1 ∪ · · · ∪ E` ), we deduce that
(5.36)
E ∗ has property Wm with respect to D.
Substitute E ∗ for E in (A); then, by the definition of B, for each z ∈ E ∗ yi (z) 6=
0, ∞. Hence in (D), gz is an entire function (cf. (5.28)); and since E ∗ contains no
points of E1 ∪ · · · ∪ E` , one has for each z ∈ E ∗ , 1 ≤ i ≤ `
(5.37)
yk,i (ωk,z + ρk,z ζ) ⇒ yi (z) on C.
50
SHAHAR NEVO
As in the proof of Theorem 5.4 (or Theorem 5.20), we may assume that for each
1 ≤ i ≤ `, ck,i → ∞ (else we get a contradiction to the induction assumption for
k→∞
` − 1).
Suppose also (as we may) that for each k ≥ 1, |ck,1 | ≥ |ck,2 | ≥ · · · ≥ |ck,` |, and
ck,i
ck,1
→ αi , 1 ≤ i ≤ `, where 0 ≤ |α` | ≤ · · · ≤ |α2 | ≤ α1 = 1. For any z ∈ E ∗ divide
(F) by ck,1 to get
ck,2
yk,2 (ωk,z + ρk,z ζ)
ck,1
ck,`
+ ··· +
yk,` (ωk,z + ρk,z ζ) ⇒ 0 on C.
ck,1
yk,1 (ωk,z + ρk,z ζ) +
(5.38)
Passing to the limit in (5.38) gives by (5.37)
y1 (z) + α2 y2 (z) + · · · + α` y` (z) = 0, z ∈ E ∗ .
(5.39)
By (5.35), it is enough to prove that E ∗ has a limit point in D∗ . That is, E ∗ has
(1)
(1)
a limit point in D which does not belong to (E1 )D ∪ · · · ∪ (E` )D . Indeed, if not,
then
(m)
(E ∗ )D
∗(1) (m−1)
= (ED )D
(1)
(1) (m−1)
⊂ ((E1 )D ∪ · · · ∪ (E` )D )D
= ∅,
violating (5.36). Thus by (5.39) we get a contradiction to the linear independence
of y1 , . . . , y` on D∗ , and the proof of Theorem 5.23 is completed.
Examples 5.8 and 5.9 show that Theorem 5.4 is sharp. (An even simpler example
is F1 = {Id}, D = ∆.) Our purpose now is to show by appropriate examples that
Theorems 5.20 and 5.22 are also sharp. First, we turn to Definition 3.33. Let
ν, m ≥ 1 be integers. By induction, it follows that an m-ple sequence σ(m, ν) of
points of D which satisfies condition (Γm,ν ) (see Definition 3.33) with respect to D
has property Wm−1,ν with respect to D. Similarly, an m-ple sequence σ(m) that
satisfies condition (Γ0m ) (see Definition 3.33) with respect to D has property Wm
with respect to D.
In Lemma 3.41, the existence of m-ple sequences which satisfy conditions (Γ0m ),
(Γm,ν ) with respect to a domain D is established for any m, ν ≥ 1. (The proof there
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
51
is with respect to a closed interval I but remains valid, mutatis mutandis, for any
domain D.)
In what follows, we show that we can obtain Lemma 3.41 in such a way that
these sequences will be ‘minimal’.
Lemma 5.25. Let D be a domain, ν ≥ 1 an integer. Then there exist sequences
of the following two kinds.
(A) For each m ≥ 1, there is an m-ple sequence of distinct points of D
σ(m, ν) : zj1 ,j2 ,...,jm (1 ≤ j1 ≤ ν, j2 ≥ 1, . . . , jm ≥ 1)
satisfying condition (Γm,ν ) with respect to D and such that
(A.1) σ(m, ν) has exactly property Wm−1,ν with respect to D (as a point set),
(1)
(A.2) σ(m, ν) ∩ σ(m, ν)D = ∅, and
(1)
(2)
(A.3) σ(m, ν)D \ σ(m, ν)D = σ(m − 1, ν)
(m ≥ 2)
(B) For each m ≥ 1, there is an m-ple sequence of distinct points of D
σ(m) : zj1 ,j2 ,...,jm (j1 ≥ 1, . . . , jm ≥ 1)
satisfying condition (Γ0m ) with respect to D and such that
(B.1) σ(m) has exactly property Wm,1 with respect to D,
(1)
(B.2) σ(m) ∩ σ(m)D = ∅, and
(1)
(2)
(B.3) σ(m)D \ σ(m)D = σ(m − 1), m ≥ 2.
Proof.
(A) Take a disk ∆(z0 , R) compactly contained in D. We shall construct σ(m, ν) by
induction on m to be contained in ∆(z0 , R).
m = 1.
Let z1 , z2 , . . . , zν be distinct points of ∆(z0 , R). Then σ(1, ν) : z1 , z2 , . . . , zν certainly satisfies (A.1) - (A.3) for m = 1.
52
SHAHAR NEVO
m ≥ 2.
Suppose we have defined an (m−1)-ple sequence of distinct points from ∆(z0 , R),
σ(m − 1, ν) : zj1 ,j2 ,...,jm−1 which satisfies condition (Γm−1,ν ) with respect to D,
together with (A.1) - (A.3) for m − 1. By (A.2), for any j1 , . . . , jm−1 , there exists
some positive number rj1 ,...,jm−1 > 0 such that
(5.40)
∆0 (zj1 ,...,jm−1 , rj1 ,...,jm−1 ) ∩ σ(m − 1, ν) = ∅
and
(5.41)
∆(zj1 ,...,jm−1 , rj1 ,...,jm−1 ) ⊂ ∆(z0 , R).
According to (5.40), the collection
(5.42)
{∆(zj1 ,...,jm−1 , rj1 ,...,jm−1 /3) : 1 ≤ j1 ≤ ν, j2 ≥ 1, . . . , jm−1 ≥ 1}
consists of strongly disjoint disks (see the proof of Lemma 2.13).
For each j1 , . . . , jm−1 (1 ≤ j1 ≤ ν, j2 ≥ 1, . . . , jm−1 ≥ 1), construct a onedimensional sequence σj1 ,...,jm−1 (1) : zj1 ,...,jm−1 ,jm (jm ≥ 1) of distinct points of
∆(zj1 ,...,jm1 , rj1 ,...,jm−1 /3) that converges to zj1 ,...,jm−1 , i.e.,
(5.43)
zj1 ,...,jm−1 ,jm
→
jm →∞
zj1 ,...,jm−1 .
The required m-ple sequence σ(m, ν) is
σ(m, ν) : zj1 ,...,jm−1 ,jm (1 ≤ j1 ≤ ν, j2 ≥ 1, . . . , jm ≥ 1).
It remains to verify that conditions (A.1) - (A.3) are satisfied. First we prove that
(5.44)
(1)
(1)
σ(m, ν)D = σ(m − 1, ν) ∪ σ(m − 1, ν)D .
By (5.43) and the definition of σ(m, ν), it is clear that the right hand term of (5.44)
is contained in the left hand term.
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
53
(1)
We show the opposite containment. Let z0 ∈ σ(m, ν)D . There exists a sequence
of distinct points of σ(m, ν),
S = {an }∞
1 with an
(n)
an = zj (n) ,j (n) ,...,j (n) , where 1 ≤ j1
1
2
m
(n)
≤ ν, j2
≥
→
n→∞
(n)
1, . . . , jm
z0 . For each n ≥ 1,
≥ 1. Denote r(n) =
rj (n) ,...,j (n) /3, z(n) = zj (n) ,...,j (n) .
1
m−1
1
m−1
If, for a specific j1 , . . . , jm−1 , S contains infinitely many points of σj1 ,...,jm−1 (1),
then by (5.43), z0 = zj1 ,...,jm−1 , which belongs to σ(m − 1, ν). Otherwise, there
exists a subsequence of S such that any disk in (5.42) contains at most one element
of S. Without loss of generality, we may assume that this subsequence is S itself.
By (5.41) and (5.42), we have
(5.45)
r(n) → 0.
n→∞
(1)
∗
∗
Since {z(n)}∞
n=0 has a limit point z in D, it follows that z ∈ σ(m − 1)D . From
(5.45), we have |an − z(n)| → 0; hence an → z ∗ , and z0 = z ∗ . Now, condition
n→∞
(A.1) follows by (5.44), P.2.9 and the correctness of this condition for m − 1. The
construction of σ(m, ν) (see (5.42)) yields that (A.2) holds as well.
Now let m ≥ 2 be an integer. Then
(1)
(2)
(1)
(1)
(2)
˙
˙
σ(m, ν)D \σ(m, ν)D = (σ(m−1, ν)∪σ(m−1,
ν)D )\(σ(m−1, ν)D ∪σ(m−1,
ν)D ).
By the correctness of (A.2) for m − 1 and P.2.8, the right term is σ(m − 1, ν) and
(A.2.8) is proved.
The existence of an m-ple sequence that satisfies (B.1) - (B.3) is a consequence of
the existence of an (m+1)-ple sequence σ(m+1, 1) that satisfies condition (Γm+1,1 )
with respect to D and (A.1) - (A.3). Indeed, the bijection (of indices)
ωj1 ,...,jm = z1,j1 ,...,jm (j1 ≥ 1, . . . , jm ≥ 1)
converts an (m + 1)-ple sequence σ(m + 1, 1) : z1,j1 ,...,jm (j + 1 ≥ 1, . . . , jm ≥ 1)
that satisfies the conditions of case (A) into an m-ple sequence
54
SHAHAR NEVO
σ ∗ (m) : ωj1 ,...,jm (j1 ≥ 1, . . . , jm ≥ 1), which satisfies condition (Γ0m ) with respect
to D; while (B.1) - (B.3) for σ ∗ (m) follow from (A.1) - (A.3) for σ(m + 1, 1),
respectively. The proof of Lemma 5.25 is completed.
Remark 5.26. Let γ : z(t), a ≤ t ≤ b be a nontrivial curve contained in D.
Then the sequences σ(m, ν), σ(m) which satisfy (A.1) - (A.3) and (B.1) - (B.3),
respectively, can be chosen such that σ(m), σ(m, ν) ⊂ γ. Moreover, since γ is a
(j)
(j)
closed set of D, then σγ = σD for each j ≥ 0 (σ = σ(m, ν), σ(m)).
The next lemma insures that under certain conditions a given point set will be
the set of non C0 -points for a sequence of meromorphic functions.
(1)
Lemma 5.27. Let D be a domain and E ⊂ D a set. Suppose that D \ ED is a
domain and
(1)
(5.46)
E ∩ ED = ∅.
Then there exists a sequence S = {fN }∞
N =1 of meromorphic functions defined on D
χ
(1)
such that fN ⇒ f on D \ ED , where f is a nonconstant meromorphic function on
(1)
(1)
D \ ED ; and ED is precisely the set of non C0 -points of any subsequence of S.
Proof. By (5.46), E is at most enumerable. The case when E is finite is obvious,
so we treat the case where E is infinite, say E = {an }∞
n=1 . The function
f (z) = e1/z
(5.47)
is holomorphic in C \ {0} and has an essential singularity at z = 0. Actually,
(5.48)
f (z) =
∞
X
1/k!
k=0
zk
and the expansion (5.48) is absolutely and uniformly convergent in any domain of
the form
∆
Ar = C \ ∆(0, r), r > 0.
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
55
We have for z ∈ Ar , n ∈ N,
¯ n
¯
∞
¯X 1/k! ¯ X
1/k!
¯
¯
= e1/r .
¯
¯≤
k
¯
z ¯
rk
k=0
k=0
Define, for each N ∈ N,
N
n
X
1 X 1/k!
.
fN (z) =
2n
(z − an )k
n=1
(5.49)
k=0
Then S = {fN }∞
N =1 is a sequence of meromorphic functions on C (and so on D).
Let us examine which points of D are non C0 -points for S. We consider several
cases separately.
(1)
(1) z0 ∈
/ E ∪ ED .
Then there exists r0 > 0 such that ∆(z0 , r0 ) ⊂ D and ∆(z0 , r0 ) ∩ E = ∅. Then
for z ∈ ∆(z0 , r0 /2) and for n ≥ 1 we get
¯
¯
n
n
¯X
1/k! ¯¯ X 1/k!
¯
≤ e2/r0 .
¯
¯≤
¯
(z − an )k ¯
(r0 /2)k
(5.50)
k=0
The series
∞
P
n=1
1 2/r0
2n e
k=0
consists of positive terms and converges to e2/r0 ; hence by
(5.49), (5.50) and Weierstrass’s Theorem, S is a sequence of holomorphic functions
in ∆(z0 , r0 /2) that converges uniformly there to an analytic function. This implies
that z0 is a C0 -point of S.
(2) z0 ∈ E.
In this case, z0 = an0 , n0 ≥ 1 and for N ≥ n0 we have
(5.51)
n0
N
n
X
1 X
1/k!
1 X 1/k!
fN (z) = n0
+
.
n
k
2
(z − an0 )k
2
(z
−
a
)
n
n=1
k=0
n6=n0
k=0
Similarly to case (1), there exists r0 > 0 such that ∆0 (z0 , r0 ) ∩ E = ∅. Set
∆
hN (z) =
N
n
X
1 X 1/k!
;
n
k
2
(z
−
a
)
n
n=1
n6=n0
k=0
56
SHAHAR NEVO
as in case (1), we deduce that {hN } is a sequence of holomorphic functions in
∆(z0 , r0 /2) that converges there uniformly to a holomorphic function. The left
n0
P
1/k!
term in (5.51), gn0 (z) = 2n10
is a meromorphic function in ∆(z0 , r0 /2)
(z−an )k
0
k=0
(with a pole of order n0 at z0 ). Then fN = hN + gn0 ; and by Lemma 3.14, S is a
C0 -sequence in ∆(z0 , r0 /2). Since fN (z0 ) = ∞, N ≥ n0 , we get by case (1) that the
convergence is to a nonconstant meromorphic function.
(1)
(3) z0 ∈ ED .
In any neighborhood of z0 contained in D, there exist infinitely many points
of E, which may be arranged as a subsequence of E, {ank }∞
k=1 . For each k ≥ 1,
fN (ank ) = ∞ for N ≥ nk . Thus if some subsequence S ∗ of S converges χ-uniformly
on a neighborhood ∆(z0 , r0 ) of z0 , then the limit must be the constant ∞. But
(1)
since z0 ∈ ED , ∆(z0 , r0 ) contains a point z1 ∈ E, so by the result of case (2), S ∗
converges χ-uniformly on some disk ∆(z1 , r1 ) ⊂ ∆(z0 , r0 ) to a nonconstant (and
hence not identically ∞) meromorphic function, a contradiction. Thus z0 is a non
C0 -point of any subsequence of S. Together, (1), (2) and (3) imply the result of the
lemma.
(1)
Remark 5.28. By (5.46), D \ ED is an open set. This fact, together with the
properties of the sequence S = {fN }, shows that if one omits the condition that
(1)
D \ ED be a domain, it is still the case that S converges locally χ-uniformly on
(1)
any connected component of D \ ED .
Remark 5.29. Instead of f (z) = e1/z in (5.47), one can take any function holomorphic in C \ {0} with an essential singularity at z = 0.
We are now ready to confirm the sharpness of Theorems 5.20 and 5.22.
Corollary 5.30. Theorem 5.20 is sharp.
Proof. Let D be a domain and let Em = σ(m) be a point set in D having the
properties described in (B) of Lemma 5.25. According to (B.2) there and by Lemma
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
57
5.27, there exists a sequence S = {fN }∞
N =1 of meromorphic functions on D having
(1)
(Em )D as its (“irreducible”) set of non C0 -points.
(1) (m−1)
By (B.1) of Lemma 5.25, |((Em )D )D
| = 1. Hence by Theorem 3.22, as a
family, S is a Qm -normal family of exact order 1 in D (in particular, it is not a
Qm−1 -normal family in D). Theorem 5.20 insures that G = {c·fN : c ∈ C, N ∈ N}
is a Qm+1 -normal family in D. We show that G is not a Qm -normal family on D.
Let us define a sequence S ∗ = { fNN }∞
1 of functions of G and check which points
are the non C0 -points of S ∗ (and its subsequences).
(1)
(1) z0 ∈
/ Em ∪ (Em )D .
By case (1) in Lemma 5.27,
fN
N
⇒ 0 on some neighborhood of z0 , so z0 is a
C0 -point of S ∗ .
(2) z0 ∈ Em .
Then z0 = an0 , n0 ≥ 1. By (5.49)
fN
N
(z0 ) = ∞ for N ≥ n0 ; but, on the other
hand, according to case (1) of Lemma 5.27,
fN
N
(z) ⇒ 0 on ∆0 (z0 , r0 ). Hence S ∗ is
irreducible with respect to the non C0 -point z0 .
(1)
(3) z0 ∈ (Em )D .
As an accumulation point of non C0 -points, z0 is one as well. Together, (1),
(2) and (3) imply that the set of the non C0 -points, for any subsequence of S ∗ is
∆
(1)
(m)
E = Em ∪ (Em )D . By P.2.9 and (B.1) of Lemma 5.25, we get |ED | = 1; hence
S ∗ as a family is not a Qm -normal family on D, and so neither is G.
Note. It is easy to verify that G is a Qm+1 -normal family of exact order 1 in D.
Remark 5.31. Take a sequence of distinct points of D, bi → b0 , b0 ∈ ∂D.
i→∞
Construct a sequence of strongly disjoint disks ∆(bi , ri ), i ≥ 1, compactly contained
in D. Now, let Em,i ⊂ ∆(bi , ri ) (i ≥ 1) be as in (B) of Lemma 5.25 with
(m)
(5.52)
∗
Set Em
=
(Em,i )D
∞
S
i=1
= {ci }, ci ∈ ∆(bi , ri ).
Em,i . By the construction of each ∆(bi , ri ), Lemma 3.40, Remark
58
SHAHAR NEVO
(1)
∗
∗
∩ (Em
)D = ∅. According to
3.39 and (B.2) of Lemma 5.25, we conclude that Em
Lemma 5.27, we then have a sequence S1 = {gN }∞
N =1 of meromorphic functions on
(1)
∗
D such that for any subsequence of S1 , (Em
)D is precisely its set of non C0 -points.
By (5.52), Lemma 3.40 and Remark 3.39, we get
(m)
∗
(Em
)D
(5.53)
= {ci : i ≥ 1},
and therefore
(m+1)
∗
(Em
)D
(5.54)
= ∅.
(1)
∗
∗
As in Corollary 5.30, Em
∪ (Em
)D will be the set of non C0 -points for any
subsequence of S2 = { gNN }. By (5.53), (5.54) and Theorem 5.20, G(S1 ) is a Qm+1 normal family of infinite order on D. Similarly, given ν ≥ 1, one can take ν distinct
points in D, c1 , . . . , cν and get a family F which is a Qm+1 -normal family of exact
order ν in D.
Corollary 5.32. Theorem 5.22 is sharp.
We demonstrate this for every ` ≥ 1.
Proof. Let D be a domain, ` ≥ 1 an integer, N1 , . . . , N` ≥ 0 not all 0. DeP̀
note N0 =
Ni . Take ` strongly disjoint disks and compactly contained in D,
i=1
∆(z, r1 ), . . . , ∆(z` , r` ). For each 1 ≤ i ≤ ` with Ni > 0, construct as described in
Lemma 5.25 (A) an (m + 1)-ple sequence of distinct points σi (m + 1, Ni ) (denote
Em,i = σi (m + 1, Ni )) in ∆(zi , ri ), having exactly property Wm,Ni with respect to
∆(zi , ri ) and also the same with respect to D.
Now, as in Lemma 5.27, construct a sequence Si = {fN,i }∞
N =1 of meromorphic
(1)
functions on D such that (Em,i )D is its set of non C0 -points, with respect to each
(1) (m−1)
of which Si is irreducible. Set Fi = Si , 1 ≤ i ≤ `. We have |(Em,i )D
1 ≤ i ≤ `.
| = Ni ,
APPLICATIONS OF ZALCMAN’S LEMMA TO Qm -NORMAL FAMILIES
59
By the discussion at the beginning of this chapter, Fi is a Qm -normal family
of order at most Ni in D, as is any of its subsequences. Theorem 5.22 guarantees that G = G(F1 , . . . , F` ) is a Qm+1 -normal family of order at most N0 in D.
Define now a sequence of functions from G, hN = N1 (fˆN,1 + · · · + fˆN,` ) where
½
fN,i
Ni > 0
ˆ
fN,i =
As in Corollary 5.30, it follows that the set of non C0 0
Ni = 0
S̀
∗
∗ (1)
∗
=
Em,i . By the construction
)D , where Em
∪(Em
points of {hN } is exactly Em
i=1
Ni >0
∗
of ∆(zi , ri ), 1 ≤ i ≤ `, and the construction of Em,i , it follows that Em
has property
(1)
∗
∗
∪ (Em
)D , too.
Wm,N0 with respect to D; and by P.2.9 this is true for Em
Again, according to the preceding discussion, {hN } as a family is not a Qm+1 normal family of order at most N0 − 1 in D, and so neither is G.
Note. If Ni = 0 for each 1 ≤ i ≤ `, then each Fi is a Qm−1 -normal family on
D; and the result of Corollary 5.32 (or the result of Theorem 5.22) follows in this
case from Theorem 5.20 (since a Qm+1 -normal family of order zero is a Qm -normal
family).
Acknowledgment. The results of this paper are taken from the author’s doctoral
dissertation, written under the direction of Professor Lawrence Zalcman, at BarIlan University. Sincere thanks to Professor Zalcman for his warm encouragement,
valuable advice and friendly criticism.
60
SHAHAR NEVO
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Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
Current address: Department of Mathematics, Weizmann Institute of Science, Rehovot 76100,
Israel
E-mail address: [email protected]