Some Normal Criteria about Shared Values with Their Multiplicity Zeros

Abstract

Let ℱ be a family of meromorphic functions in the domain 𝐷, all of whose zeros are multiple. Let 𝑛(𝑛≥2) be an integer and let 𝑎, 𝑏 be two nonzero finite complex numbers. If 𝑓+𝑎(𝑓^′)^𝑛 and 𝑔+𝑎(𝑔^′)^𝑛 share 𝑏 in 𝐷 for every pair of functions 𝑓,𝑔∈ℱ, then ℱ is normal in 𝐷.

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. With renewed interest in normal families of analytic and meromorphic functions in plane domains, mainly because of their role in complex dynamics, it has become quite interesting to talk about normal families in their own right.

We will be concerned with the analytic maps (i.e., meromorphic functions) 𝑓∶𝐷,|⋅|_𝑅²⟶ℂ,𝑋 (1.1) from 𝐷 (endowed with the Euclidean metric) to the extended complex plane ℂ endowed with the spherically metric 𝑋 given by 𝑋𝑧,𝑧^′=𝑧−𝑧^′1+|𝑧|²1+𝑧^′2𝑧,𝑧^′∈ℂ. (1.2)

A family ℱ of meromorphic functions defined in 𝐷 is said to be normal, in the sense of Montel, if for any sequence {𝑓_𝑛}⊂ℱ, there exists a subsequence {𝑓_𝑛𝑗} such that 𝑓_𝑛𝑗 converges spherically locally and uniformly in 𝐷 to a meromorphic function or ∞. Clearly, ℱ is normal in 𝐷 if and only if it is normal at every point of 𝐷 (see [1, 2]).

Let 𝑓 and 𝑔 be two nonconstant meromorphic functions in 𝐷, and 𝑎∈ℂ. We say that 𝑓 and 𝑔 share the value 𝑎 in 𝐷, if 𝑓−𝑎 and 𝑔−𝑎 have the same zeros (ignoring multiplicities). When 𝑎=∞ the zeros of 𝑓−𝑎 means the poles of 𝑓 (see [3]).

Influenced from Bloch's principle [4], every condition which reduces a meromorphic function in the plane ℂ to a constant, makes a family of meromorphic functions in a domain 𝐷 normal. Although the principle is false in general (see [5]), many authors proved normality criterion for families of meromorphic functions corresponding to Liouville-Picard type theorem (see [1, 2, 6]).

It is also more interesting to find normality criteria from the point of view of shared values. In this area, Schiff [1] first proved an interesting result that a family of meromorphic functions in a domian is normal if in which every function shares three distinct finite complex numbers with its first derivative. And later, Sun [7] proved that a family of meromorphic functions in a domian is normal if in which each pair of functions share three fixed distinct valus, which is an improvement of the famous Montel's Normal Criterion [8] by the ideas of shared values. More results about normality criteria concerning shared values can be found, for instance, (see [9–12]) and so on.

In 1989, Schwick [13] proved that let ℱ be a family of meromorphic functions in a domain 𝐷, if (𝑓^𝑛)^(𝑘)≠1 for every 𝑓∈ℱ, where 𝑛, 𝑘 are two positive integers and 𝑛≥𝑘+3, then ℱ is normal in 𝐷.

Recently, by the ideas of shared values, Li and Gu [14] proved the following.

Theorem A. Let ℱ be a family of meromorphic functions in 𝐷. If (𝑓^𝑛)^(𝑘) and (𝑔^𝑛)^(𝑘) share 𝑎 in 𝐷 for each pair of functions 𝑓, 𝑔 in ℱ, where 𝑛, 𝑘 are two positive integers such that 𝑛≥𝑘+2 and 𝑎 is a finite nonzero complex number, then ℱ is normal in 𝐷.

In 1998, Wang and Fang [15] obtained the following result.

Theorem B. Let ℱ be a family of meromorphic functions in 𝐷. Let 𝑘 be a positive integer and 𝑏 be a nonzero finite complex number. If, for each 𝑓∈ℱ, all zeros of 𝑓 have multiplicity at least 𝑘+2, and 𝑓^(𝑘)(𝑧)≠𝑏 on D, then ℱ is normal in 𝐷.

Remark 1.1. By a counter-example, Wang and Fang [15] show Theorem B is not valid if all zeros of 𝑓 have multiplicity less than 𝑘+2.

It is natural to ask whether Theorem B can be improved by the idea of shared values. In this paper we investigate this problem and obtain the following result.

Theorem 1.2. Let 𝑘 (𝑘≥2) be an integer and 𝑏 be a nonzero finite complex number, and let ℱ be a family of meromorphic functions in 𝐷, all of whose zeros have multiplicity at least 𝑘+2. If, for every pair 𝑓,𝑔∈ℱ, all zeros of 𝑓^(𝑘)(𝑧),𝑔^(𝑘)(𝑧) are multiple, 𝑓^(𝑘)(𝑧) and 𝑔^(𝑘)(𝑧) share 𝑏 in 𝐷, then ℱ is normal in 𝐷.

Remark 1.3. Comparing with Theorem A, Theorem 1.2 releases the condition that the poles of 𝑓(𝑧) have multiplicity at least 𝑘+2, which improves Theorem A in some sense.

Example 1.4. Take 𝐷={𝑧∶|𝑧|<1} and ℱ=𝑓_𝑚=𝑚𝑧^𝑘+2∣𝑚=1,2,… (1.3) or ℱ=𝑓_𝑚=𝑒^𝑚𝑧∣𝑚=1,2,…. (1.4) Obviously, any 𝑓_𝑚 has zeros of multiplicity at least 𝑘+2 and 𝑓_𝑚^(𝑘)(𝑧) has zeros of multiplicity at least 2. For any 𝑓_𝑚 and 𝑔_𝑚 in ℱ, we have 𝑓_𝑚^(𝑘)(𝑧) and 𝑔_𝑚^(𝑘)(𝑧) share 0. However the families ℱ is not normal at 𝑧=0.

Remark 1.5. Example 1.4 shows that the condition 𝑏≠0 in Theorem 1.2 is inevitable.

In 1959, Hayman [16] proved that let 𝑓 be a meromorphic function in ℂ, if 𝑓^′−𝑎𝑓^𝑛≠𝑏, where 𝑛 is a positive integer and 𝑎, 𝑏 are two finite complex numbers such that 𝑛≥5 and 𝑎≠0, then 𝑓 is a constant. On the other hand, Mues [17] showed that for 𝑛=3,4 the conclusion is not valid.

The following theorem which confirmed a Hayman's well-known conjecture about normal families in [18].

Theorem C. Let ℱ be a family of meromorphic functions in 𝐷, 𝑛 be a positive integer and 𝑎, 𝑏 be two finite complex numbers such that 𝑎≠0. If 𝑛≥3 and for each function 𝑓∈ℱ, 𝑓^′−𝑎𝑓^𝑛≠𝑏, then ℱ is normal in 𝐷.

In 2008, by the ideas of shared values, Zhang [12] proved the following.

Theorem D. Let ℱ be a family of meromorphic functions in 𝐷, 𝑛 be a positive integer and 𝑎, 𝑏 be two finite complex numbers such that 𝑎≠0, If 𝑛≥4 and for every pair of functions 𝑓 and 𝑔 in ℱ, If 𝑓^′−𝑎𝑓^𝑛 and 𝑔^′−𝑎𝑔^𝑛 share the value 𝑏, then ℱ is normal in 𝐷.

In 1994, Ye [19] considered a similar problem and obtained that if 𝑓 is a transcendental meromorphic function and 𝑎 is a nonzero finite complex number, then 𝑓+𝑎(𝑓^′)^𝑛 assumes every finite complex value infinitely often for 𝑛≥3. Ye [19] also asked whether the conclusion remains valid for 𝑛=2.

In 2008, Fang and Zalcman [20] solved this problem and obtained the following result.

Theorem E. Let 𝑓 be a transcendental meromorphic function and 𝑎 be a nonzero complex number. Then 𝑓+𝑎(𝑓^′)^𝑛 assumes every complex value infinitely often for each positive integer 𝑛≥2.

Remark 1.6. By a special example, Fang and Zalcman [20] show Theorem E is not valid for 𝑛=1.

On the basis of the above results, Fang and Zalcman [20] obtained the normality criterion corresponding to Theorem E.

Theorem F. Let ℱ be a family of meromorphic functions in 𝐷. Let 𝑛 (𝑛≥2) be an integer, and let 𝑎 (𝑎≠0), 𝑏 be two finite complex numbers. If, for each 𝑓∈ℱ, all zeros of 𝑓 are multiple and 𝑓+𝑎(𝑓^′)^𝑛≠𝑏 in 𝐷, then ℱ is normal in 𝐷.

Likewise Theorem D, it is natural to ask whether Theorem F can be improved by the ideas of shared values. In this paper we investigate this problem and obtain the following result.

Theorem 1.7. Let ℱ be a family of meromorphic functions in 𝐷, all of whose zeros are multiple and let 𝑛 (𝑛≥2) be an integer and 𝑎, 𝑏 be two nonzero finite complex numbers. If 𝑓+𝑎(𝑓^′)^𝑛 and 𝑔+𝑎(𝑔^′)^𝑛 share 𝑏 in 𝐷 for every pair of functions 𝑓,𝑔∈ℱ, then ℱ is normal in 𝐷.

Here we will generalize above results by allowing 𝑓(𝑧)+𝑎(𝑓^′(𝑧))^𝑛−𝑏 to have zeros.

For the sake of convenience, we give the following notations:𝐸(ℱ,𝑛)∶=𝑧∈𝐷∣𝑓(𝑧)+𝑎𝑓^′(𝑧)^𝑛−𝑏=0,𝑓(𝑧)∈ℱ;𝑈(𝑧₀,𝑟)∶=𝑧∈𝐷∣𝑧−𝑧₀;𝐸𝑧≤𝑟₀,𝑟,ℱ,𝑛∶=_𝑓∈ℱ𝑧∈𝑈𝑧₀,𝑟∣𝑓(𝑧)+𝑎𝑓^′(𝑧)^𝑛−𝑏=0,𝑓(𝑧)∈ℱ; (1.5)♯𝐸 denotes the number of the elements in the set 𝐸.

Corollary 1.8. Let ℱ be a family of meromorphic functions in 𝐷, all of whose zeros are multiple and let 𝑛 (𝑛≥2) be an integer, and let 𝑎, 𝑏 be two nonzero finite complex numbers. If, for each 𝑓∈ℱ, there exists a positive constant 𝑀 such that |𝑓(𝑧)|≥𝑀 whenever 𝑧∈𝐸(ℱ,𝑛), then ℱ is normal in 𝐷.

Corollary 1.9. Let ℱ be a family of meromorphic functions in 𝐷, all of whose zeros are multiple and let 𝑛(𝑛≥2) be an integer, and let 𝑎 (𝑎≠0), 𝑏 be two finite complex numbers. Suppose that(i)there are two distinct complex numbers 𝑐,𝑑∈ℂ∪{∞} such that 𝑓(𝑧)≠𝑐,𝑑 for every 𝑓(𝑧)∈ℱ;(ii)for arbitrary 𝑧₀∈𝐷, there is 𝑟>0 such that 𝑈(𝑟,𝑧₀)⊂𝐷 and ♯𝐸(𝑧₀,𝑟,ℱ,𝑛)<∞. Then ℱ is normal in 𝐷.

Example 1.10 (see [20]). Let 𝐷={𝑧∶|𝑧|<1} and ℱ={𝑓_𝑚} where 𝑓_𝑚∶=−(1/4)𝑧²+𝑚𝑧+1/2 and 𝑚 is a positive integer. Then for every pair of functions 𝑓,𝑔∈ℱ, 𝑓_𝑚+(𝑓^′_𝑚)²≡𝑚²+1/2 and 𝑔_𝑚+(𝑔^′_𝑚)²≡𝑚²+1/2 share any point in 𝐷. However ℱ is not normal at 𝑧=0.

Remark 1.11. Example 1.10 shows the condition that all zeros of 𝑓∈ℱ are multiple is necessary in Theorem 1.2.

Example 1.12. Take 𝐷={𝑧∶|𝑧|<1}, ℱ=𝑓_𝑚=𝑚𝑧²∣𝑚=1,2,…. (1.6)

Remark 1.13. For 𝑛=2, we obtain 𝑓_𝑚+𝑎(𝑓^′_𝑚)²=𝑚𝑧²+(2𝑚)²𝑎𝑧². So 𝑓+𝑎(𝑓^′)² and 𝑔+𝑎(𝑔^′)² share 0 in 𝐷 for every pair functions 𝑓,𝑔∈ℱ. But ℱ is not normal in 𝐷.
For 𝑛=2, from 𝑚𝑧²+𝑎(2𝑚𝑧)²=𝑏, we deduce (𝑚+4𝑎𝑚²)𝑧²=𝑏. Obviously, If 𝑚 sufficiently large, then 𝑚+4𝑎𝑚²≠0, |𝑓_𝑚(𝑧)|=|𝑚𝑧²|=|𝑚𝑏/(𝑚+4𝑎𝑚²)|=|𝑏/(1+4𝑎𝑚)| and then |𝑓_𝑚(𝑧)|→0. Example 1.12 shows that the condition 𝑏≠0 in Theorem 1.7 is inevitable for 𝑛=2. Example 1.12 also shows the hypothesis there exists a positive constant 𝑀 such that |𝑓(𝑧)|≥𝑀 for all 𝑓∈ℱ whenever 𝑧∈𝐸(ℱ,𝑛) cannot be omitted in Corollary 1.8.

Remark 1.14. Some ideas of this paper are based on [9, 12, 21–23].

2. Preliminary Lemmas

In order to prove our theorems, we need the following lemmas.

First, we need the following well-known Pang-Zalcman lemma, which is the local version of [10, 24].

Lemma 2.1. Let ℱ be a family of meromorphic functions in the unit disc Δ with the property that for each 𝑓(𝑧)∈ℱ, all zeros of multiplicity at least 𝑘. Suppose that there exists a number 𝐴≥1 such that |𝑓^(𝑘)(𝑧)|≤𝐴 whenever 𝑓(𝑧)∈ℱ and 𝑓(𝑧)=0. If ℱ is not normal at a point 𝑧₀∈Δ, then for 0≤𝛼≤𝑘, there are(1)a sequence of complex numbers 𝑧_𝑛∈Δ, 𝑧_𝑛→𝑧₀;(2)a sequence of functions 𝑓_𝑛∈ℱ;(3)a sequence of positive numbers 𝜌_𝑛→0⁺;such that 𝑔_𝑛(𝜉)=𝜌_𝑛^−𝛼𝑓_𝑛(𝑧_𝑛+𝜌_𝑛𝜉) converge locally uniformly (with respect to the spherical metric) to a nonconstant meromorphic function 𝑔(𝜉) on ℂ, and moreover, the zeros of 𝑔(𝜉) are of multiplicity at least 𝑘, 𝑔^♯(𝜉)≤𝑔^♯(0)=𝑘𝐴+1. In particular, 𝑔 has order at most 2.

In Lemma 2.1, the order of 𝑔 is defined by using the Nevanlinna's characteristic function 𝑇(𝑟,𝑔): 𝜌𝑔=lim_𝑟→∞suplog𝑇𝑟,𝑔.log𝑟 (2.1) Here 𝑔^♯(𝜉) denotes the spherical derivative 𝑔^♯(𝜉)=𝑔^′(𝜉)1+𝑔(𝜉)². (2.2)

Lemma 2.2 (see [25]). Let 𝑓(𝑧) be a transcendental meromorphic function of finite order in ℂ. If 𝑓(𝑧) has no simple zero, then 𝑓^′(𝑧) assumes every nonzero finite value infinitely often.

Lemma 2.3 (see [15]). Let 𝑓(𝑧) be a transcendental meromorphic function in ℂ. If all zeros of 𝑓(𝑧) have multiplicity at least 3, for any positive integer 𝑘, then 𝑓^(𝑘)(𝑧) assumes every nonzero finite value infinitely often.

Lemma 2.4. Let 𝑘≥2 be an integer and 𝑏 be a nonzero finite complex number and let 𝑓(𝑧) be a nonconstant rational meromorphic function in ℂ, all zeros of 𝑓(𝑧) have multiplicity at least 𝑘+2. If all zeros of 𝑓^(𝑘)(𝑧) are multiple, then 𝑓^(𝑘)(𝑧)−𝑏 has at least two distinct zeros.

Proof. In the following, we consider two cases.Case 1. Assume, to the contrary, that 𝑓^(𝑘)(𝑧)−𝑏 has at most one zero 𝑧₀.Subcase 1.1 (𝑓(𝑧) is a nonconstant polynomial). Set 𝑓^(𝑘)(𝑧)−𝑏=𝐾(𝑧−𝑧₀)^𝑙, where 𝐾 is a nonzero constant, 𝑙 is a positive integer. Because all zeros of 𝑓^(𝑘)(𝑧) are multiple, we obtain 𝑙≥2 and 𝑓^(𝑘+1)(𝑧)=𝐾𝑙(𝑧−𝑧₀)^𝑙−1 (𝑙−1≥1). Thus, 𝑓^(𝑘+1)(𝑧) has exactly one zero. Since all zeros of 𝑓(𝑧) have multiplicity at least 𝑘+2, we derive 𝑓^(𝑘)(𝑧) has the same zero 𝑧₀. Hence 𝑓^(𝑘)(𝑧₀)=0, which contradicts with 𝑓^(𝑘)(𝑧₀)=𝑏≠0.Subcase 1.2 (𝑓(𝑧) is rational but not a polynomial). By the assumption of Lemma 2.4, we may set 𝑓^(𝑘)(𝑧)=𝐴𝑧−𝛼₁^𝑚1𝑧−𝛼₂^𝑚2⋯𝑧−𝛼_𝑠^𝑚𝑠𝑧−𝛽₁^𝑛1𝑧−𝛽₂^𝑛2⋯𝑧−𝛽_𝑡^𝑛𝑡=𝑃(𝑧)𝑄(𝑧), (2.3) where 𝐴 is a nonzero constant. Since all zeros of 𝑓^(𝑘)(𝑧) are multiple, we find 𝑚_𝑖≥2 (𝑖=1,2,…,𝑠), 𝑛_𝑗≥(𝑘+1) (𝑗=1,2,…,𝑡) (𝑘≥2).
For simplicity, we denote𝑚₁+𝑚₂+⋯+𝑚_𝑠𝑛=𝑀≥2𝑠,₁+𝑛₂+⋯+𝑛_𝑡=𝑁≥(𝑘+1)𝑡, (2.4) where 𝑃(𝑧) and 𝑄(𝑧) are coprime polynomials of degree 𝑀, 𝑁, respectively, in (2.3).
Since 𝑓^(𝑘)(𝑧)−𝑏 has just a zero 𝑧₀, from (2.3) we obtain𝑓^(𝑘)(𝑧)𝐵=𝑏+𝑧−𝑧₀^𝑙𝑧−𝛽₁^𝑛1𝑧−𝛽₂^𝑛2⋯𝑧−𝛽_𝑡^𝑛𝑡=𝑃(𝑧)𝑄(𝑧). (2.5) By 𝑏≠0, we deduce 𝑧₀≠𝛼_𝑖 (𝑖=1,…,𝑠), where 𝐵 is a nonzero constant.
From (2.3), we get𝑓^(𝑘+1)(𝑧)=𝑧−𝛼₁^𝑚1−1𝑧−𝛼₂^𝑚2−1⋯𝑧−𝛼_𝑠^𝑚𝑠−1𝑔₁(𝑧)𝑧−𝛽₁^𝑛1+1⋯𝑧−𝛽_𝑡^𝑛𝑡+1, (2.6) where 𝑔₁(𝑧) is polynomial of degree at most 𝑠+𝑡−1.
Differentiating (2.5) yields 𝑓^(𝑘+1)(𝑧)=𝑧−𝑧₀^𝑙−1𝑔₂(𝑧)𝑧−𝛽₁^𝑛1+1⋯𝑧−𝛽_𝑡^𝑛𝑡+1, (2.7) where 𝑔₂(𝑧)=𝐵(𝑙−𝑁)𝑧^𝑡+𝑏_𝑡−1𝑧^𝑡−1+⋯+𝑏₀, (𝑏_𝑡−1⋯𝑏₀ are constants).
Next we distinguish two subcases.
Subcase 1.2.1 (𝑙≠𝑁). By using (2.5), we deduce deg𝑃(𝑧)≥deg𝑄(𝑧), that is, 𝑀≥𝑁. Since 𝑧₀≠𝛼_𝑖, that (2.6) and (2.7) imply 𝑀−𝑠≤deg𝑔₂=𝑡. By using (2.4), we get 𝑀𝑀≤𝑠+𝑡≤2+𝑁≤𝑀𝑘+12+𝑀().𝑘+1<𝑀𝑘≥2 (2.8) Which is a contradiction since 𝑘≥2.Subcase 1.2.2 (𝑙=𝑁). We further distinguish two subcases:Subcase 1.2.2.1 (𝑀≥𝑁). By using (2.6) and (2.7), we obtain 𝑀−𝑠≤deg𝑔₂≤𝑡−1<𝑡. Similar to Subcase 1.2.1, we obtain a contradiction 𝑀<𝑀.Subcase 1.2.2.2 (𝑀<𝑁). By using (2.6) and (2.7) again, we deduce 𝑙−1≤deg𝑔₁≤𝑠+𝑡−1, and hence 𝑀𝑁=𝑙≤𝑠+𝑡−1+1=𝑠+𝑡≤2+𝑁𝑘+1<𝑁 (2.9) this is impossible for 𝑘≥2.Case 2 (𝑓^(𝑘)(𝑧)−𝑏≠0 (𝑘≥2)). By Nevanlinna's second fundamental theorem, we have 𝑇𝑟,𝑓^(𝑘)≤𝑁𝑟,𝑓^(𝑘)+𝑁1𝑟,𝑓^(𝑘)+𝑁1𝑟,𝑓^(𝑘)−𝑏+𝑆𝑟,𝑓^(𝑘)≤1𝑁𝑘+1𝑟,𝑓^(𝑘)+12𝑁1𝑟,𝑓^(𝑘)+𝑆𝑟,𝑓^(𝑘)≤12+1𝑇𝑘+1𝑟,𝑓^(𝑘)+𝑆𝑟,𝑓^(𝑘). (2.10) It follows that 𝑇(𝑟,𝑓^(𝑘))=𝑆(𝑟,𝑓^(𝑘)), a contradiction.
The proof of Lemma 2.4 is complete.

Example 2.5. Take 𝑓(𝑧)=(𝑧+1)⁴𝑧². (2.11)

Remark 2.6. By a simple computation, for 𝑘=𝑏=2, we deduce 𝑓^′′(𝑧)−2 has only one zero at 𝑧=−3/4. Example 2.5 shows that the condition all of zeros of 𝑓^(𝑘)(𝑧) are the multiple seems not to be omitted.

Lemma 2.7. Let 𝑔(𝑧) be a nonconstant rational meromorphic function in ℂ and let 𝑛 (𝑛≥2) be an integer. If all zeros of 𝑔(𝑧) are multiple, then (𝑔^′(𝑧))^𝑛−𝑐=0 has at least two distinct zeros (where 𝑐 is a nonzero constant).

Proof. Suppose that (𝑔^′(𝑧))^𝑛−𝑐≠0, we set 𝑐₁,𝑐₂,…,𝑐_𝑛 the distinct solutions of 𝑤^𝑛=𝑐 (𝑛≥2). By Nevanlinna's second fundamental theorem, 𝑇𝑟,𝑔^′≤𝑁𝑟,𝑔^′+𝑁1𝑟,𝑔^′−𝑐₁+⋯+𝑁1𝑟,𝑔^′−𝑐_𝑛+𝑆𝑟,𝑔^′≤𝑁𝑟,𝑔^′+𝑆𝑟,𝑔^′≤12𝑁𝑟,𝑔^′+𝑆𝑟,𝑔^′≤12𝑇𝑟,𝑔^′+𝑆𝑟,𝑔^′. (2.12) It follows that 𝑇(𝑟,𝑔^′)=𝑆(𝑟,𝑔^′), a contradiction.
Assume, to the contrary, that (𝑔^′(𝑧))^𝑛−𝑐 has exactly one zero 𝑧₀.
If 𝑔(𝑧) is a nonconstant polynomial, then we set (𝑔^′(𝑧))^𝑛−𝑐=𝐾(𝑧−𝑧₀)^𝑙, where 𝐾 is nonzero constant and 𝑙 is a positive integer. Because all zeros of 𝑔(𝑧) are multiple and 𝑛≥2, we have 𝑙≥2 and ((𝑔^′(𝑧))^𝑛)^′=𝐾𝑙(𝑧−𝑧₀)^𝑙−1 (𝑙−1≥1). Thus, ((𝑔^′(𝑧))^𝑛)^′ has exactly one zero. Noting that all zeros of (𝑔^′(𝑧))^𝑛 are multiple, we deduce (𝑔^′(𝑧))^𝑛 has the same zero 𝑧₀. Hence (𝑔^′(𝑧₀))^𝑛=0, which contradicts with (𝑔^′(𝑧₀))^𝑛=𝑐≠0.
If 𝑔 is a rational function but not a polynomial, by all zeros of 𝑔 are multiple, then we know 𝑔^′ is not a constant. For 𝑛≥2, we obtain (𝑔^′−𝑐₁)(𝑔^′−𝑐₂)⋯(𝑔^′−𝑐_𝑛)=0 (where 𝑐₁,…,𝑐_𝑛 are distinct roots of 𝑤^𝑛=𝑐). So there exists a 𝑧₀ such that 𝑔^′(𝑧₀)=𝑐_𝑗 and 𝑔^′(𝑧₀)≠𝑐_𝑘 (where 1≤𝑗<𝑘≤𝑛). We have𝑔^′(𝑧)=𝑐_𝑗+𝐴𝑧−𝑧₀^𝑛𝑃(𝑧)≡𝑐_𝑘+𝐵𝑃(𝑧)(𝑛≥2,𝐴,𝐵∈ℂ⧵{0}). (2.13) So we obtain 𝑐_𝑘−𝑐_𝑗𝑃(𝑧)+𝐵≡𝐴𝑧−𝑧₀^𝑛, (2.14) for some nonconstant polynomial 𝑃(𝑧) with 𝑃(𝑧₀)≠0. Then 𝑐₀𝑃(𝑧)+𝐵≡𝐴(𝑧−𝑧₀)^𝑛, where 𝑐₀=𝑐_𝑗−𝑐_𝑘 is a nonzero constant. Furthermore, 𝑐₀𝑃^′(𝑧)≡𝐴𝑛𝑧−𝑧₀^𝑛−1. (2.15) Observing that 𝑔^′ has only multiple poles, and by (2.13), we obtain 𝑃(𝑧) has multiple zeros and 𝑃^′(𝑧₀)≠0. Putting 𝑧₀ into (2.15), we get that 𝑃^′(𝑧₀)=0, which is a contradiction.
The proof of Lemma 2.7 is complete.

3. Proof of Theorems

Proof of Theorem 1.2. We may assume that 𝐷={|𝑧|<1}. Suppose that ℱ is not normal in 𝐷. Without loss of generality, we assume that ℱ is not normal at 𝑧₀=0. Then, by Lemma 2.1, there are a sequence of complex numbers 𝑧_𝑗, 𝑧_𝑗→0 (𝑗→∞); a sequence of functions 𝑓_𝑗∈ℱ; and a sequence of positive numbers 𝜌_𝑗→0⁺ such that 𝑔_𝑗(𝜉)=𝜌_𝑗^−𝑘𝑓_𝑗(𝑧_𝑗+𝜌_𝑗𝜉) converges uniformly with respect to the spherical metric to a nonconstant mermorphic function 𝑔(𝜉) in ℂ and all zeros of 𝑔(𝜉) have the multiplicity at least 𝑘+2. Moreover, 𝑔(𝜉) is at most of order 2.
From the above, we get 𝑔_𝑗^(𝑘)(𝜉)=𝑓_𝑗^(𝑘)(𝑧_𝑗+𝜌_𝑗𝜉). Noting that all zeros of 𝑓_𝑗^(𝑘)(𝑧) are multiple, by Hurwitz's theorem, then all zeros of 𝑔_𝑗^(𝑘)(𝜉) have the multiplicity at least 2.
On every compact subsets of ℂ which contains no poles of 𝑔, we have𝑓_𝑗^(𝑘)𝑧_𝑗+𝜌_𝑗𝜉−𝑏=𝑔_𝑗^(𝑘)(𝜉)−𝑏 (3.1) converges uniformly with respect to the spherical metric to 𝑔^(𝑘)(𝜉)−𝑏(𝑘≥2).
If 𝑔^(𝑘)(𝜉)−𝑏≡0, then 𝑔 is a polynomial of degree 𝑘. This contradicts with that all zeros of 𝑓 have multiplicity at least 𝑘+2.
Since 𝑔 is a nonconstant meromorphic function, by Lemmas 2.3 and 2.4, we deduce that 𝑔^(𝑘)(𝜉)−𝑏 has at least two distinct zeros.
Next we will prove that 𝑔^(𝑘)(𝜉)−𝑏 has just a unique zero. On the contrary, let 𝜉₀ and 𝜉^∗₀ be two distinct solutions of 𝑔^(𝑘)(𝜉)−𝑏, and choose 𝛿(>0) small enough such that 𝐷(𝜉₀,𝛿)∩𝐷(𝜉^∗₀,𝛿)=∅ where 𝐷(𝜉₀,𝛿)={𝜉∶|𝜉−𝜉₀|<𝛿} and 𝐷(𝜉^∗₀,𝛿)={𝜉∶|𝜉−𝜉^∗₀|<𝛿}. From (3.1) and Hurwitz's theorem, there are points 𝜉_𝑗∈𝐷(𝜉₀,𝛿), 𝜉^∗_𝑗∈𝐷(𝜉^∗₀,𝛿) such that for sufficiently large 𝑗𝑓_𝑗^(𝑘)𝑧_𝑗+𝜌_𝑗𝜉_𝑗𝑓−𝑏=0,_𝑗(𝑘)𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗−𝑏=0. (3.2)
By the hypothesis that for each pair of functions 𝑓 and 𝑔 in ℱ, 𝑓^(𝑘)(𝜉) and 𝑔^(𝑘)(𝜉) share 𝑏 in 𝐷, we know that for any positive integer 𝑚𝑓_𝑚^(𝑘)𝑧_𝑗+𝜌_𝑗𝜉_𝑗𝑓−𝑏=0,_𝑚(𝑘)𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗−𝑏=0. (3.3) Fix 𝑚, take 𝑗→∞, and in view of that 𝑧_𝑗+𝜌_𝑗𝜉_𝑗→0, 𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗→0, we have 𝑓_𝑚^(𝑘)(0)−𝑏=0. (3.4) Since the zeros of 𝑓_𝑚^(𝑘)−𝑏 have no accumulation point, we have that 𝑧_𝑗+𝜌_𝑗𝜉_𝑗=0 and 𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗=0.
Hence 𝜉_𝑗𝑧=−^𝑗𝜌_𝑗,𝜉^∗_𝑗𝑧=−^𝑗𝜌_𝑗. (3.5) This contradicts with 𝜉_𝑗∈𝐷(𝜉₀,𝛿), 𝜉^∗_𝑗∈𝐷(𝜉^∗₀,𝛿) and 𝐷(𝜉₀,𝛿)∩𝐷(𝜉^∗₀,𝛿)=∅ So 𝑔^(𝑘)(𝜉)−𝑏 has just a unique zero, which contradicts the fact that 𝑔^(𝑘)(𝜉)−𝑏 has at least two distinct zeros.
The proof of Theorem 1.2 is complete.

Proof of Theorem 1.7. Likewise the proof of Theorem 1.2, we assume that ℱ is not normal at 𝑧₀=0. Then, by Lemma 2.1, there are a sequence of complex numbers 𝑧_𝑗, 𝑧_𝑗→0(𝑗→∞), a sequence of functions 𝑓_𝑗∈ℱ, and a sequence of positive numbers 𝜌_𝑗→0⁺ such that 𝑔_𝑗(𝜉)=𝜌_𝑗⁻¹𝑓_𝑗(𝑧_𝑗+𝜌_𝑗𝜉) converges uniformly with respect to the spherical metric to a nonconstant mermorphic functions 𝑔(𝜉), all zeros of 𝑔(𝜉) are multiple. Moreover, 𝑔(𝜉) is of order at most 2.
On every compact subsets of ℂ which contains no poles of 𝑔, we have𝑓_𝑗𝑧_𝑗+𝜌_𝑗𝜉+𝑎𝑓^′_𝑗𝑧_𝑗+𝜌_𝑗𝜉^𝑛−𝑏=𝜌_𝑗𝑔_𝑗(𝜉)+𝑎𝑔^′_𝑗(𝜉)^𝑛−𝑏 (3.6) converges uniformly with respect to the spherical metric to 𝑎(𝑔^′(𝜉))^𝑛−𝑏(𝑛≥2).
If 𝑎(𝑔^′(𝜉))^𝑛−𝑏≡0, then 𝑔 is a polynomial of degree 1, which contradicts with the zeros of 𝑔(𝜉) are multiple.
By Lemmas 2.2 and 2.7, we derive that 𝑎(𝑔^′(𝜉))^𝑛−𝑏 has at least two distinct zeros.
Next we will prove that 𝑎(𝑔^′(𝜉))^𝑛−𝑏 has just a unique zero. On the contrary, let 𝜉₀ and 𝜉^∗₀ be two distinct solutions of (𝑔^′(𝜉))^𝑛−𝑐=0 (where 𝑐=𝑏/𝑎≠0), and choose 𝛿(>0) small enough such that 𝐷(𝜉₀,𝛿)∩𝐷(𝜉^∗₀,𝛿)=∅ where 𝐷(𝜉₀,𝛿)={𝜉∶|𝜉−𝜉₀|<𝛿} and 𝐷(𝜉^∗₀,𝛿)={𝜉∶|𝜉−𝜉^∗₀|<𝛿}. From (3.6), by Hurwitz's theorem, there are points 𝜉_𝑗∈𝐷(𝜉₀,𝛿), 𝜉^∗_𝑗∈𝐷(𝜉^∗₀,𝛿) such that for sufficiently large 𝑗𝑓_𝑗𝑧_𝑗+𝜌_𝑗𝜉_𝑗+𝑎𝑓^′_𝑗𝑧_𝑗+𝜌_𝑗𝜉_𝑗^𝑛𝑓−𝑏=0,_𝑗𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗+𝑎𝑓^′_𝑗𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗^𝑛−𝑏=0. (3.7)
By the hypothesis that for each pair of functions 𝑓 and 𝑔 in ℱ, 𝑓+𝑎(𝑓^′)^𝑛 and 𝑔+𝑎(𝑔^′)^𝑛 share 𝑏 in 𝐷, we know that for any positive integer 𝑚𝑓_𝑚𝑧_𝑗+𝜌_𝑗𝜉_𝑗+𝑎𝑓^′_𝑚𝑧_𝑗+𝜌_𝑗𝜉_𝑗^𝑛𝑓−𝑏=0,_𝑚𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗+𝑎𝑓^′_𝑚𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗^𝑛−𝑏=0. (3.8) Fix 𝑚, take 𝑗→∞, and in view of that 𝑧_𝑗+𝜌_𝑗𝜉_𝑗→0, 𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗→0, then 𝑓_𝑚(0)+𝑎𝑓^′_𝑚(0)^𝑛−𝑏=0. (3.9) Since the zeros of 𝑓_𝑚+𝑎(𝑓^′_𝑚)^𝑛−𝑏 have no accumulation point, we have that 𝑧_𝑗+𝜌_𝑗𝜉_𝑗=0, 𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗=0.
Hence 𝜉_𝑗𝑧=−^𝑗𝜌_𝑗,𝜉^∗_𝑗𝑧=−^𝑗𝜌_𝑗. (3.10) This contradicts with 𝜉_𝑗∈𝐷(𝜉₀,𝛿), 𝜉^∗_𝑗∈𝐷(𝜉^∗₀,𝛿) and 𝐷(𝜉₀,𝛿)∩𝐷(𝜉^∗₀,𝛿)=∅. So (𝑔^′(𝜉))^𝑛−𝑐 has just a unique zero which contradicts with the fact (𝑔^′(𝜉))^𝑛−𝑐 has at least two distinct zeros.
The proof of Theorem 1.7 is complete.

Proof of Corollary 1.8. Similar to the proof of Theorem 1.2, we assume that ℱ is not normal at 𝑧₀=0. Then by Lemma 2.1, there exist a sequence of functions 𝑓_𝑗∈ℱ, a sequence of complex numbers 𝑧_𝑗→𝑧₀ and a sequence of positive numbers 𝜌_𝑗→0, such that 𝑔_𝑗(𝜉)=𝜌_𝑗⁻¹𝑓_𝑗𝑧_𝑗+𝜌_𝑗𝜉⟶𝑔(𝜉), (3.11) locally uniformly with respect to the spherical metric, where 𝑔(𝜉) is nonconstant meromorphic function on ℂ, all of whose zeros have multiplicity at least 2.
On every compact subsets of ℂ which contains no poles of 𝑔, we have𝑓_𝑗𝑧_𝑗+𝜌_𝑗𝜉_𝑗+𝑎𝑓^′_𝑗𝑧_𝑗+𝜌_𝑗𝜉_𝑗^𝑛−𝑏=𝜌_𝑗𝑔_𝑗(𝜉)+𝑎𝑔^′_𝑗(𝜉)^𝑛−𝑏⟶𝑎𝑔^′(𝜉)^𝑛−𝑏. (3.12) If 𝑎(𝑔^′)^𝑛−𝑏≡0, that is (𝑔^′)^𝑛≡𝑏/𝑎. Then 𝑔 is a polynomial with degree 1. It contradicts the fact that all zeros of 𝑔 have multiplicity at least 2.
By Lemmas 2.2 and 2.7, we know there exists some 𝜉₀∈ℂ such that 𝑎𝑔^′𝜉₀^𝑛−𝑏=0. (3.13) From the above discussion, we get 𝑔(𝜉₀)≠∞. Since 𝑔_𝑗(𝜉)→𝑔(𝜉) uniformly in a closed disk 𝑈(𝜉₀;𝛿), then by Hurwitz's theorem, there is 𝜉_𝑗→𝜉₀ (𝑗→∞) such that 𝑓_𝑗𝑧_𝑗+𝜌_𝑗𝜉_𝑗+𝑎𝑓^′_𝑗𝑧_𝑗+𝜌_𝑗𝜉_𝑗^𝑛−𝑏=𝜌_𝑗𝑔_𝑗𝜉_𝑗+𝑎𝑔^′_𝑗𝜉_𝑗^𝑛−𝑏⟶𝑎𝑔^′𝜉₀^𝑛−𝑏=0. (3.14) For 𝑗 sufficiently large, we obtain 𝑓_𝑗(𝑧_𝑗+𝜌_𝑗𝜉_𝑗)+𝑎(𝑓^′_𝑗(𝑧_𝑗+𝜌_𝑗𝜉_𝑗))^𝑛−𝑏=0. By the assumption, we get |𝑓_𝑗(𝑧_𝑗+𝜌_𝑗𝜉_𝑗)|≥𝑀. This implies 𝑔_𝑗𝜉_j≥𝜌_𝑗⁻¹𝑀. (3.15)
Noting that 𝑔 is holomorphic at 𝜉₀, then |𝑔(𝜉)|≤𝐶 for some positive constant 𝐶, and for all 𝜉∈𝑈(𝜉₀;𝜂). Again by 𝑔_𝑗→𝑔 and for all 𝜀>0, there are some 𝑗₀ such that for all 𝜉∈𝑈(𝜉₀;𝜂), we have 𝑔_𝑗(𝜉)−𝑔(𝜉)<𝜀. (3.16) For all 𝑗≥𝑗₀, by (3.15), we find that 𝐶≥𝑔𝜉_𝑗≥𝑔_𝑗𝜉_𝑗−𝑔_𝑗𝜉_𝑗−𝑔𝜉_𝑗≥𝜌_𝑗⁻¹𝑀−𝜀. (3.17) That is 𝐶≥𝜌_𝑗⁻¹𝑀−𝜀 (3.18) for all 𝑗≥𝑗₀, which is impossible.
Thus, the proof of Corollary 1.8 is proved.

Proof of Corollary 1.9 (For any 𝑧₀∈𝐷, we will prove that ℱ is normal at 𝑧₀). From the assumptions in the theorem, we can choose a sufficiently small 𝑟>0 such that 𝐸(𝑧₀,𝑟,ℱ,𝑛)=∅ or {𝑧₀}. Thus we see that 𝑓+𝑎(𝑓^′)^𝑛≠𝑏 in 𝑈(𝑧₀,𝑟)⧵{𝑧₀} for any 𝑓(𝑧)∈ℱ. Theorem F gives that ℱ is normal in 𝑈(𝑧₀,𝑟)⧵{𝑧₀}. Next we will prove that ℱ is normal at 𝑧₀.
Since the Möbius transformation for ℱ does not change the normality, without loss of generality, we may assume that (𝑐,𝑑)=(0,∞).
Set 𝐶_𝑟/2∶={𝑧∈𝑈(𝑧₀,𝑟)∶|𝑧−𝑧₀|=𝑟/2}. By the given condition, we see that ℱ is normal on 𝐶_𝑟/2. Hence for any sequence of functions {𝑓_𝑛}^∞_𝑛=1⊂ℱ, there is a subsequence, say {𝑓_𝑛𝑘(𝑧)}⊂{𝑓_𝑛(𝑧)}, such that as 𝑘→∞,𝑓_𝑛𝑘(𝑧)⟶𝑔(𝑧) (3.19) uniformly convergence on 𝐶_𝑟/2.
If 𝑔(𝑧)≢∞, by 𝑓_𝑛𝑘(𝑧)≠0,∞, we have 𝑔(𝑧) is analytic on 𝐶_𝑟/2. Hence for all 𝜀>0, there is an integer 𝐾 such that𝑓_𝑛𝑘(𝑧)−𝑓_𝑛𝑚(𝑧)<𝜀 (3.20) for all 𝑧∈𝐶_𝑟/2. For arbitrary numbers 𝑘,𝑚>𝐾, by the maximum modular theorem, we have 𝑓_𝑛𝑘(𝑧)−𝑓_𝑛𝑚(𝑧)<𝜀 (3.21) for all |𝑧|≤𝐶_𝑟/2. Thus ℱ is normal at 𝑧=𝑧₀.
If 𝑔(𝑧)≡∞, then there is an integer 𝐾 and a positive number 𝑀 such that𝑓_𝑛𝑘(𝑧)≥𝑀 (3.22) for all 𝑘≥𝐾, 𝑧∈𝐶_𝑟/2. Noting that 𝑓_𝑛𝑘≠0 in 𝑈(𝑧₀,𝑟), we obtain 𝑓_𝑛𝑘(𝑧)≥𝑀 (3.23) for all 𝑘≥𝐾, |𝑧|≤𝐶_𝑟/2 by the minimum modular theorem. Hence, we have 𝑓_𝑛𝑘⟶∞(𝑘⟶∞) (3.24) uniformly on |𝑧|≤𝑟/2. Therefore ℱ is normal at 𝑧=𝑧₀.

Appendix

Using exactly the same argument as in the proof of Lemma 2.4, we can show that the following result.

Theorem A.1. Let 𝑘 be a positive integer and 𝑏 be a nonzero finite complex number and let 𝑓(𝑧) be a nonconstant rational meromorphic function in ℂ. If all zeros of 𝑓(𝑧) have multiplicity at least 𝑘+2 and all zeros of 𝑓^(𝑘)(𝑧) have multiplicity at least 3, then 𝑓^(𝑘)(𝑧)−𝑏 has at least two distinct zeros.

Also with the same method as in the proof of Theorem 1.2, we obtain the next conclusion.

Theorem A.2. Let 𝑘 be a positive integer and 𝑏 be a nonzero finite complex number, and let ℱ be a family of meromorphic functions in 𝐷, all of whose zeros are of multiplicity at least 𝑘+2. If, for every pair 𝑓,𝑔∈ℱ, all zeros of 𝑓^(𝑘)(𝑧),𝑔^(𝑘)(𝑧) have multiplicity at least 3, 𝑓^(𝑘)(𝑧) and 𝑔^(𝑘)(𝑧) share 𝑏 on 𝐷, then ℱ is normal in 𝐷.

For further study, we pose three questions.

Question 1. Whether the condition all zeros of 𝑓^(𝑘)(𝑧) have multiplicity at least 2 in Theorem 1.2 can be weakened?

Question 2. Whether the conclusion of Theorem 1.2 still holds for 𝑘=1?

Question 3. Whether the condition 𝑏≠0 in Theorem 1.7 is necessary for 𝑛>2?

Acknowledgments

The authors are grateful to the referee for his(her) valuable suggestions and comments. The author would like to express hearty thanks to Professor Hongxun Yi for his valuable advice and helpful information. This research is supported by the NSFC (no. 10771121), the NSF of Shandong (no. Z2008A01), the RFDP (no. 20060422049), the NSFC (no. 10871130) and the EEMM (no. 10XKJ01).