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Normality of meromorphic functions and differential polynomials share values
Advances in Difference Equations volume 2014, Article number: 120 (2014)
Abstract
In this paper, we discuss the normality of meromorphic functions which involves differential polynomial sharing values. We obtain two results: Let k be a positive integer, b (≠0) be a complex number, and be a polynomial with degree at least 2, and be a differential polynomial with . Let ℱ be a family of meromorphic functions defined in D, all of whose zeros have multiplicity at least . If has at least two distinct zeros, has at most one distinct zero in D for each , then ℱ is normal in D. If has at least two distinct zeros and for each pair of functions f and g in ℱ, and share b in D, then ℱ is normal in D, too. Two examples show that a condition in our results is necessary and our results improve Fang and Hong’s, and Zeng’s corresponding results.
MSC:30D35, 34A05.
1 Introduction and main results
Let D be a domain in ℂ, and ℱ be a family of meromorphic functions defined in the domain D. ℱ is said to be normal in D, in the sense of Montel, if for every sequence contains a subsequence such that converges spherically uniformly on compact subsets of D.
ℱ is said to be normal at a point if there exists a neighborhood of in which ℱ is normal. It is well known that ℱ is normal in a domain D if and only if it is normal at each of its points.
Let f and g be meromorphic functions defined in a domain D, and a and b be complex numbers. If whenever , we write . If and , we write ; If , we say that f and g share the value a in D.
Let be non-negative integers and one of them nonzero at least, and set
is called the differential monomial of f, the degree of and the weight of .
Let be differential monomials of f, and let be analytic in D. Set
is called a differential polynomial of f, the degree of and the weight of . If , then is called a homogeneous differential polynomial of degree t. Set
The following theorem was proved by Fang and Hong [1].
Theorem 1.1 [1]
Let ℱ be a family of meromorphic functions defined in D, k and q (≥2) be two positive integers, and be a differential polynomial with . If the zeros of are of multiplicity at least and for each , then ℱ is normal in D.
It is natural to ask whether the condition in Theorem 1.1 that can be relaxed. In this paper we investigate this problem and prove the following result.
Theorem 1.2 Let ℱ be a family of meromorphic functions defined in D, k be a positive integer, let be a polynomial with degree at least 2, and be a differential polynomial with . If has at least two distinct zeros, the zeros of are of multiplicity at least and has at most one distinct zero in D for each , then ℱ is normal in D.
By the idea of shared values, very recently, Zeng [2] proved the following theorem.
Theorem 1.3 [2]
Let k and q (≥2) be two positive integers, be a complex number, and let be a differential polynomial with . Let ℱ be a family of meromorphic functions defined in D, all of whose zeros have multiplicity at least . If for each pair of functions f and g in ℱ, and share b in D, then ℱ is normal in D.
It is natural to ask whether Theorem 1.3 can be improved. In this paper, we study this problem and obtain the following theorem.
Theorem 1.4 Let k be a positive integer, b (≠0) be a complex number, be a polynomial, and let be a differential polynomial with . Let ℱ be a family of meromorphic functions defined in D, all of whose zeros have multiplicity at least . If has at least two distinct zeros and for each pair of functions f and g in ℱ, and share b in D, then ℱ is normal in D.
Example 1.1 Let , and . Then
We can see that , has only one distinct zero in D for each function in ℱ, and and share 1 in D for each pair of functions and in ℱ. On the other hand, , , for any . This implies that the family ℱ fails to be equicontinuous at 0, and thus ℱ is not normal at 0.
Remark 1.5 This example shows that to have at least two distinct zeros ( to have at least two distinct zeros) is necessary in Theorem 1.2 (Theorem 1.4).
Example 1.2 Let , and . Then
We can see that if , and for each pair of functions and in ℱ, and share 1. Therefore, ℱ is normal in D by our Theorem 1.4.
Remark 1.6 From this example we also know that has only one solution in D for each in ℱ. The case of shared b includes the case of ≠b, that is to say, Theorem 1.2 is a generalization of Theorem 1.1 and Theorem 1.4 is a generalization of Theorem 1.3.
2 Preliminary lemmas
In order to prove our results, we need the following lemmas. The first one is Zalcman’s Theorem.
Lemma 2.1 [3]
Let , let ℱ be a family of functions meromorphic on the unit disc Δ, all of whose zeros have multiplicity at least k, and suppose that there exists such that whenever . Then if ℱ is not normal at , there exist, for each ,
-
(a)
functions ;
-
(b)
points , , and
-
(c)
positive numbers
such that locally uniformly with respect to the spherical metric, where g is a nonconstant meromorphic function on ℂ, all of whose zeros have multiplicity at least k, such that . In particular, g has order at most 2.
Here denotes the spherical derivative
Lemma 2.2 [4]
Let , where are constants with , and q and p are two co-prime polynomials, neither of which vanishes identically, with , and let k be a positive integer and b a nonzero complex number. If , and the zeros of f all have multiplicity at least , then
where c and d are distinct complex numbers.
Lemma 2.3 Let g be a nonconstant meromorphic function, and be a polynomial. If has at least two distinct zeros and all zeros of g have multiplicity at least , then has at least two distinct zeros.
Proof Case 1. If has only one zero α, then , and , .
Suppose that () are two distinct zeros of . Without loss of generality, we may assume that , then for .
Firstly, we will show that is not a transcendental meromorphic function. By Nevanlinna Theory, we have
Hence, we get , it follows that is not a transcendental meromorphic function.
If is a polynomial, then
where Q is a polynomial such that and the degree of Q is not less than 1. Thus there exists an , such that . That is to say, there exists an , such that , which is a contradiction.
Therefore is rational but not a polynomial. Under the conditions of Lemma 2.2 on the rational functions g, we have
where c and d are distinct complex numbers, , and then
where is a complex number.
Hence has distinct zeros, which contradicts having only the zero .
Case 2. . Since , by Nevanlinna Theory once more, we have
where () are two distinct zeros of . Hence, we get , and it follows that is a constant. This together with the fact that the zeros of g have multiplicity at least shows that g is a constant, a contradiction. □
3 Proofs of theorems
Proof of Theorem 1.2 We show that ℱ is normal in D. Otherwise, there exists at least one point such that ℱ is not normal at . Then by Lemma 2.1, we can find a subsequence of ℱ, which we may denote by , , and such that converges local uniformly with respect to the spherical metric to a nonconstant meromorphic function g on ℂ, all of whose zeros have multiplicity at least .
It is easily seen that
Noting that all () are analytic on D implies
for sufficiently large n, we deduce from that
converges uniformly to 0 on ℂ.
Thus we find that
converges local uniformly to on ℂ.
Hence, by Hurwitz’s Theorem, the hypothesis of the theorem, and Lemma 2.3, we see that or has at least two distinct zeros on ℂ.
Case 1. If on ℂ.
Then by having at least two distinct zeros, we find that has at least two distinct zeros except for at most one complex number c. Therefore Lemma 2.3 tells us that has zero for at least two distinct c except that is a constant function. This is also impossible.
Case 2. If has at least two distinct zeros on ℂ.
Then, without out loss generality, let and be two distinct zeros of , and choose δ (>0) small enough such that , where , and . By Hurwitz’s Theorem, there exist two sequences of points and such that for sufficiently large n
Hence, we have and for sufficiently large n. Thus each has two distinct zeros for large enough n, which contradicts our hypothesis.
This contradiction shows that ℱ is normal in D and hence Theorem 1.2 is proved. □
Proof of Theorem 1.4 Suppose that ℱ is a family meromorphic and not normal in D. Then there exists at least one point such that ℱ is not normal at the point . By Lemma 2.1, there exist:
-
(a)
functions ;
-
(b)
points , , and
-
(c)
positive numbers
such that locally uniformly with respect to the spherical metric, where g is a nonconstant meromorphic function on ℂ, all of whose zeros have multiplicity at least .
It is easily seen that
Noting that all () are analytic on D implies
for sufficiently large n, we deduce from that
converges uniformly to 0 on ℂ.
Thus we know that
converges local uniformly to on ℂ.
Take , we consider two cases.
Case 1. .
Then there exists a positive number such that
for all z in , by sharing condition.
Hence, by Hurwitz’s Theorem, the hypothesis of the theorem, and Lemma 2.3, we see that or on ℂ.
If , then by Lemma 2.3 and the hypothesis of the theorem, we see that has at least two distinct zeros except that is a constant function, a contradiction.
If , the same arguments of the proof of Case 1 in the proof of Theorem 1.3 implies that it does not hold.
Case 2. .
Next we consider two subcases.
Subcase 2.1. for all z in . From the discussion above, we have in ℂ. This is impossible.
Subcase 2.2. There exists a such that in . By the supposition and discussion above, this means that
for , and
We claim that has just a unique zero.
Suppose that there exist two distinct zeros and , choose δ (>0) small enough such that , where and .
By Hurwitz’s Theorem, there exist points , , such that for sufficiently large j
By the assumption that and share b in D for each pair of functions f and g in ℱ, we see that for any integer m
We fix m and note that , if . From this we deduce
Since
if , and
noting that the zeros of
have no accumulation point, for sufficiently large n, we have
Hence
This contradicts the fact that , , and . So has just a unique zero ignoring multiplicity. This contradicts the conclusion of Lemma 2.3 that has at least two distinct zeros.
Hence ℱ is normal at , and then ℱ is normal in D. The proof of Theorem 1.4 is complete. □
References
Fang ML, Hong W: Some results on normal family of meromorphic functions. Bull. Malays. Math. Soc. 2000, 23: 143–151.
Zeng, CP: Normality and shared values. Indian J. Math. Reprint
Pang XC, Zalcman L: Normal families and shared values. Bull. Lond. Math. Soc. 2000, 32: 325–331. 10.1112/S002460939900644X
Wang YF, Fang ML: Picard values and normal families of meromorphic functions with zeros. Acta Math. Sin. New Ser. 1998, 14: 17–26. 10.1007/BF02563879
Acknowledgements
This work was supported by the Visiting Academic Sponsor Project of Department of Mathematics and Statistics at Curtin University of Technology (200001807894), the first author would like to express his hearty thanks to Curtin University of Technology for providing him with very comfortable research environment. This work was completed with the support with the NSF of China (11271090) and NSF of Guangdong Province (S2012010010121). The authors wish also specially to thank the managing editor and referees for their very helpful comments and useful suggestions.
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JL and ZH carried out the design of the study and performed the analysis. WY and ZL participated in its design and coordination. All authors read and approved the final manuscript.
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Yuan, W., Lai, J., Huang, Z. et al. Normality of meromorphic functions and differential polynomials share values. Adv Differ Equ 2014, 120 (2014). https://doi.org/10.1186/1687-1847-2014-120
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DOI: https://doi.org/10.1186/1687-1847-2014-120