Normal Criterion Concerning Shared Values

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 𝑏_𝑓,𝑐_𝑓 depending on 𝑓 satisfying (i)𝑏_𝑓/𝑐_𝑓 is a constant;  (ii)min{𝜎(0,𝑏_𝑓),𝜎(0,𝑐_𝑓),𝜎(𝑏_𝑓,𝑐_𝑓)≥𝑚} for some 𝑚>0;  (iii)(1/𝑐^𝑓𝑘−1)(𝑓^)^𝑘(𝑧)+𝑓(𝑧)≠𝑏^𝑘𝑓/𝑐^𝑓𝑘−1 or (1/𝑐^𝑓𝑘−1)(𝑓^)^𝑘(𝑧)+𝑓(𝑧)=𝑏^𝑘𝑓/𝑐^𝑓𝑘−1⇒𝑓(𝑧)=𝑏_𝑓, 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 {𝑓_𝑛}⊂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 𝑎∈𝐶, set 𝐸_𝑓(𝑎)=𝑓⁻¹({𝑎})∩𝐷={𝑧∈𝐷∶𝑓(𝑧)=𝑎}.(1.1)

Two meromorphic functions 𝑓 and 𝑔 on 𝐷 are said to share the value 𝑎 if 𝐸_𝑓(𝑎)=𝐸_𝑔(𝑎). Let 𝑎 and 𝑏 be complex numbers. If 𝑔(𝑧)=𝑏 whenever 𝑓(𝑧)=𝑎, we write 𝑓(𝑧)=𝑎⇒𝑔(𝑧)=𝑏.(1.2) If 𝑓(𝑧)=𝑎⇒𝑔(𝑧)=𝑏 and 𝑔(𝑧)=𝑏⇒𝑓(𝑧)=𝑎, we write 𝑓(𝑧)=𝑎⟺𝑔(𝑧)=𝑏.(1.3)

According to Bloch’s principle [2], every condition which reduces a meromorphic function in the plane 𝐶 to 𝑎 constant forces a family of meromorphic functions in 𝑎 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 𝜎(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 𝑎(≠0) and 𝑏 be complex numbers, and let 𝑛(≥2),𝑘 be positive integers, then 𝑓+𝑎(𝑓^)^𝑛 assumes every value 𝑏∈𝐶 infinitely often.

Theorem 1.2 (see [11]). Let 𝐹 be a transcendental function. Let 𝑎(≠0) and 𝑏 be complex numbers, and let 𝑛(≥2),𝑘 be positive integers. If for every 𝑓∈𝐹 has multiple zeros, and 𝑓+𝑎(𝑓^)^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 𝑎(≠0) and let 𝑏 be complex numbers, and 𝑛,𝑘 be positive integers, which satisfy 𝑛≥𝑘+1, then 𝑓+𝑎(𝑓^(𝑘))^𝑛 assumes each value 𝑏∈𝐶 infinitely often.

Theorem 1.4 (see [12]). Let 𝑓 be a transcendental function. Let 𝑎(≠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+𝑎(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 𝑏_𝑓,𝑐_𝑓 depending on 𝑓 satisfying that(i)𝑏_𝑓/𝑐_𝑓 is a constant; (ii)min{𝜎(0,𝑏_𝑓),𝜎(0,𝑐_𝑓),𝜎(𝑏_𝑓,𝑐_𝑓)≥𝑚} for some 𝑚>0; (iii)(1/𝑐^𝑓𝑘−1)(𝑓^)^𝑘(𝑧)+𝑓(𝑧)≠𝑏^𝑘𝑓/𝑐^𝑓𝑘−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 𝑏_𝑓,𝑐_𝑓 depending on 𝑓 satisfying that(i)𝑏_𝑓/𝑐_𝑓is a constant; (ii)min{𝜎(0,𝑏_𝑓),𝜎(0,𝑐_𝑓),𝜎(𝑏_𝑓,𝑐_𝑓)≥𝑚} for some 𝑚>0; (iii)(1/𝑐^𝑓𝑘−1)(𝑓^)^𝑘(𝑧)+𝑓(𝑧)=𝑏^𝑘𝑓/𝑐^𝑓𝑘−1⇒𝑓(𝑧)=𝑏_𝑓.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 𝑧_𝑛∈𝐷,𝑧_𝑛→𝑧₀, a sequence of positive numbers 𝜌_𝑛→0⁺, and a subsequence of functions 𝑓_𝑛∈𝐹 such that 𝑔_𝑛(𝜁)=𝑓^𝑛𝑧_𝑛+𝜌_𝑛𝜍𝜌^𝛼𝑛→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)=|𝑔^(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 𝜎(𝑔(𝑎),𝑔(𝑏))≥𝑚,𝜎(𝑔(𝑏),𝑔(𝑐))≥𝑚,𝜎(𝑔(𝑐),𝑔(𝑎))≥𝑚, for some constants 𝑎,𝑏, and 𝑐 also satisfy the uniform Lipschitz condition 𝜎(𝑔(𝑧),𝑔(𝑤))≤𝑘_𝑚𝜎(𝑧,𝑤),(2.2) where 𝑘_𝑚 is a constant depending on 𝑚.

3. Proof of Theorems

Proof of Theorem A. Let 𝑀=𝑏_𝑓/𝑐_𝑓. We can find nonzero constants 𝑏 and 𝑐 satisfying 𝑀=𝑏/𝑐. For each 𝑓∈𝐹, define a Möbius map 𝑔_𝑓 by 𝑔_𝑓=𝑐_𝑓𝑧/𝑐, thus 𝑔^𝑓−1=𝑐𝑧/𝑐_𝑓.
Next we will show 𝐺={(𝑔^𝑓−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 𝑔_𝑛∈𝐺,𝑧_𝑛∈Δ, and 𝜌_𝑛→0⁺, such that 𝑇_𝑛(𝜁)=𝑔_𝑛(𝑧_𝑛+𝜌_𝑛𝜁)/𝜌^{𝑛1/(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/𝑐^𝑘−1)(𝑇^)^𝑘(𝜁)≡𝑏^𝑘/𝑐^𝑘−1, then 𝑇(𝜁) is a polynomial with degree at most 1, a contradiction.
Case 2. If there exists 𝜁₀ such that (1/𝑐^𝑘−1)(𝑇^)^𝑘(𝜁₀)=𝑏^𝑘/𝑐^𝑘−1. Noting that 𝜌_𝑛𝑇_𝑛(𝜁)+(1/𝑐^𝑘−1)(𝑇^𝑛)^𝑘(𝜁)−(𝑏^𝑘/𝑐^𝑘−1)→(1/𝑐^𝑘−1)(𝑇^)^𝑘(𝜁)−(𝑏^𝑘/𝑐^𝑘−1). By Hurwitz’s theorem, there exist a sequence of points 𝜁_𝑛→𝜁₀ such that (for large enough 𝑛) 0=𝜌_𝑛𝑇_𝑛𝜁_𝑛+1𝑐^𝑘−1𝑇^𝑛^𝑘𝜁_𝑛−𝑏^𝑘𝑐^𝑘−1=𝑔_𝑛𝑧_𝑛+𝜌_𝑛𝜁_𝑛+1𝑐^𝑘−1𝑔^𝑛^𝑘𝑧_𝑛+𝜁_𝑛−𝑏^𝑘𝑐^𝑘−1=𝑐𝑐_𝑓𝑓_𝑛𝑧_𝑛+𝜌_𝑛𝜁_𝑛+1𝑐^𝑘−1𝑐^𝑘𝑐^𝑘𝑓𝑓^𝑛^𝑘𝑧_𝑛+𝜁_𝑛−𝑏^𝑘𝑐^𝑘−1.(3.1) Hence 𝑓_𝑛(𝑧_𝑛+𝜌_𝑛𝜁_𝑛)+(1/𝑐^𝑓𝑘−1)(𝑓^𝑛)^𝑘(𝑧_𝑛+𝜁_𝑛)=𝑏^𝑘𝑓/𝑐^𝑓𝑘−1. This contradicts with the suppose of Theorem A.
Case 3. If (1/𝑐^𝑘−1)(𝑇^)^𝑘(𝜁)≠𝑏^𝑘/𝑐^𝑘−1. Let 𝑐₁,𝑐₂,…,𝑐_𝑘 be the solution of the equation 𝑤^𝑘=𝑐^𝑘, then 𝑇^(𝜁)≠𝑐_𝑖(𝑖=1,2,…,𝑘). When 𝑇(𝜁) is a rational function, then 𝑇^(𝜁) is also a rational function. By Picard Theorem we can deduce that 𝑇^(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,𝑇^≤𝑁𝑟,𝑇^+^𝑘_𝑖=1𝑁𝑟,1𝑇^≤−𝑐_𝑖+𝑠𝑟,𝑇^𝑁𝑟,𝑇^+𝑠𝑟,𝑇^≤12𝑁𝑟,𝑇^+𝑠𝑟,𝑇^≤12𝑇𝑟,𝑇^+𝑠𝑟,𝑇^.(3.2) Hence, 𝑇(𝑟,𝑇^)≤𝑠(𝑟,𝑇^), a contradiction.
Hence 𝐺={(𝑔^𝑓−1∘𝑓)∣𝑓∈𝐹} is normal and equicontinuous in Δ. There given (𝜀/𝑘_𝑚>0), where 𝑘_𝑚 is the constant of Lemma 2.2, there exists 𝛿>0 such that for the spherical distance 𝜎(𝑥,𝑦)<𝛿, 𝜎𝑔^𝑓−1∘𝑓(𝑥),𝑔^𝑓−1(𝑦)<𝜀𝑘_𝑚(3.3) for each 𝑓∈𝐹. Hence by Lemma 2.2. 𝜎(𝑓(𝑥),𝑓(𝑦))==𝑘𝜎_𝑚𝑔_𝑓∘𝑔^𝑓−1∘𝑓(𝑥),𝑔_𝑓∘𝑔^𝑓−1∘𝑓(𝑦)𝜎𝑔^𝑓−1∘𝑓(𝑥),𝑔^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=𝑏_𝑓/𝑐_𝑓. We can find nonzero constants 𝑏 and 𝑐 satisfying 𝑀=𝑏/𝑐. For each 𝑓∈𝐹, define a Möbius map 𝑔_𝑓 by 𝑔_𝑓=𝑐_𝑓𝑧/𝑐, thus 𝑔^𝑓−1=𝑐𝑧/𝑐_𝑓.
Next we will show 𝐺={(𝑔^𝑓−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 𝑔_𝑛∈𝐺,𝑧_𝑛∈Δ, and 𝜌_𝑛→0⁺, such that 𝑇_𝑛(𝜁)=𝑔_𝑛(𝑧_𝑛+𝜌_𝑛𝜁)/𝜌^{𝑛1/(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/𝑐^𝑘−1)(𝑇^)^𝑘(𝜁)≡𝑏^𝑘/𝑐^𝑘−1, then 𝑇(𝜁) is a polynomial with degree at most 1, a contradiction.
Case 2. If there exists 𝜁₀ such that (1/𝑐^𝑘−1)(𝑇^)^𝑘(𝜁₀)=𝑏^𝑘/𝑐^𝑘−1. Noting that 𝜌_𝑛𝑇_𝑛(𝜁)+(1/𝑐^𝑘−1)(𝑇^𝑛)^𝑘(𝜁)−(𝑏^𝑘/𝑐^𝑘−1)→(1/𝑐^𝑘−1)(𝑇^)^𝑘(𝜁)−(𝑏^𝑘/𝑐^𝑘−1). By Hurwitz’s theorem, there exist a sequence of points 𝜁_𝑛→𝜁₀ such that (for large enough 𝑛) 0=𝜌_𝑛𝑇_𝑛𝜁_𝑛+1𝑐^𝑘−1𝑇^𝑛^𝑘𝜁_𝑛−𝑏^𝑘𝑐^𝑘−1=𝑔_𝑛𝑧_𝑛+𝜌_𝑛𝜁_𝑛+1𝑐^𝑘−1𝑔^𝑛^𝑘𝑧_𝑛+𝜁_𝑛−𝑏^𝑘𝑐^𝑘−1=𝑐𝑐_𝑓𝑓_𝑛𝑧_𝑛+𝜌_𝑛𝜁_𝑛+1𝑐^𝑘−1𝑐^𝑘𝑐^𝑘𝑓𝑓^𝑛^𝑘𝑧_𝑛+𝜁_𝑛−𝑏^𝑘𝑐^𝑘−1.(3.5) Hence 𝑓_𝑛(𝑧_𝑛+𝜌_𝑛𝜁_𝑛)+(1/𝑐^𝑓𝑘−1)(𝑓^𝑛)^𝑘(𝑧_𝑛+𝜁_𝑛)=𝑏^𝑘𝑓/𝑐^𝑓𝑘−1, then we have 𝑓_𝑛(𝑧_𝑛+𝜌_𝑛𝜁_𝑛)=𝑏_𝑓 by the condition (iii)(1/𝑐^𝑓𝑘−1)(𝑓^)^𝑘(𝑧)+𝑓(𝑧)=𝑏^𝑘𝑓/𝑐^𝑓𝑘−1⇒𝑓(𝑧)=𝑏_𝑓.
Thus 𝑇𝜁₀=lim_𝑛→∞𝑔_𝑛𝑧_𝑛+𝜌_𝑛𝜁_𝑛𝜌_𝑛=lim_𝑛→∞𝑐𝑓𝑧_𝑛+𝜌_𝑛𝜁_𝑛𝑐_𝑓𝜌_𝑛=lim_𝑛→∞𝑏𝜌_𝑛=∞.(3.6) This is a contradiction.
Case 3. If (1/𝑐^𝑘−1)(𝑇^)^𝑘(𝜁)≠𝑏^𝑘/𝑐^𝑘−1. Let 𝑐₁,𝑐₂,…,𝑐_𝑘 be the solution of the equation 𝑤^𝑘=𝑐^𝑘, then 𝑇^(𝜁)≠𝑐_𝑖(𝑖=1,2,…,𝑘). When 𝑇(𝜁) is a rational function, then 𝑇^(𝜁) is also a rational function. By Picard theorem we can deduce that 𝑇^(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,𝑇^≤𝑁𝑟,𝑇^+^𝑘_𝑖=1𝑁𝑟,1𝑇^≤−𝑐_𝑖+𝑠𝑟,𝑇^𝑁𝑟,𝑇^+𝑠𝑟,𝑇^≤12𝑁𝑟,𝑇^+𝑠𝑟,𝑇^≤12𝑇𝑟,𝑇^+𝑠𝑟,𝑇^.(3.7) Hence, 𝑇(𝑟,𝑇^)≤𝑠(𝑟,𝑇^), a contradiction.
Hence 𝐺={(𝑔^𝑓−1∘𝑓)∣𝑓∈𝐹} is normal and equicontinuous in Δ. There given (𝜀/𝑘_𝑚>0), where 𝑘_𝑚 is the constant of Lemma 2.2, there exists 𝛿>0 such that for the spherical distance 𝜎(𝑥,𝑦)<𝛿, 𝜎𝑔^𝑓−1∘𝑓(𝑥),𝑔^𝑓−1(𝑦)<𝜀𝑘_𝑚(3.8) for each 𝑓∈𝐹. Hence by Lemma 2.2. 𝜎(𝑓(𝑥),𝑓(𝑦))==𝑘𝜎_𝑚𝑔_𝑓∘𝑔^𝑓−1∘𝑓(𝑥),𝑔_𝑓∘𝑔^𝑓−1∘𝑓(𝑦)𝜎𝑔^𝑓−1∘𝑓(𝑥),𝑔^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)=𝑏_𝑓 when (1/𝑐^𝑓𝑘−1)(𝑓^)^𝑘(𝑧)+𝑓(𝑧)=𝑏^𝑘𝑓/𝑐^𝑓𝑘−1 is replaced by (iii)|𝑓(𝑧)|≥|𝑏_𝑓| when (1/𝑐^𝑓𝑘−1)(𝑓^)^𝑘(𝑧)+𝑓(𝑧)=𝑏^𝑘𝑓/𝑐^𝑓𝑘−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.