Some Normality Criteria of Meromorphic Functions

Abstract

This paper studies some normality criteria for a family of meromorphic functions, which improve some results of Lahiri, Lu and Gu, as well as Charak and Rieppo.

1. Introduction and Main Results

Let be a nonconstant meromorphic function in the complex plane . We shall use the standard notations in Nevanlinna's value distribution theory of meromorphic functions such as , , and (see, e.g., [1, 2]). The notation is defined to be any quantity satisfying as possibly outside a set of of finite linear measure.

Let be a family of meromorphic functions on a domain . We say that is normal in if every sequence of functions contains either a subsequence which converges to a meromorphic function uniformly on each compact subset of or a subsequence which converges to uniformly on each compact subset of . (See [1, 3].)

The Bloch principle [3] is the hypothesis that a family of analytic (meromorphic) functions which have a common property in a domain will in general be a normal family if reduces an analytic (meromorphic) function in the open complex plane to a constant. Unfortunately the Bloch principle is not universally true. But it is also very difficult to find some counterexamples about the converse of the Bloch principle, and hence it is interesting to study the problem.

In 2005, Lahiri [4] proved the following criterion for the normality, and gave a counterexample to the converse of the Bloch principle by using the result.

Theorem A

Let be a family of meromorphic functions in a domain , and let , be two finite constants. Define

(1.1)

If there exists a positive number such that for every , one has whenever , then is normal.

In this direction, Lahiri and Dewan [5] as well as Xu and Zhang [6] proved the following result.

Theorem B.

Let be a family of meromorphic functions in a domain , and let , be two finite constants. Suppose that

(1.2)

where and are positive integers.

If for every

(i)all zeros of have multiplicity at least ,

(ii)there exists a positive number such that for every one has whenever ,

then is normal in so long as ; or and .

Here, we also give a counterexample to the converse of the Bloch principle by considering Theorem B, which is similar to an example in [7].

Example 1.1.

Let , then for all . Now we can see that

(1.3)

but Theorem B is true especially when is an empty set for every in the family.

In the following, we continue to study the normal family when and in Theorem B.

Theorem 1.2.

Let be a family of meromorphic functions in a domain , and , be two finite constants. Suppose that

(1.4)

where is a positive integer.

If for every

(i)all zeros of have multiplicity at least ,

(ii)there exists a positive number such that for every , one has whenever , then is normal in .

Corollary 1.3.

Let be a family of meromorphic functions in a domain , all of whose zeros have multiplicity at least , and let , be two finite constants. Suppose that , where is a positive integer. Then is normal in .

Recently, Lu and Gu [8] considered two related normal families.

Theorem C.

Let be a family of meromorphic functions in a domain ; all of whose zeros have multiplicity at least . Suppose that, for each , for , then is a normal family on , where is a nonzero finite complex number and is an integer number.

Theorem D.

Let be a family of meromorphic functions in a domain ; all of whose zeros have multiplicity at least , and all of whose poles are multiple. Suppose that, for each , for , then is a normal family on , where is a nonzero finite complex number and is an integer number.

In this paper, we give a simple proof and improve the above results.

Theorem 1.4.

Let be a family of meromorphic functions in a domain ; all of whose zeros have multiplicity at least . Suppose that, for each , for , then is a normal family on , where is a nonzero finite complex number and is an integer number.

In 2009, Charak and Rieppo [7] generalized Theorem A and obtained two normality criteria of Lahiri's type.

Theorem E.

Let be a family of meromorphic functions in a complex domain . Let such that . Let , , , be nonnegative integers such that , , and , and put

(1.5)

If there exists a positive constant such that for all whenever , then is a normal family.

Theorem F.

Let be a family of meromorphic functions in a complex domain . Let such that . Let , , , be nonnegative integers such that , and put

(1.6)

If there exists a positive constant such that for all whenever , then is a normal family.

Naturally, we ask whether the above results are still true when is replaced by in Theorems E and F. We obtain the following results.

Theorem 1.5.

Let be a family of meromorphic functions in a complex domain ; all of whose zeros have multiplicity at least . Let such that . Let , , , be nonnegative integers such that , , and (if , ), and put

(1.7)

If there exists a positive constant such that for all whenever , then is a normal family.

Theorem 1.6.

Let be a family of meromorphic functions in a complex domain ; all of whose zeros have multiplicity at least . Let such that . Let , , , be nonnegative integers such that , and put

(1.8)

If there exists a positive constant such that for all whenever , then is a normal family.

2. Some Lemmas

Lemma 2.1 (see [9]).

Let be a family of functions meromorphic on the unit disc, all of whose zeros have multiplicity at least , then if is not normal, there exist, for each

(a)a number

(b)points

(c)functions

(d)positive number such that locally uniformly, where is a nonconstant meromorphic on , all of whose zeros have multiplicity at least , such that

Here, as usual, is the spherical derivative.

Lemma 2.2.

Let be rational in the complex plane and positive integers. If has only zero with multiplicity at least , then takes on each nonzero value .

Proof.

In Lemma 6 of [7], the case of is proved. We just consider the case of by a different way which comes from [10].

If is a polynomial, obviously the conclusion holds. If is a nonpolynomial rational function, then we can set

(2.1)

where is a nonzero constant. Since has only zero with multiplicity at least , we find that

(2.2)

For convenience, we denote

(2.3)

Differentiating (2.1), we obtain

(2.4)

where is a polynomial with .

Suppose that has no zero, then we can write

(2.5)

where is a nonzero constant.

Differentiating (2.5), we obtain

(2.6)

where is a polynomial of the form , in which , , are constants.

Comparing (2.1) and (2.5), we can obtain . From (2.4) and (2.6), we have

(2.7)

It is a contradiction with and . This proves the lemma.

Lemma 2.3 (see [11]).

Let be a transcendental meromorphic function all of whose zeros have multiplicity at least , then assumes every finite nonzero value infinitely often, where if , and if .

Remark 2.4.

The lemma was first proved by Wang as if and if in [12]. Recently, the result is improved by [11].

Lemma 2.5.

Let be a meromorphic function all of whose zeros have multiplicity with at least in the complex plane, then must have zeros for any constant .

Proof.

If is rational, then by Lemma 2.2 the conclusion holds.

If is transcendental, supposing that has no zeros, then by Lemma 2.3, we can get a contradiction. This completes the proof of the lemma.

Lemma 2.6.

Let be meromorphic in the complex plane, and let be a constant, for any positive integer ; if , then is a constant.

Proof.

If is not a constant, and from , we know that , then with the identity , we can get that, if ,

(2.8)

and with being a set of values of finite linear measure. It is a contradiction.

Lemma 2.7 (see [13]).

Let be a transcendental meromorphic function, and let , be two integers. Then for any nonzero value , the function has infinitely many zeros.

Lemma 2.8 (see [14]).

Let be a transcendental meromorphic function, and let be an integer. Then for any nonzero value , the function has infinitely many zeros.

Lemma 2.9.

Let be a family of meromorphic functions in a complex domain . Let such that . Let , , , be nonnegative integers such that , and put

(2.9)

has a finite zero.

Proof.

The algebraic complex equation

(2.10)

has always a nonzero solution; say . By [14, Corollary ] or [15], Lemmas 2.2, 2.7, and 2.8, the meromorphic function cannot avoid it and thus there exists such that .

By assumption, we may write and . Consequently

(2.11)

and we complete the proof of the lemma.

Remark 2.10.

If , we need when by Lemma 2.3. We can get a similar result.

3. Proof of Theorems

Proof of Theorem 1.2.

Let . Suppose that is not normal at . Then by Lemma 2.1, there exist a sequence of functions , a sequence of complex numbers and such that

(3.1)

converges spherically and locally uniformly to a nonconstant meromorphic function in . Also the zeros of are of multiplicity at least . So . Applying Lemma 2.5 to the function , we know that

(3.2)

for some . Clearly is neither a zero nor a pole of . So in some neighborhood of , converges uniformly to . Now in some neighborhood of we see that is the uniform limit of

(3.3)

By (3.2) and Hurwitz's theorem, there exists a sequence such that for all large values of

(3.4)

Therefore for all large values of , it follows from the given condition that .

Since is not a pole of , there exists a positive number such that in some neighborhood of we get .

Since converges uniformly to in some neighborhood of , we get for all large values of and for all in that neighborhood of

(3.5)

Since , we get for all large values of

(3.6)

which is a contradiction. This proves the theorem.

Proof of Theorem 1.4.

If is not normal at . We assume without loss of generality that , then by Lemma 2.1, for , there exist a sequence of points , a sequence of positive numbers and a sequence of functions of such that

(3.7)

spherically uniformly on compact subsets of , where is a nonconstant meromorphic function on ; all of whose zeros have multiplicity at least. By (3.7),

(3.8)

It follows that or by Hurwitz's theorem. From Lemma 2.6, we obtain that . By Lemma 2.5, we get a contradiction. This completes the proof of the theorem.

Proof of Theorem 1.5.

Suppose that is not normal at . Then by Lemma 2.1, for , there exist a sequence of functions , a sequence of complex number , and such that

(3.9)

converges spherically and locally uniformly to a nonconstant meromorphic function in . Also the zeros of are of multiplicity at least . So . By Lemmas 2.2, 2.3, 2.7, and 2.8, we get

(3.10)

for some . Clearly is neither a zero nor a pole of . So in some neighborhood of , converges uniformly to . Now in some neighborhood of we have

(3.11)

where is replaced by and , .

Taking and using the assumption , we see that

(3.12)

is the uniform limit of

(3.13)

in some neighborhood of . By (3.10) and Hurwitz's theorem, there exists a sequence such that for all large values of

(3.14)

Hence, for all large , it follows from the given condition that

(3.15)

In the following, we can get a contradiction in a similar way with the proof of the last part of Theorem 1.2. This completes the proof of the theorem.

Proof of Theorem 1.6.

Suppose that is not normal at . Then by Lemma 2.1, for , there exist a sequence of functions , a sequence of complex numbers , and such that

(3.16)

converges spherically and locally uniformly to a nonconstant meromorphic function in . Also the zeros of are of multiplicity at least . So . By Lemma 2.9, we get

(3.17)

for some .

In the following, we can get a contradiction in a similar way with the proof of the last part of Theorem 1.5. This completes the proof of the theorem.