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Some Normality Criteria of Meromorphic Functions

  • Junfeng Xu
  • Wensheng Cao
Open Access
Research Article

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.

Keywords

Rational Function Complex Number Compact Subset Meromorphic Function Nonnegative Integer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction and Main Results

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

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

The Bloch principle [3] is the hypothesis that a family of analytic (meromorphic) functions which have a common property Open image in new window in a domain Open image in new window will in general be a normal family if Open image in new window reduces an analytic (meromorphic) function in the open complex plane Open image in new window 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 Open image in new window be a family of meromorphic functions in a domain Open image in new window , and let Open image in new window , Open image in new window be two finite constants. Define

If there exists a positive number Open image in new window such that for every Open image in new window , one has Open image in new window whenever Open image in new window , then Open image in new window is normal.

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

Theorem B.

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

where Open image in new window and Open image in new window are positive integers.

If for every Open image in new window

(i)all zeros of Open image in new window have multiplicity at least Open image in new window ,

(ii)there exists a positive number Open image in new window such that for every Open image in new window one has Open image in new window whenever Open image in new window ,

then Open image in new window is normal in Open image in new window so long as Open image in new window Open image in new window ; or Open image in new window Open image in new window and Open image in new window .

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.

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

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

Theorem 1.2.

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

where Open image in new window is a positive integer.

If for every Open image in new window

(i)all zeros of Open image in new window have multiplicity at least Open image in new window ,

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

Corollary 1.3.

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

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

Theorem C.

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

Theorem D.

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

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

Theorem 1.4.

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

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

Theorem E.

If there exists a positive constant Open image in new window such that Open image in new window for all Open image in new window whenever Open image in new window , then Open image in new window is a normal family.

Theorem F.

If there exists a positive constant Open image in new window such that Open image in new window for all Open image in new window whenever Open image in new window , then Open image in new window is a normal family.

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

Theorem 1.5.

If there exists a positive constant Open image in new window such that Open image in new window for all Open image in new window whenever Open image in new window , then Open image in new window is a normal family.

Theorem 1.6.

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

If there exists a positive constant Open image in new window such that Open image in new window for all Open image in new window whenever Open image in new window , then Open image in new window is a normal family.

2. Some Lemmas

Lemma 2.1 (see [9]).

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

(a)a number Open image in new window

(b)points Open image in new window Open image in new window

(c)functions Open image in new window

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

Here, as usual, Open image in new window is the spherical derivative.

Lemma 2.2.

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

Proof.

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

If Open image in new window is a polynomial, obviously the conclusion holds. If Open image in new window is a nonpolynomial rational function, then we can set
where Open image in new window is a nonzero constant. Since Open image in new window has only zero with multiplicity at least Open image in new window , we find that
For convenience, we denote
Differentiating (2.1), we obtain

where Open image in new window is a polynomial with Open image in new window .

Suppose that Open image in new window has no zero, then we can write

where Open image in new window is a nonzero constant.

Differentiating (2.5), we obtain

where Open image in new window is a polynomial of the form Open image in new window , in which Open image in new window , Open image in new window , Open image in new window are constants.

Comparing (2.1) and (2.5), we can obtain Open image in new window . From (2.4) and (2.6), we have

It is a contradiction with Open image in new window and Open image in new window . This proves the lemma.

Lemma 2.3 (see [11]).

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

Remark 2.4.

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

Lemma 2.5.

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

Proof.

If Open image in new window is rational, then by Lemma 2.2 the conclusion holds.

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

Lemma 2.6.

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

Proof.

If Open image in new window is not a constant, and from Open image in new window , we know that Open image in new window , then with the identity Open image in new window , we can get that, if Open image in new window ,

and Open image in new window with Open image in new window being a set of Open image in new window values of finite linear measure. It is a contradiction.

Lemma 2.7 (see [13]).

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

Lemma 2.8 (see [14]).

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

Lemma 2.9.

has a finite zero.

Proof.

The algebraic complex equation

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

By assumption, we may write Open image in new window and Open image in new window . Consequently

and we complete the proof of the lemma.

Remark 2.10.

If Open image in new window , we need Open image in new window when Open image in new window by Lemma 2.3. We can get a similar result.

3. Proof of Theorems

Proof of Theorem 1.2.

Let Open image in new window . Suppose that Open image in new window is not normal at Open image in new window . Then by Lemma 2.1, there exist a sequence of functions Open image in new window , a sequence of complex numbers Open image in new window and Open image in new window such that
converges spherically and locally uniformly to a nonconstant meromorphic function Open image in new window in Open image in new window . Also the zeros of Open image in new window are of multiplicity at least Open image in new window . So Open image in new window . Applying Lemma 2.5 to the function Open image in new window , we know that
for some Open image in new window . Clearly Open image in new window is neither a zero nor a pole of Open image in new window . So in some neighborhood of Open image in new window , Open image in new window converges uniformly to Open image in new window . Now in some neighborhood of Open image in new window we see that Open image in new window is the uniform limit of
By (3.2) and Hurwitz's theorem, there exists a sequence Open image in new window such that for all large values of Open image in new window

Therefore for all large values of Open image in new window , it follows from the given condition that Open image in new window .

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

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

which is a contradiction. This proves the theorem.

Proof of Theorem 1.4.

If Open image in new window is not normal at Open image in new window . We assume without loss of generality that Open image in new window , then by Lemma 2.1, for Open image in new window , there exist a sequence of points Open image in new window , a sequence of positive numbers Open image in new window and a sequence of functions Open image in new window of Open image in new window such that
spherically uniformly on compact subsets of Open image in new window , where Open image in new window is a nonconstant meromorphic function on Open image in new window ; all of whose zeros have multiplicity Open image in new window at least. By (3.7),

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

Proof of Theorem 1.5.

Suppose that Open image in new window is not normal at Open image in new window . Then by Lemma 2.1, for Open image in new window , there exist a sequence of functions Open image in new window , a sequence of complex number Open image in new window , and Open image in new window such that
converges spherically and locally uniformly to a nonconstant meromorphic function Open image in new window in Open image in new window . Also the zeros of Open image in new window are of multiplicity at least Open image in new window . So Open image in new window . By Lemmas 2.2, 2.3, 2.7, and 2.8, we get
for some Open image in new window . Clearly Open image in new window is neither a zero nor a pole of Open image in new window . So in some neighborhood of Open image in new window , Open image in new window converges uniformly to Open image in new window . Now in some neighborhood of Open image in new window we have

where Open image in new window is replaced by Open image in new window and Open image in new window , Open image in new window .

Taking Open image in new window and using the assumption Open image in new window , we see that
is the uniform limit of
in some neighborhood of Open image in new window . By (3.10) and Hurwitz's theorem, there exists a sequence Open image in new window such that for all large values of Open image in new window
Hence, for all large Open image in new window , it follows from the given condition that

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 Open image in new window is not normal at Open image in new window . Then by Lemma 2.1, for Open image in new window , there exist a sequence of functions Open image in new window , a sequence of complex numbers Open image in new window , and Open image in new window such that
converges spherically and locally uniformly to a nonconstant meromorphic function Open image in new window in Open image in new window . Also the zeros of Open image in new window are of multiplicity at least Open image in new window . So Open image in new window . By Lemma 2.9, we get

for some Open image in new window .

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.

Notes

Acknowledgments

The authors would like to thank Professor Lahiri for supplying the electronic file of the paper [4]. The authors were supported by NSF of China (No. 10771121, No. 10801107), NSF of Guangdong Province (No. 9452902001003278, No. 8452902001000043), and Department of Education of Guangdong (No. LYM08097).

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Copyright information

© J. Xu andW. Cao. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of MathematicsWuyi UniversityJiangmenChina

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