Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 312324 | https://doi.org/10.1155/2012/312324

Wei Chen, Yingying Zhang, Jiwen Zeng, Honggen Tian, "Normal Criterion Concerning Shared Values", Journal of Applied Mathematics, vol. 2012, Article ID 312324, 7 pages, 2012. https://doi.org/10.1155/2012/312324

Normal Criterion Concerning Shared Values

Academic Editor: Yansheng Liu
Received12 Jun 2012
Accepted21 Aug 2012
Published06 Sep 2012

Abstract

We study normal criterion of meromorphic functions shared values, we obtain the following. Let 𝐹 be a family of meromorphic functions in a domain 𝐷, such that function 𝑓∈𝐹 has zeros of multiplicity at least 2, there exists nonzero complex numbers 𝑏𝑓,𝑐𝑓 depending on 𝑓 satisfying (i)𝑏𝑓/𝑐𝑓 is a constant;  (ii)min{ğœŽ(0,𝑏𝑓),ğœŽ(0,𝑐𝑓),ğœŽ(𝑏𝑓,𝑐𝑓)≥𝑚} for some 𝑚>0;  (iii)(1/𝑐𝑓𝑘−1)(ğ‘“î…ž)𝑘(𝑧)+𝑓(𝑧)≠𝑏𝑘𝑓/𝑐𝑓𝑘−1 or (1/𝑐𝑓𝑘−1)(ğ‘“î…ž)𝑘(𝑧)+𝑓(𝑧)=𝑏𝑘𝑓/𝑐𝑓𝑘−1⇒𝑓(𝑧)=𝑏𝑓, then 𝐹 is normal. These results improve some earlier previous results.

1. Introduction and Main Results

We use 𝐶 to denote the open complex plane, 𝐶(=𝐶∪{∞}) to denote the extended complex plane and 𝐷 to denote a domain in 𝐶. A family 𝐹 of meromorphic functions defined in 𝐷⊂𝐶 is said to be normal, if for any sequence {𝑓𝑛}⊂𝐹 contains a subsequence which converges spherically, and locally, uniformly in 𝐷 to a meromorphic function or ∞. Clearly 𝐹 is said to be normal in 𝐷 if and only if it is normal at every point of 𝐷 see [1].

Let 𝐷 be a domain in 𝐶. For 𝑓 meromorphic on 𝐶 and ğ‘Žâˆˆğ¶, set 𝐸𝑓(ğ‘Ž)=𝑓−1({ğ‘Ž})∩𝐷={𝑧∈𝐷∶𝑓(𝑧)=ğ‘Ž}.(1.1)

Two meromorphic functions 𝑓 and 𝑔 on 𝐷 are said to share the value ğ‘Ž if 𝐸𝑓(ğ‘Ž)=𝐸𝑔(ğ‘Ž). Let ğ‘Ž and 𝑏 be complex numbers. If 𝑔(𝑧)=𝑏 whenever 𝑓(𝑧)=ğ‘Ž, we write 𝑓(𝑧)=ğ‘Žâ‡’ğ‘”(𝑧)=𝑏.(1.2) If 𝑓(𝑧)=ğ‘Žâ‡’ğ‘”(𝑧)=𝑏 and 𝑔(𝑧)=𝑏⇒𝑓(𝑧)=ğ‘Ž, we write 𝑓(𝑧)=ğ‘ŽâŸºğ‘”(𝑧)=𝑏.(1.3)

According to Bloch’s principle [2], every condition which reduces a meromorphic function in the plane 𝐶 to ğ‘Ž constant forces a family of meromorphic functions in ğ‘Ž domain 𝐷 normal. Although the principle is false in general (see [3]), many authors proved normality criterion for families of meromorphic functions by starting from Liouville-Picard type theorem (see [4]). It is also more interesting to find normality criteria from the point of view of shared values. In this area, Schwick [5] first proved an interesting result that a family of meromorphic functions in a domain is normal if in which every function shares three distinct finite complex numbers with its first derivative. And later, more results about normality criteria concerning shared values have emerged [6–9]. In recent years, this subject has attracted the attention of many researchers worldwide.

In this paper, we use ğœŽ(𝑥,𝑦) to denote the spherical distance between 𝑥 and 𝑦 and the definition of the spherical distance can be found in [10].

In 2008, Fang and Zalcman [11] proved the following results.

Theorem 1.1 (see [11]). Let 𝑓 be a transcendental function. Let ğ‘Ž(≠0) and 𝑏 be complex numbers, and let 𝑛(≥2),𝑘 be positive integers, then 𝑓+ğ‘Ž(ğ‘“î…ž)𝑛 assumes every value 𝑏∈𝐶 infinitely often.

Theorem 1.2 (see [11]). Let 𝐹 be a transcendental function. Let ğ‘Ž(≠0) and 𝑏 be complex numbers, and let 𝑛(≥2),𝑘 be positive integers. If for every 𝑓∈𝐹 has multiple zeros, and 𝑓+ğ‘Ž(ğ‘“î…ž)𝑛≠𝑏, then 𝐹 is normal in 𝐷.

In 2009, Xu et al. [12] proved the following results.

Theorem 1.3 (see [12]). Let 𝑓 be a transcendental function. Let ğ‘Ž(≠0) and let 𝑏 be complex numbers, and 𝑛,𝑘 be positive integers, which satisfy 𝑛≥𝑘+1, then 𝑓+ğ‘Ž(𝑓(𝑘))𝑛 assumes each value 𝑏∈𝐶 infinitely often.

Theorem 1.4 (see [12]). Let 𝑓 be a transcendental function. Let ğ‘Ž(≠0) and 𝑏 be complex numbers, and let 𝑛,𝑘 be positive integers, which satisfy 𝑛≥𝑘+1. If for every 𝑓∈𝐹 has only zeros of multiplicity at least 𝑘+1, and satisfies 𝑓+ğ‘Ž(𝑓(𝑘))𝑛≠𝑏, then 𝐹 is normal in 𝐷.

In Theorems 1.2 and 1.4, the constants are the same for each 𝑓∈𝐹. Now we will prove the condition for the constants be the same can be relaxed to some extent.

Theorem A. Let 𝐹 be a family of meromorphic functions in the unit disc Δ, and 𝑘 be a positive integer and 𝑘≥3. For every 𝑓∈𝐹, such that all zeros of 𝑓 have multiplicity at least 2, there exist finite nonzero complex numbers 𝑏𝑓,𝑐𝑓 depending on 𝑓 satisfying that(i)𝑏𝑓/𝑐𝑓 is a constant; (ii)min{ğœŽ(0,𝑏𝑓),ğœŽ(0,𝑐𝑓),ğœŽ(𝑏𝑓,𝑐𝑓)≥𝑚} for some 𝑚>0; (iii)(1/𝑐𝑓𝑘−1)(ğ‘“î…ž)𝑘(𝑧)+𝑓(𝑧)≠𝑏𝑘𝑓/𝑐𝑓𝑘−1.Then 𝐹 is normal in Δ.

Theorem B. Let 𝐹 be a family of meromorphic functions in the unit disc Δ, and 𝑘(≥3) be a positive integer. For every 𝑓∈𝐹, such that all zeros of 𝑓 have multiplicity at least 2, there exist finite nonzero complex numbers 𝑏𝑓,𝑐𝑓 depending on 𝑓 satisfying that(i)𝑏𝑓/𝑐𝑓is a constant; (ii)min{ğœŽ(0,𝑏𝑓),ğœŽ(0,𝑐𝑓),ğœŽ(𝑏𝑓,𝑐𝑓)≥𝑚} for some 𝑚>0; (iii)(1/𝑐𝑓𝑘−1)(ğ‘“î…ž)𝑘(𝑧)+𝑓(𝑧)=𝑏𝑘𝑓/𝑐𝑓𝑘−1⇒𝑓(𝑧)=𝑏𝑓.Then 𝐹 is normal in Δ.

2. Some Lemmas

In order to prove our theorems, we require the following results.

Lemma 2.1 (see [7]). Let 𝐹 be a family of meromorphic functions in a domain 𝐷, and 𝑘 be a positive integer, such that each function 𝑓∈𝐹 has only zeros of multiplicity at least 𝑘, and suppose that there exists 𝐴≥1 such that |𝑓(𝑘)(𝑧)|≤𝐴 whenever 𝑓(𝑧)=0,𝑓∈𝐹. If 𝐹 is not normal at 𝑧0∈𝐷, then for each 0≤𝛼≤𝑘, there exist a sequence of points 𝑧𝑛∈𝐷,𝑧𝑛→𝑧0, a sequence of positive numbers 𝜌𝑛→0+, and a subsequence of functions 𝑓𝑛∈𝐹 such that 𝑔𝑛𝑓(𝜁)=𝑛𝑧𝑛+𝜌𝑛𝜍𝜌𝛼𝑛→𝑔(𝜁)(2.1) locally uniformly with respect to the spherical metric in 𝐶, where 𝑔 is a nonconstant meromorphic function, all of whose zeros have multiplicity at least 𝑘, such that 𝑔#(𝜁)≤𝑔#(0)=𝑘𝐴+1. Morever,  𝑔 has order at most 2.
Here as usual, 𝑔#(𝜁)=|ğ‘”î…ž(𝜁)|/(1+|𝑔(𝜁)|2) is the spherical derivative.

Lemma 2.2 (see [10]). Let 𝑚 be any positive number. Then, Möbius transformation 𝑔 satisfies ğœŽ(𝑔(ğ‘Ž),𝑔(𝑏))≥𝑚,ğœŽ(𝑔(𝑏),𝑔(𝑐))≥𝑚,ğœŽ(𝑔(𝑐),𝑔(ğ‘Ž))≥𝑚, for some constants ğ‘Ž,𝑏, and 𝑐 also satisfy the uniform Lipschitz condition ğœŽ(𝑔(𝑧),𝑔(𝑤))â‰¤ğ‘˜ğ‘šğœŽ(𝑧,𝑤),(2.2) where 𝑘𝑚 is a constant depending on 𝑚.

3. Proof of Theorems

Proof of Theorem A. Let 𝑀=𝑏𝑓/𝑐𝑓. We can find nonzero constants 𝑏 and 𝑐 satisfying 𝑀=𝑏/𝑐. For each 𝑓∈𝐹, define a Möbius map 𝑔𝑓 by 𝑔𝑓=𝑐𝑓𝑧/𝑐, thus 𝑔𝑓−1=𝑐𝑧/𝑐𝑓.
Next we will show 𝐺={(𝑔𝑓−1∘𝑓)∣𝑓∈𝐹} is normal in Δ. Suppose to the contrary, 𝐺 is not normal in Δ. Then by Lemma 2.1. We can find 𝑔𝑛∈𝐺,𝑧𝑛∈Δ, and 𝜌𝑛→0+, such that 𝑇𝑛(𝜁)=𝑔𝑛(𝑧𝑛+𝜌𝑛𝜁)/𝜌𝑛1/(𝑘+1) converges locally uniformly with respect to the spherical metric to a nonconstant meromorphic function 𝑇(𝜁) whose zeros of multiplicity at least 2 and spherical derivative is limited and 𝑇 has order at most 2.
We now consider three cases.
Case 1. If (1/𝑐𝑘−1)(ğ‘‡î…ž)𝑘(𝜁)≡𝑏𝑘/𝑐𝑘−1, then 𝑇(𝜁) is a polynomial with degree at most 1, a contradiction.
Case 2. If there exists 𝜁0 such that (1/𝑐𝑘−1)(ğ‘‡î…ž)𝑘(𝜁0)=𝑏𝑘/𝑐𝑘−1. Noting that 𝜌𝑛𝑇𝑛(𝜁)+(1/𝑐𝑘−1)(ğ‘‡î…žğ‘›)𝑘(𝜁)−(𝑏𝑘/𝑐𝑘−1)→(1/𝑐𝑘−1)(ğ‘‡î…ž)𝑘(𝜁)−(𝑏𝑘/𝑐𝑘−1). By Hurwitz’s theorem, there exist a sequence of points 𝜁𝑛→𝜁0 such that (for large enough 𝑛) 0=𝜌𝑛𝑇𝑛𝜁𝑛+1𝑐𝑘−1î€·ğ‘‡î…žğ‘›î€¸ğ‘˜î€·ğœğ‘›î€¸âˆ’ğ‘ğ‘˜ğ‘ğ‘˜âˆ’1=𝑔𝑛𝑧𝑛+𝜌𝑛𝜁𝑛+1𝑐𝑘−1î€·ğ‘”î…žğ‘›î€¸ğ‘˜î€·ğ‘§ğ‘›+𝜁𝑛−𝑏𝑘𝑐𝑘−1=𝑐𝑐𝑓𝑓𝑛𝑧𝑛+𝜌𝑛𝜁𝑛+1𝑐𝑘−1ğ‘ğ‘˜ğ‘ğ‘˜ğ‘“î€·ğ‘“î…žğ‘›î€¸ğ‘˜î€·ğ‘§ğ‘›+𝜁𝑛−𝑏𝑘𝑐𝑘−1.(3.1) Hence 𝑓𝑛(𝑧𝑛+𝜌𝑛𝜁𝑛)+(1/𝑐𝑓𝑘−1)(ğ‘“î…žğ‘›)𝑘(𝑧𝑛+𝜁𝑛)=𝑏𝑘𝑓/𝑐𝑓𝑘−1. This contradicts with the suppose of Theorem A.
Case 3. If (1/𝑐𝑘−1)(ğ‘‡î…ž)𝑘(𝜁)≠𝑏𝑘/𝑐𝑘−1. Let 𝑐1,𝑐2,…,𝑐𝑘 be the solution of the equation 𝑤𝑘=𝑐𝑘, then ğ‘‡î…ž(𝜁)≠𝑐𝑖(𝑖=1,2,…,𝑘). When 𝑇(𝜁) is a rational function, then ğ‘‡î…ž(𝜁) is also a rational function. By Picard Theorem we can deduce that ğ‘‡î…ž(𝜁) is a constant (𝑘≥3). Hence 𝑇(𝜁) is a polynomial with degree at most 1. This contradicts with 𝑇(𝜁) has zeros of multiplicity at least 2. When 𝑇(𝜁) is a transcendental function, combining with the second main theorem, we have 𝑇𝑟,ğ‘‡î…žî€¸â‰¤ğ‘î€·ğ‘Ÿ,ğ‘‡î…žî€¸+𝑘𝑖=1𝑁1𝑟,ğ‘‡î…žâˆ’ğ‘ğ‘–î‚î€·+𝑠𝑟,ğ‘‡î…žî€¸â‰¤ğ‘î€·ğ‘Ÿ,ğ‘‡î…žî€¸î€·+𝑠𝑟,ğ‘‡î…žî€¸â‰¤12𝑁𝑟,ğ‘‡î…žî€¸î€·+𝑠𝑟,ğ‘‡î…žî€¸â‰¤12𝑇𝑟,ğ‘‡î…žî€¸î€·+𝑠𝑟,ğ‘‡î…žî€¸.(3.2) Hence, 𝑇(𝑟,ğ‘‡î…ž)≤𝑠(𝑟,ğ‘‡î…ž), a contradiction.
Hence 𝐺={(𝑔𝑓−1∘𝑓)∣𝑓∈𝐹} is normal and equicontinuous in Δ. There given (𝜀/𝑘𝑚>0), where 𝑘𝑚 is the constant of Lemma 2.2, there exists 𝛿>0 such that for the spherical distance ğœŽ(𝑥,𝑦)<𝛿, ğœŽğ‘”î‚€î‚€ğ‘“âˆ’1𝑔∘𝑓(𝑥),𝑓−1<𝜀(𝑦)𝑘𝑚(3.3) for each 𝑓∈𝐹. Hence by Lemma 2.2. ğ‘”ğœŽ(𝑓(𝑥),𝑓(𝑦))=ğœŽî‚€î‚€ğ‘“âˆ˜ğ‘”ğ‘“âˆ’1𝑔∘𝑓(𝑥),𝑓∘𝑔𝑓−1∘𝑓(𝑦)=ğ‘˜ğ‘šğœŽğ‘”î‚€î‚€ğ‘“âˆ’1𝑔∘𝑓(𝑥),𝑓−1∘𝑓(𝑦)<𝜀.(3.4) Therefore, the family is equicontinuous in Δ. This completes the proof of Theorem A.

Proof of Theorem B. Let 𝑀=𝑏𝑓/𝑐𝑓. We can find nonzero constants 𝑏 and 𝑐 satisfying 𝑀=𝑏/𝑐. For each 𝑓∈𝐹, define a Möbius map 𝑔𝑓 by 𝑔𝑓=𝑐𝑓𝑧/𝑐, thus 𝑔𝑓−1=𝑐𝑧/𝑐𝑓.
Next we will show 𝐺={(𝑔𝑓−1∘𝑓)∣𝑓∈𝐹} is normal in Δ. Suppose to the contrary, 𝐺 is not normal in Δ. Then by Lemma 2.1. We can find 𝑔𝑛∈𝐺,𝑧𝑛∈Δ, and 𝜌𝑛→0+, such that 𝑇𝑛(𝜁)=𝑔𝑛(𝑧𝑛+𝜌𝑛𝜁)/𝜌𝑛1/(𝑘+1) converges locally uniformly with respect to the spherical metric to a nonconstant meromorphic function 𝑇(𝜁) whose spherical derivate is limited and 𝑇 has order at most 2.
We will also consider three cases.
Case 1. If (1/𝑐𝑘−1)(ğ‘‡î…ž)𝑘(𝜁)≡𝑏𝑘/𝑐𝑘−1, then 𝑇(𝜁) is a polynomial with degree at most 1, a contradiction.
Case 2. If there exists 𝜁0 such that (1/𝑐𝑘−1)(ğ‘‡î…ž)𝑘(𝜁0)=𝑏𝑘/𝑐𝑘−1. Noting that 𝜌𝑛𝑇𝑛(𝜁)+(1/𝑐𝑘−1)(ğ‘‡î…žğ‘›)𝑘(𝜁)−(𝑏𝑘/𝑐𝑘−1)→(1/𝑐𝑘−1)(ğ‘‡î…ž)𝑘(𝜁)−(𝑏𝑘/𝑐𝑘−1). By Hurwitz’s theorem, there exist a sequence of points 𝜁𝑛→𝜁0 such that (for large enough 𝑛) 0=𝜌𝑛𝑇𝑛𝜁𝑛+1𝑐𝑘−1î€·ğ‘‡î…žğ‘›î€¸ğ‘˜î€·ğœğ‘›î€¸âˆ’ğ‘ğ‘˜ğ‘ğ‘˜âˆ’1=𝑔𝑛𝑧𝑛+𝜌𝑛𝜁𝑛+1𝑐𝑘−1î€·ğ‘”î…žğ‘›î€¸ğ‘˜î€·ğ‘§ğ‘›+𝜁𝑛−𝑏𝑘𝑐𝑘−1=𝑐𝑐𝑓𝑓𝑛𝑧𝑛+𝜌𝑛𝜁𝑛+1𝑐𝑘−1ğ‘ğ‘˜ğ‘ğ‘˜ğ‘“î€·ğ‘“î…žğ‘›î€¸ğ‘˜î€·ğ‘§ğ‘›+𝜁𝑛−𝑏𝑘𝑐𝑘−1.(3.5) Hence 𝑓𝑛(𝑧𝑛+𝜌𝑛𝜁𝑛)+(1/𝑐𝑓𝑘−1)(ğ‘“î…žğ‘›)𝑘(𝑧𝑛+𝜁𝑛)=𝑏𝑘𝑓/𝑐𝑓𝑘−1, then we have 𝑓𝑛(𝑧𝑛+𝜌𝑛𝜁𝑛)=𝑏𝑓 by the condition (iii)(1/𝑐𝑓𝑘−1)(ğ‘“î…ž)𝑘(𝑧)+𝑓(𝑧)=𝑏𝑘𝑓/𝑐𝑓𝑘−1⇒𝑓(𝑧)=𝑏𝑓.
Thus 𝑇𝜁0=limğ‘›â†’âˆžğ‘”ğ‘›î€·ğ‘§ğ‘›+𝜌𝑛𝜁𝑛𝜌𝑛=limğ‘›â†’âˆžî€·ğ‘§ğ‘ğ‘“ğ‘›+𝜌𝑛𝜁𝑛𝑐𝑓𝜌𝑛=limğ‘›â†’âˆžğ‘ğœŒğ‘›=∞.(3.6) This is a contradiction.
Case 3. If (1/𝑐𝑘−1)(ğ‘‡î…ž)𝑘(𝜁)≠𝑏𝑘/𝑐𝑘−1. Let 𝑐1,𝑐2,…,𝑐𝑘 be the solution of the equation 𝑤𝑘=𝑐𝑘, then ğ‘‡î…ž(𝜁)≠𝑐𝑖(𝑖=1,2,…,𝑘). When 𝑇(𝜁) is a rational function, then ğ‘‡î…ž(𝜁) is also a rational function. By Picard theorem we can deduce that ğ‘‡î…ž(𝜁) is a constant (𝑘≥3). Hence 𝑇(𝜁) is a polynomial with degree at most 1. This contradicts with 𝑇(𝜁) has zeros of multiplicity at least 2. When 𝑇(𝜁) is a transcendental function, combining with the second main theorem, we have 𝑇𝑟,ğ‘‡î…žî€¸â‰¤ğ‘î€·ğ‘Ÿ,ğ‘‡î…žî€¸+𝑘𝑖=1𝑁1𝑟,ğ‘‡î…žâˆ’ğ‘ğ‘–î‚î€·+𝑠𝑟,ğ‘‡î…žî€¸â‰¤ğ‘î€·ğ‘Ÿ,ğ‘‡î…žî€¸î€·+𝑠𝑟,ğ‘‡î…žî€¸â‰¤12𝑁𝑟,ğ‘‡î…žî€¸î€·+𝑠𝑟,ğ‘‡î…žî€¸â‰¤12𝑇𝑟,ğ‘‡î…žî€¸î€·+𝑠𝑟,ğ‘‡î…žî€¸.(3.7) Hence, 𝑇(𝑟,ğ‘‡î…ž)≤𝑠(𝑟,ğ‘‡î…ž), a contradiction.
Hence 𝐺={(𝑔𝑓−1∘𝑓)∣𝑓∈𝐹} is normal and equicontinuous in Δ. There given (𝜀/𝑘𝑚>0), where 𝑘𝑚 is the constant of Lemma 2.2, there exists 𝛿>0 such that for the spherical distance ğœŽ(𝑥,𝑦)<𝛿, ğœŽğ‘”î‚€î‚€ğ‘“âˆ’1𝑔∘𝑓(𝑥),𝑓−1<𝜀(𝑦)𝑘𝑚(3.8) for each 𝑓∈𝐹. Hence by Lemma 2.2. ğ‘”ğœŽ(𝑓(𝑥),𝑓(𝑦))=ğœŽî‚€î‚€ğ‘“âˆ˜ğ‘”ğ‘“âˆ’1𝑔∘𝑓(𝑥),𝑓∘𝑔𝑓−1∘𝑓(𝑦)=ğ‘˜ğ‘šğœŽğ‘”î‚€î‚€ğ‘“âˆ’1𝑔∘𝑓(𝑥),𝑓−1∘𝑓(𝑦)<𝜀.(3.9) Therefore, the family is equicontinuous in Δ. This completes the proof of Theorem B.

Remark 3.1. Using the similar argument, if the condition (iii)𝑓(𝑧)=𝑏𝑓 when (1/𝑐𝑓𝑘−1)(ğ‘“î…ž)𝑘(𝑧)+𝑓(𝑧)=𝑏𝑘𝑓/𝑐𝑓𝑘−1 is replaced by (iii)|𝑓(𝑧)|≥|𝑏𝑓| when (1/𝑐𝑓𝑘−1)(ğ‘“î…ž)𝑘(𝑧)+𝑓(𝑧)=𝑏𝑘𝑓/𝑐𝑓𝑘−1, then 𝐹 is normal too.

Authors’ Contribution

W. Chen performed the proof and drafted the paper. All authors read and approved the final paper.

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgment

This paper is supported by Nature Science Foundation of Fujian Province (2012J01022). The authors wish to thank the referee for some valuable corrections.

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Copyright © 2012 Wei Chen et al. 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.


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