Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 21, 1996, 89–100

NORMAL FAMILIES, ORDERS OF ZEROS, AND OMITTED VALUES

Chen Huaihui and Peter Lappan

Nanjing Normal University, Department of Mathematics Nanjing, Jiangsu 210024, Peoples Republic of China

Michigan State University, Department of Mathematics East Lansing, Michigan 48824, U.S.A.

Abstract. For k a positive integer, Ω a region in the complex plane, and α a complex number, let ℳ_{k}(Ω, α) denote the collection of all functions f meromorphic in Ω such that each zero of f − α has multiplicity at least k. Let D denote the unit disk in the complex plane. We give three conditions, each of which is sufficient for a subset ℱ of ℳ_{k}(D, α) to be a normal family. These conditions are: (1) for each compact subset K of D and for some β > 0 there exists a constant M_{K}(β) (depending on both K and β) such that, for each f ∈ ℱ ,

{z ∈ K : |f(z)| < β} ⊂ {z ∈ K : |f^{(k)}(z)| ≤ M_{K}(β)};

(2) for λ > 2/k and for each compact subset K of D there exists a constant C_{K,λ} (depending on both K and λ) such that

∫∫_{{z∈K:|f(z)|<1}} |f^{(k)}(z)|^{λ} dx dy < C_{K,λ}

for each f ∈ ℱ ; (3) for each compact subset K of D there is a constant M_{k}(K) such that the product of the spherical derivatives of two or three consecutive derivatives of f, up to the derivative of order k − 1, is uniformly bounded by M_{k}(K) for z ∈ K, f ∈ ℱ . These results are suggested by and build on previous results of the first author and Gu Yong-Xing, Rauno Aulaskari and the second author, and the second author alone.

1. Preliminaries

Let C denote the complex plane and let D = {z ∈ C : |z| < 1} be the unit disk. A family ℱ of functions meromorphic in a region Ω ⊂ C is called a normal family if each sequence in ℱ contains a subsequence which converges uniformly on each compact subset of Ω. (Here, we allow the subsequence to converge uniformly on each compact subset to the function which is identically ∞ in the sense that the subsequence of reciprocals converges uniformly on each compact subset to the function which is identically zero.) It is well known that a family ℱ of meromorphic functions is a normal family if and only if it is locally a normal family, that is, it is a normal family on a neighborhood of each point of Ω.

1991 Mathematics Subject Classification: Primary 30D45.

Thus, all theorems about whether a family of functions is a normal family or not can be formulated in terms of families defined on disks. Although some of our results are formulated in general regions, the proofs are all “localized” to disks.

Let f^{#}(z) = |f′(z)|/(1 + |f(z)|^{2}) denote the spherical derivative of f. A well known result due to F. Marty [8] says that a family ℱ of functions meromorphic in D is a normal family if and only if for each compact subset K of D there exists a constant M_{K} such that f^{#}(z) ≤ M_{K} for each f ∈ ℱ and each z ∈ K. A single function f meromorphic in D is called a normal function if sup{(1 − |z|^{2})f^{#}(z) : z ∈ D} < ∞ (see [6]).

We will make repeated use of the following results, due, repectively, to J. Lohwater and Ch. Pommerenke [7], L. Zalcman [11], X. Pang [9], and H.H. Chen and Y.X. Gu [3].

Theorem LP [7, Theorem 1]. If a function f meromorphic in D is not a normal function, then there exist sequences {z_{n}} and {ϱ_{n}} such that z_{n} ∈ D, |z_{n}| → 1, ϱ_{n} > 0, ϱ_{n}/(1 − |z_{n}|) → 0, and the sequence {g_{n}(t) = f(z_{n} + ϱ_{n}t)} converges uniformly on each compact subset of C to a non-constant meromorphic function g.

Theorem Z [11, Lemma, p. 814]. Let ℱ be a family of functions meromorphic on D. If ℱ is not a normal family, then there exist a compact subset K of D, a sequence of functions {f_{n}} in ℱ, a sequence of points {z_{n}} in K, and a sequence of positive real numbers {ϱ_{n}}, where ϱ_{n} → 0, such that the sequence of functions {g_{n}(t) = f_{n}(z_{n} + ϱ_{n}t)} converges uniformly on each compact subset of C to a non-constant meromorphic function g.

Theorem P [9]. Let ℱ be a family of functions meromorphic on D. If ℱ is not a normal family and −1 < λ < 1, then there exist a compact subset K of D, a sequence of functions {f_{n}} in ℱ, a sequence of points {z_{n}} in K, and a sequence of positive real numbers {ϱ_{n}}, where ϱ_{n} → 0, such that the sequence of functions {g_{n}(t) = (ϱ_{n})^{λ}f_{n}(z_{n} + ϱ_{n}t)} converges uniformly on each compact subset of C to a non-constant meromorphic function g.

Theorem CG [3, Theorem 2, p. 677]. Let k be a positive number and let ℱ be a family of meromorphic functions in D with the property that each function in ℱ has only zeros of degree at least k. If ℱ is not a normal family in each neighborhood of the point z_{0} ∈ D, then, for any positive number α with α < k, there exist a sequence of points {z_{n}} in D, a sequence {ϱ_{n}} of positive numbers, and a sequence of functions {f_{n}} in ℱ such that z_{n} → z_{0} , ϱ_{n} → 0, and the sequence of functions {(ϱ_{n})^{−α}f_{n}(z_{n} +ϱ_{n}t)} converges uniformly on each compact subset of C to a non-constant meromorphic function g.

If Ω is an open set, k is a positive integer, and α ∈ C, we denote by ℳ_{k}(Ω, α) the collection of all functions f meromorphic on Ω such that each zero of f − α has multiplicity at least k. We will make use of the following three properties of the class ℳ_{k}(Ω, α):

(a) if f omits the value α on Ω then f ∈ ℳ_{k}(Ω, α) for each positive integer k,

(b) if Ω is a region on the complex plane and f ∈ M_{k}(Ω, α) then either f^{(k)} , the k th derivative of f, is not identically zero or f ≡ α, and

(c) if {f_{n}} ⊂ ℳ_{k}(Ω, α) is a sequence such that {f_{n}} converges uniformly on each compact subset of the region Ω to a function f, then f ∈ ℳ_{k}(Ω, α).

In Section 2 below, Theorem 1 gives a slight improvement of a result of the first author and Gu Yong-Xing [3, Corollary 1], using methods different from those used in [3]. Theorem 2 is a modification of Theorem 1 to normal functions. In Section 3, our Theorems 3 and 4 give some improvements to results of R. Aulaskari and the second author [1, 2]. Finally, in Section 4, our Theorems 5, 6, and 7 give some extensions to results by the second author [5].

2. Sufficient conditions for normal families and functions

We begin with a slight improvement of a result of the first author and Gu Yong-Xing [3, Corollary 1]. The method of proof here is quite different from that appearing in [3].

Theorem 1. Let ℱ be a family of functions meromorphic in D, let k be a positive integer, let α ∈ C, and let ℱ ⊂ ℳ_{k}(D, α). If, for each compact subset K in D and for some β > 0 there exists a constant M_{K}(β) such that

{z ∈ K : |f(z)| < β} ⊂ {z ∈ K : |f^{(k)}(z)| < M_{K}(β)} for each f ∈ ℱ ,

then ℱ is a normal family in D.

Proof. Suppose ℱ is not a normal family in D. By Theorem Z there exist a sequence of functions {f_{n}} in ℱ, a number r, 0 < r < 1, a sequence of points {z_{n}} in D satisfying |z_{n}| < r for each n, and a sequence of positive numbers {ϱ_{n}} for which ϱ_{n} → 0 and the sequence {g_{n}(t) = f_{n}(z_{n} + ϱ_{n}t)} converges uniformly on each compact subset of C to a non-constant meromorphic function g(t). The hypotheses of the theorem guarantee that all the zeros of g −α, if any, are of order at least k. This means that g^{(k)} is not identically zero.

Let K = \{z ∈ D : |z| ≤ \frac{1}{2}(1 + r)\}. If t ∈ C and |g(t)| < β, then we have

|g^{(k)}(t)| = lim_{n→∞}(ϱ_{n})^{k}|f^{(k)}(z_{n} + ϱ_{n}t)| < lim_{n→∞}(ϱ_{n})^{k}M_{K}(β) → 0,

and hence g^{(k)}(t) = 0. But the set {t ∈ C : |g(t)| < β} is a non-empty open subset of C, and so our reasoning in the previous sentence leads to the conclusion that g^{(k)} ≡ 0, in violation of the last sentence of the previous paragraph. Thus, the assumption that ℱ is not a normal family leads to a contradiction, and the theorem is proved.

A similar result giving a sufficient condition for a function to be a normal function is the following.

Theorem 2. Let f be a function meromorphic in D, let k be a positive integer, let α ∈ C, and let f ∈ ℳ_{k}(D, α). If there exists a number β > 0 and a constant M_{β} such that

{z ∈ D : |f(z)| < β} ⊂ {z ∈ D : |f^{(k)}(z)|(1 − |z|)^{k} < M_{β}},

then f is a normal function.

Proof. Suppose that f is not a normal function. By Theorem LP there exist a sequence of points {z_{n}} in D, and a sequence of positive numbers {ϱ_{n}}, where ϱ_{n}/(1 − |z_{n}|) → 0, such that the sequence {g_{n}(t) = f(z_{n} + ϱ_{n}t)} converges uniformly on each compact subset of C to a non-constant meromorphic function g(t). The hypotheses on f require that all the zeros of g − α are of order at least k. Thus g^{(k)}(t) is not identically zero.

Now suppose that t is fixed and |g(t)| < β. This means that |g_{n}(t)| < β for n sufficiently large, and thus,

|g_{n}^{(k)}(t)| = (ϱ_{n})^{k}|f^{(k)}(z_{n} + ϱ_{n}t)|

= \frac{(ϱ_{n})^{k}}{(1 − |z_{n} + ϱ_{n}t|)^{k}} |f^{(k)}(z_{n} + ϱ_{n}t)|(1 − |z_{n} + ϱ_{n}t|)^{k}

≤ \frac{(ϱ_{n})^{k}}{(1 − |z_{n}|)^{k}} \frac{(1 − |z_{n}|)^{k}}{(1 − |z_{n} + ϱ_{n}t|)^{k}} |f^{(k)}(z_{n} + ϱ_{n}t)|(1 − |z_{n} + ϱ_{n}t|)^{k}

≤ (\frac{ϱ_{n}}{1 − |z_{n}|})^{k}(\frac{1 − |z_{n}|}{1 − |z_{n} + ϱ_{n}t|})^{k}M_{β} → 0.

It follows that g^{(k)}(t) = 0 whenever |g(t)| < β, and this means that g^{(k)}(t) ≡ 0, contradicting the last sentence of the previous paragraph. Thus, it is not possible for f to be a non-normal function, and this proves the theorem.

3. Integral criteria

Our next result is suggested by a result due to R. Aulaskari and the second author [1, Theorem 1].

Theorem AL [1, Theorem 1, p. 29]. Let ℱ be a family of functions meromorphic in D. Then ℱ is a normal family if and only if, for each R, 0 < R < 1, and each λ > 2, there exists a constant C_{λ,R} such that

∫∫_{|z|≤R}(f^{#}(z))^{λ} dx dy < C_{λ,R} for each f ∈ ℱ .

Theorem AL requires that the exponent λ be greater than 2. By imposing the condition that ℱ ⊂ ℳ_{k}(D, α) for some fixed k and α, we can change this restriction on λ and, at the same time, perform the integration over a potentially much smaller set, as follows.

Theorem 3. Let k be a positive number and let α ∈ C. Let ℱ be a family of functions meromorphic on D such that ℱ ⊂ ℳ_{k}(D, α). Let λ be a real number such that λ > 2/k. If for each compact subset K of D there exists a constant C_{K,λ} such that ∫∫_{K(f)} |f^{(k)}(z)|^{λ} dx dy < C_{K,λ} for each f ∈ ℱ, where K(f) = {z ∈ K : |f(z)| < 1}, then ℱ is a normal family.

We remark that the set K(f) varies with the function f.

Proof. Suppose that ℱ is not a normal family. By Theorem Z there exists a sequence {f_{n}} in ℱ, a compact subset K_{0} of D together with a sequence {z_{n}} of points in K_{0} , and a sequence of positive numbers {ϱ_{n}} where ϱ_{n} → 0 such that the sequence of functions {g_{n}(t) = f_{n}(z_{n} + ϱ_{n}t)} converges uniformly on each compact subset of C to a non-constant meromorphic function g(t). Fix s > 0 and let z = z_{n} + ϱ_{n}t. For |t| ≤ s and n sufficiently large, we have that z ∈ K_{0} , so we may assume in what follows that z ∈ K_{0} . Also, we have g_{n}^{(k)}(t) = (ϱ_{n})^{k}f_{n}^{(k)}(z_{n} + ϱ_{n}t). Using the notation z = x + iy = z_{n} + ϱ_{n}t and t = u + iv, we have dx dy = (ϱ_{n})^{2} du dv, so that if

G_{s,n} = {t ∈ C : |g_{n}(t)| < 1, |t| ≤ s}

and

F_{s,n} = {z = z_{n} + ϱ_{n}t : |t| ≤ s, |f_{n}(z)| < 1},

we have

∫∫_{G_{s,n}} |g_{n}^{(k)}(t)|^{λ} du dv = ∫∫_{G_{s,n}} (ϱ_{n})^{λk}|f_{n}^{(k)}(z_{n} + ϱ_{n}t)|^{λ} du dv

= ∫∫_{F_{s,n}} (ϱ_{n})^{λk−2}|f_{n}^{(k)}(z)|^{λ} dx dy

≤ (ϱ_{n})^{λk−2} ∫∫_{K_{0}(f_{n})} |f_{n}^{(k)}(z)|^{λ} dx dy ≤ (ϱ_{n})^{λk−2}C_{K_{0},λ} → 0,

since ϱ_{n} → 0 and λk − 2 > 0. But, we also have that

∫∫_{G_{s,n}} |g_{n}^{(k)}(t)|^{λ} du dv → ∫∫_{K_{s}} |g^{(k)}(t)|^{λ} du dv

where K_{s} = {t ∈ C : |g(t)| < 1, |t| ≤ s}. Thus, the integral ∫∫_{K_{s}} |g^{(k)}(t)|^{λ} du dv must be zero for each choice of s, and it follows that g^{(k)} ≡ 0. But the hypotheses about the family ℱ guarantee that g^{(k)} cannot be identically zero. It follows that ℱ must be a normal family.

Next, we give a version of the result of Theorem 3 for a normal function. This is an extension of both [1, Theorem 2, p. 32] and [2, Theorem, p. 438].

Theorem 4. Let k be a positive integer, let α ∈ C, and let f be a function meromorphic in D. If f ∈ ℳ_{k}(D, α), if λ > 2/k, and if there exists a constant C_{λ} such that

∫∫_{F (f)} |f^{(k)}(z)|^{λ}(1 − |z|)^{λk−2} dx dy < C_{λ},

where F (f) = {z ∈ D : |f(z)| < 1}, then f is a normal function.

Proof. Suppose that f is not a normal function. By Theorem LP there exists a sequence {z_{n}} of points in D, and a sequence of positive numbers {ϱ_{n}} where ϱ_{n}/(1 − |z_{n}|) → 0 such that the sequence of functions {g_{n}(t) = f(z_{n} + ϱ_{n}t)} converges uniformly on each compact subset of C to a non-constant meromorphic function g(t). Fix s > 0 and let z = z_{n} +ϱ_{n}t. For |t| ≤ s and n sufficiently large, we will have that z ∈ D, so we may assume in what follows that z ∈ D. Also, we have g_{n}^{(k)}(t) = (ϱ_{n})^{k}f^{(k)}(z_{n} + ϱ_{n}t). Using the notation z = x + iy = z_{n} + ϱ_{n}t and t = u + iv, we have dx dy = (ϱ_{n})^{2} du dv, so that if G_{s,n} = {t ∈ C : |g_{n}(t)| < 1, |t| ≤ s} and F_{s,n} = {z = z_{n} + ϱ_{n}t : |t| ≤ s, |f(z)| < 1}, we have

∫∫_{G_{s,n}} |g_{n}^{(k)}(t)|^{λ} du dv = ∫∫_{G_{s,n}} (ϱ_{n})^{λk}|f_{n}^{(k)}(z_{n} + ϱ_{n}t)|^{λ} du dv

= ∫∫_{G_{s,n}} (\frac{ϱ_{n}}{1 − |z_{n} + ϱ_{n}t|})^{λk−2}|f^{(k)}(z_{n} + ϱ_{n}t)|^{λ}

× (1 − |z_{n} + ϱ_{n}t|)^{λk−2}(ϱ_{n})^{2} du dv

≤ ∫∫_{G_{s,n}} (a_{n})^{λk−2}|f^{(k)}(z_{n} + ϱ_{n}t)|^{λ}(1 − |z_{n} + ϱ_{n}t|)^{λk−2}(ϱ_{n})^{2} du dv

= (a_{n})^{λk−2} ∫∫_{F_{s,n}} |f_{n}^{(k)}(z)|^{λ}(1 − |z|)^{λk−2} dx dy ≤ (a_{n})^{λk−2}C_{λ} → 0,

where

a_{n} = sup\{\frac{ϱ_{n}}{1 − |z_{n} + ϱ_{n}t|} : |t| ≤ s\}

(and a_{n} → 0 since ϱ_{n}/(1 − |z_{n}|) → 0 and (1 − |z_{n}|)/(1 − |z_{n} + ϱ_{n}t|) → 1 for |t| ≤ s). But, we also have that

∫∫_{G_{s,n}} |g_{n}^{(k)}(t)|^{λ} du dv → ∫∫_{K_{s}} |g^{(k)}(t)|^{λ} du dv

where K_{s} = {t ∈ C : |g(t)| < 1, |t| ≤ s}. Thus, the integral ∫∫_{K_{s}} |g^{(k)}(t)|^{λ} du dv must be zero for each choice of s, and it follows that g^{(k)} ≡ 0. But the hypotheses about the function f guarantee that g^{(k)} cannot be identically zero. It follows that f must be a normal function.

4. Extensions of the spherical derivative

For a function meromorphic in D, define a product of spherical derivatives by

P_{n}(f)(z) = ∏_{j=0}^{n−1}(f^{(j)})^{#}(z).

In [5], the second author proved that if ℱ is a normal family of meromorphic functions in D, then for each positive integer n and each compact subset K of D, there exists a constant M_{n}(K) such that P_{n}(f)(z) ≤ M_{n}(K) for each f ∈ ℱ and each z ∈ K. (The case n = 2 was suggested by Yamashita [10].) The converse of this result is not true as the example ℱ = {nz} shows, since P_{2}(f) ≡ 0 for each f ∈ ℱ but ℱ is not a normal family. However, if we can guarantee that, for each function f ∈ ℱ, P_{n}(f)(z) is not identically zero, then the converse has at least a chance of being valid. We show here that such a converse is valid for n ≤ 3, but is not valid for n ≥ 4.

Theorem 5. Let α be a complex number and let k be a positive integer. Suppose that ℱ is a family of functions meromorphic in D such that ℱ ⊂ ℳ_{k}(D, α). Further, suppose that for each compact subset K of D there exists a constant M_{k}(K) such that P_{k}(f)(z) ≤ M_{k}(K) for each f ∈ ℱ and each z ∈ D. If k ≤ 3, then ℱ is a normal family. If k ≥ 4, then there are examples for which ℱ is not a normal family.

Proof. Let k = 2 and suppose that ℱ is not a normal family in D. By Theorem Z, there exists a sequence of functions {f_{n}} in ℱ, a real number r, where 0 < r < 1, a sequence of points {z_{n}} in D such that |z_{n}| < r for each n, and a sequence of positive real numbers {ϱ_{n}}, where ϱ_{n} → 0, such that the sequence of functions {g_{n}(t) = f_{n}(z_{n}+ϱ_{n}t)} converges uniformly on each compact subset of C to a non-constant meromorphic function g(t). The hypotheses imply that g ∈ ℳ_{2}(C, α) and that g′′ is not identically zero.

Fix s > 0. Since ϱ_{n} → 0, we have that

\{w = z_{n} + ϱ_{n}t : |t| ≤ s\} ⊂ K_{0} = \{z ∈ D : |z| ≤ \frac{1}{2}(1 + r)\}

for each n sufficiently large. Then

P_{2}(g)(t) = lim_{n→∞} P_{2}(g_{n})(t) = lim_{n→∞} \frac{ϱ_{n}|f′_{n}(z_{n} + ϱ_{n}t)|}{1 + |f_{n}(z_{n} + ϱ_{n}t)|^{2}} · \frac{(ϱ_{n})^{2}|f′′_{n}(z_{n} + ϱ_{n}t)|}{1 + |ϱ_{n}f′_{n}(z_{n} + ϱ_{n}t)|^{2}}

= lim_{n→∞} ϱ_{n}P_{2}(f_{n})(z_{n} + ϱ_{n}t)\frac{(ϱ_{n})^{2}(1 + |f′_{n}(z_{n} + ϱ_{n}t)|^{2})}{1 + |ϱ_{n}f′_{n}(z_{n} + ϱ_{n}t)|^{2}} = 0

because ϱ_{n} → 0 implies that, for n sufficiently large,

\frac{(ϱ_{n})^{2}(1 + |f′_{n}(z_{n} + ϱ_{n}t)|^{2})}{1 + |ϱ_{n}f′_{n}(z_{n} + ϱ_{n}t)|^{2}} = \frac{(ϱ_{n})^{2} + (ϱ_{n})^{2}|f′_{n}(z_{n} + ϱ_{n}t)|^{2}}{1 + |ϱ_{n}f′_{n}(z_{n} + ϱ_{n}t)|^{2}} < 1,

and P_{2}(f_{n})(z_{n} + ϱ_{n}t) ≤ M_{2}(K_{0}), by hypothesis. But this calculation says that P_{2}(g) ≡ 0, which implies g′′ ≡ 0, so we have contradicted the last sentence of the previous paragraph. Thus, we must have that ℱ is a normal function when k = 2.

Now let k = 3, and let 0 < λ < 1. If ℱ is not a normal family on D, then by Theorem P there exists a sequence of functions {f_{n}} in ℱ, a sequence of points {z_{n}} in D such that z_{n} → z_{0} ∈ D, and a sequence {ϱ_{n}} of positive real numbers with ϱ_{n} → 0 such that the sequence of functions {g_{n}(t) = (ϱ_{n})^{−λ}f_{n}(z_{n} + ϱ_{n}t)} converges uniformly on each compact subset of the complex plane to a non-constant meromorphic function g(t). As in the previous case, g ∈ ℳ_{3}(C, α) and g^{(3)} is not identically zero on C.

Fix t_{0} on the complex plane such that g(t_{0}) is finite. Let r be such that |z_{0}| < r < 1 and fix s > |t_{0}|. Since ϱ_{n} → 0, we have that

K_{n,s} = \{w = z_{n} + ϱ_{n}t : |t| ≤ s\} ⊂ K_{0} = \{z ∈ D : |z| ≤ \frac{1}{2}(1 + r)\}

for n sufficiently large. Then, in some neighborhood of t_{0} , we have that the first three derivatives of g are all finite, g^{(3)} is not identically zero, and f_{n}(z_{n} + ϱ_{n}t) = (ϱ_{n})^{λ}g_{n}(t) → 0, since ϱ → 0 and g_{n}(t) converges uniformly on compact subsets to g(t). It follows that {F_{n}(t) = f_{n}(z_{n} + ϱ_{n}t)} converges uniformly to zero in a neighborhood of t_{0} . Now, letting u_{n} = z_{n} + ϱ_{n}t so that g_{n}(t) = (ϱ_{n})^{−λ}f_{n}(u_{n}), we have

\begin{aligned}P_{3}(g_{n})(t) & = (ϱ_{n})^{6−3λ}P_{3}(f_{n})(u_{n}) × \frac{1 + |f_{n}(u_{n})|^{2}}{1 + |(ϱ_{n})^{−λ}f_{n}(u_{n})|^{2}} \\ & × \frac{1 + |f′_{n}(u_{n})|^{2}}{1 + |(ϱ_{n})^{1−λ}f′_{n}(u_{n})|^{2}} × \frac{1 + |f′′_{n}(u_{n})|^{2}}{1 + |(ϱ_{n})^{2−λ}f′′_{n}(u_{n})|^{2}} . \end{aligned}

Of these quantities, we have that P_{3}(f_{n}(u_{n})) ≤ M_{3}(K_{0}) by hypothesis, and {F_{n}(t) = f_{n}(u_{n})} converges to zero in a neighborhood of t_{0} . It follows that

H_{n}(t) = \frac{1 + |f_{n}(u_{n})|^{2}}{1 + |(ϱ_{n})^{−λ}f_{n}(u_{n})|^{2}}

is uniformly bounded in a closed neighborhood of t_{0} . Thus, in a neighborhood of t_{0} ,

P_{3}(g_{n}(t)) ≤ C(ϱ_{n})^{6−3λ} \frac{1 + |f′_{n}(u_{n})|^{2}}{1 + |(ϱ_{n})^{1−λ}f′_{n}(u_{n})|^{2}} · \frac{1 + |f′′_{n}(u_{n})|^{2}}{1 + |(ϱ_{n})^{2−λ}f′′_{n}(u_{n})|^{2}} ,

where C is a constant not greater than M_{3}(K_{0}). But

\begin{aligned}(ϱ_{n})^{6−3λ} & \frac{1 + |f′_{n}(u_{n})|^{2}}{1 + |(ϱ_{n})^{1−λ}f′_{n}(u_{n})|^{2}} · \frac{1 + |f′′_{n}(u_{n})|^{2}}{1 + |(ϱ_{n})^{2−λ}f′′_{n}(u_{n})|^{2}} \\ & = (ϱ_{n})^{λ} \frac{(ϱ_{n})^{2−2λ}(1 + |f′_{n}(u_{n})|^{2})}{1 + |(ϱ_{n})^{1−λ}f′_{n}(u_{n})|^{2}} · \frac{(ϱ_{n})^{4−2λ}(1 + |f′′_{n}(u_{n})|^{2})}{1 + |(ϱ_{n})^{2−λ}f′′_{n}(u_{n})|^{2}} → 0 \end{aligned}

since ϱ_{n} → 0 and all of λ, 2 − 2λ, and 4 − 2λ are all positive, which means that each of the last two factors is bounded when t is confined to a neighborhood of t_{0} . Thus, we conclude that P_{3}(g_{n})(t) → P_{3}(g)(t) ≡ 0 on a neighborhood of t_{0} , which means that g is a constant function on the complex plane, in violation of the supposition that g is a non-constant function. It follows that ℱ must be a normal family.

To give a counterexample for the cases k ≥ 4, we can use the family ℱ = {h_{n}(z) = e^{nz}}, which is not a normal family. Each function in ℱ omits the value zero. If f_{n}(z) = e^{nz}, we claim that P_{k}(f_{n}) is bounded for each k ≥ 4. First suppose k ≠ 5. Let T_{k} denote the k th triangular number, that is, T = \frac{1}{2}k(k + 1), let q_{k} = [\frac{1}{2}k], where [·] denotes the greatest integer function, and let N_{k} = {2, 4, …, 2k − 2}. Then there exists a subset S_{k} ⊂ N_{k}, where S_{k} contains q_{k} elements, such that ∑_{j∈S_{k}} j is equal to either T_{k} or T_{k} + 1, whichever is an even number. Then we can write P_{k}(f_{n}) in the form

∏_{2j∈S_{k}} \frac{|n^{j}e^{nz}|^{2}}{1 + |n^{j}e^{nz}|^{2}} × ∏_{2j∈(N_{k}−S_{k})} \frac{1}{1 + |n^{j}e^{nz}|^{2}} × \frac{a_{k}}{1 + |e^{nz}|^{2}} ,

where a_{k} = |e^{nz}| if T_{k} is even and k is odd, a_{k} = 1/n if k is even and T_{k} is odd, a_{k} = 1 if both T_{k} and k are even, and a_{k} = |e^{nz}|/n if both T_{k} and k are odd. Thus, in the product, each of the k factors is less than 1, so it follows that P_{k}(f_{n}) is less than 1 whenever k ≥ 4, k ≠ 5. (The set S_{k} does not exist for k = 5.) To cover the case k = 5, we write P_{5}(f_{n}) in the form

∏_{j=3}^{4} \frac{|n^{j}e^{nz}|^{2}}{1 + |n^{j}e^{nz}|^{2}} × ∏_{{j=0 \atop j≠1}}^{2} \frac{1}{1 + |n^{j}e^{nz}|^{2}} × \frac{|ne^{nz}|}{1 + |ne^{nz}|^{2}} .

Each of the factors indicated is less than 1, so we have shown that P_{5}(f_{n}) is less than 1 for each n. This shows the counter-example. We note that for this counter-example, the functions f_{n} have no zeros. For a counter-example where the functions have zeros, let 𝒢 = {nz^{k}}. The reasoning for this counter-example is similar to that for the family ℱ = {e^{nz}}.

We note that there are no results similar to the results of Theorem 5 for normal functions, as the example in [5, Theorem 3] shows.

The technique used to prove the case k = 3 in Theorem 5 yields a generalization involving three consecutive factors from the product P_{k}(f)(z) for k ≥ 3, as follows.

Theorem 6. Let α be a complex number, let ν ∈ {2, 3}, let k be a positive integer with k ≥ ν, and let p be a non-negative integer such that 0 ≤ p ≤ k − ν. Suppose that ℱ is a family of functions meromorphic in a domain D such that ℱ ⊂ ℳ_{k}(D, α). Further, suppose that for each compact subset K of D there exists a constant M_{k}(K) such that

∏_{j=p}^{p+ν−1}(f^{(j)})^{#}(z) ≤ M_{k}(K)

for each f ∈ ℱ and z ∈ K. Then ℱ is a normal family on D.

We remark that the number of factors of the product under consideration is exactly ν.

Proof. Assume that ℱ is not a normal family on D. Let λ be a real number such that 0 < λ < 1. According to Theorem CG, there exist a sequence of functions {f_{n}} in ℱ, a sequence of points {z_{n}} in D with z_{n} → z_{0} ∈ D, and a sequence of positive numbers {ϱ_{n}} with ϱ_{n} → 0, such that the sequence {g_{n}(t) = (ϱ_{n})^{−p−λ}f_{n}(z_{n} +ϱ_{n}t)} converges uniformly on each compact subset of C to a non-constant meromorphic function g. Using this sequence {g_{n}(t)} in place of the ones used in the proof of Theorem 5, the argument used in the proof of Theorem 5 yield the desired conclusions in the two cases ν = 2 and ν = 3.

We can also prove a result of a related but somewhat different character.

Theorem 7. Let ℱ be a family of functions meromorphic on D, and let k be a positive integer greater than 1. Suppose both

(1) ℱ ⊂ ℳ_{k}(D, 0), and

(2) for each compact subset K of D, there exist a number δ > 0 and a positive number M such that |f^{(k−1)}(z)f^{(k)}(z)| ≤ M whenever ∑_{j=0}^{k−1} |f^{(j)}(z)| ≤ δ, z ∈ K, and f ∈ ℱ.

Then ℱ is a normal family on D.

Proof. Let 0 < β < \frac{1}{2} . If ℱ is not a normal family in a neighborhood of the point z_{0} , then, by Theorem CG, there exist sequences {f_{n}} of functions in ℱ, {z_{n}} of points in D, and {ϱ_{n}} of positive numbers such that ϱ_{n} → 0, z_{n} → z_{0} , and the sequence {g_{n}(t) = (ϱ_{n})^{1−k−β}f(z_{n} + ϱ_{n}t)} converges uniformly on each compact subset of the complex plane to a non-constant function g(t) meromorphic in the complex plane. By (1), g^{(k)} cannot be identically zero. Also, we may assume that ϱ_{n} < 1 for each n, and that, given a number s > 0 there exists a compact set K_{0} of D such that z_{n} + ϱ_{n}t ∈ K_{0} whenever |t| ≤ s. Then, we have

g_{n}^{(j)}(t) = (ϱ_{n})^{1−k−β+j}f_{n}^{(j)}(z_{n} + ϱ_{n}t), 0 ≤ j ≤ k − 1

and hence |g_{n}^{(j)}(t)| ≥ |f_{n}^{(j)}(z_{n} + ϱ_{n}t)|, 0 ≤ j ≤ k − 1. Let δ > 0 and M be the constants given by (2) corresponding to the compact set K_{0} . Then, for |t| ≤ s, ∑_{j=0}^{k−1} |g_{n}^{(j)}(t)| ≤ δ implies that ∑_{j=0}^{k−1} |f_{n}^{(j)}(z_{n} + ϱ_{n}t)| ≤ δ, and thus,

|f_{n}^{(k−1)}(z_{n} + ϱ_{n}t)f_{n}^{(k)}(z_{n} + ϱ_{n}t)| ≤ M.

Thus, if |t| ≤ s and ∑_{j=0}^{k−1} |g_{n}^{(j)}(t)| ≤ δ, then

|g_{n}^{(k−1)}(t)g_{n}^{(k)}(t)| = (ϱ_{n})^{1−2β}|f_{n}^{(k−1)}(z_{n} + ϱ_{n}t)f^{(k)}(z_{n} + ϱ_{n}t)| ≤ (ϱ_{n})^{1−2β}M → 0

as n → ∞. Hence it follows that g must satisfy on K_{0} a “strong version of (2)”, that is, ∑_{j=0}^{k−1} |g^{(j)}(t)| ≤ \frac{1}{2}δ implies that |g^{(k−1)}(t)g^{(k)}(t)| = 0. (This is actually independent of the choice of s, since the s considered above was arbitrary. Also, K_{0} can be any compact subset of D having z_{0} as an interior point.) Thus, if the hypothesis of this “strong version of (2)” ever occurs on an open set—as would happen, for example, if g ever assumed the value zero (because of condition (1))—then we would have that g^{(k)} ≡ 0 on an open set, which contradicts the implication built into condition (1) that the k th derivative of g cannot be identically zero (since g ∈ ℳ_{k}(D, 0) and g is not identically zero). Thus, to avoid this inconsistency, we must have that ∑_{j=0}^{k−1} |g^{(j)}(t)| ≥ \frac{1}{2}δ for all t ∈ C. Then g never assumes the value zero, and, for t ∈ C,

∑_{j=0}^{k−1}|\frac{g^{(j)}(t)}{g(t)}| ≥ \frac{δ}{2|g(t)|}

which means that

\frac{1}{|g(t)|} ≤ \frac{2}{δ}(1 + ∑_{j=1}^{k−1}|\frac{g^{(j)}(t)}{g(t)}|)

and hence m(r, 1/g) ≤ ∑_{j=1}^{k−1} m(r, g^{(j)}/g) + O(1) = o(T (r, g)) where this last equality, an extension of Nevalinna’s “Lemma on the logarithmic derivative”, is valid outside a set of r with finite measure (see for example, [4, Theorem 3.11, pp. 67–68]). Thus, outside this set, we have

T (r, g) = O(1) + T (r, \frac{1}{g}) = O(1) + m(r, \frac{1}{g}) = o(T (r, g))

and this last is not possible, since g is a non-constant function. This completes the proof.

We remark that the boundedness of the product (f^{(k)})^{#}(z)(f^{(k−1)})^{#}(z) implies (2).

100 Chen Huaihui and Peter Lappan

Finally, we remark that we cannot replace the product of the two factors f^{(k−1)}(z)f^{(k)}(z) by the product of the three factors f^{(k−2)}(z)f^{(k−1)}(z)f^{(k)}(z) in (2) of Theorem 7. For example, if we let f_{n}(z) = nz^{3} , then

|f_{n}(z)| + |f′_{n}(z)| + |f′′_{n}(z)| = n|z|^{3} + 3n|z|^{2} + 6n|z|.

If this sum is less than some δ > 0, then 6n|z| is also less than δ, and this means that

|f′_{n}(z)f′′_{n}(z)f_{n}^{(3)}(z)| = 108n^{3}|z|^{3} ≤ \frac{1}{2}(6n|z|)^{3} ≤ \frac{1}{2}δ^{3},

which implies that both (1) and the version of (2) using three factors are valid, but the family {f_{n}(z)} is not a normal family on the unit disk D.

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Received 29 June 1994