A Note on Normal Families of Meromorphic Functions Concerning Shared Values

Abstract

We study the normality of families of meromorphic functions related to a Hayman conjecture. We consider whether a family of meromorphic functions ℱ is normal in 𝐷 if, for every pair of functions 𝑓 and 𝑔 in ℱ, 𝑓^′−𝑎𝑓^𝑛 and 𝑔^′−𝑎𝑔^𝑛 share the value 𝑏 for 𝑛=1,2, and 3, where 𝑎 and 𝑏≠0 are two finite complex numbers. Some examples show that the conditions in our results are the best possible.

1. Introduction and Main Results

Let 𝑓(𝑧) and 𝑔(𝑧) be two nonconstant meromorphic functions in a domain 𝐷⊆ℂ, and let 𝑎 be a finite complex value. We say that 𝑓 and 𝑔 share 𝑎 CM (or IM) in 𝐷 provided that 𝑓−𝑎 and 𝑔−𝑎 have the same zeros counting (or ignoring) multiplicity in 𝐷. When 𝑎=∞ the zeros of 𝑓−𝑎 mean the poles of 𝑓 (see [1]). It is assumed that the reader is familiar with the standard notations and the basic results of Nevanlinna’s value-distribution theory ([2–4] or [1]).

Bloch’s principle [5] states that every condition which reduces a meromorphic function in the plane ℂ to be a constant forces a family of meromorphic functions in a domain 𝐷 normal. Although the principle is false in general (see [6]), many authors proved the normality criterion for families of meromorphic functions corresponding to Liouville-Picard-type theorem (see [7] or [4]).

It is also more interesting to find normality criteria from the point of view of shared values. In this area, Schwick [8] first proved an interesting result that a family of meromorphic functions in a domain is normal if it in every function in it shares three distinct finite complex numbers with its first derivative. Later, more results about normality criteria concerning shared values can be found, for instance, see [9–11] and so on. In recent years, this subject has attracted the attention of many researchers worldwide.

We now first introduce a normality criterion related to a Hayman normal conjecture [12].

Theorem 1.1. Let ℱ be a family of holomorphic (meromorphic) functions defined in a domain 𝐷, 𝑛∈ℕ,𝑎≠0,𝑏∈ℂ. If 𝑓^′(𝑧)+𝑎𝑓^𝑛(𝑧)−𝑏≠0 for each function 𝑓(𝑧)∈ℱ and 𝑛≥2(𝑛≥3), then ℱ is normal in 𝐷.

The results for the holomorphic case are due to Drasin [7] for 𝑛≥3, Pang [13] for 𝑛=3, and Chen and Fang [14] for 𝑛=2, Ye [15] for 𝑛=2, and Chen and Gu [16] for the generalized result with 𝑎 and 𝑏 replaced by meromorphic functions. The results for the meromorphic case are due to Li [17], Li [18], and Langley [19] for 𝑛≥5, Pang [13] for 𝑛=4, Chen and Fang [14] for 𝑛=3, and Zalcman [20] for 𝑛=3, obtained independently.

When 𝑛=2 and ℱ is meromorphic, Theorem 1.1 is not valid in general. Fang and Yuan [21] gave an example to this, and moreover a result added other conditions below.

Example 1.2. The family of meromorphic functions ℱ={𝑓_𝑗(𝑧)=𝑗𝑧/(𝑗𝑧−1)²∶𝑗=1,2,…,} is not normal in 𝐷={𝑧∶|𝑧|<1}. This is deduced by 𝑓^#_𝑗(0)=𝑗→∞, as 𝑗→∞, and Marty’s criterion [2], although, for any 𝑓_𝑗(𝑧)∈ℱ,𝑓^′_𝑗+𝑓²_𝑗=𝑗(𝑗𝑧−1)⁻⁴≠0.

Here 𝑓^#(𝜉) denotes the spherical derivative 𝑓^#(𝜉)=𝑓^′(𝜉)1+𝑓(𝜉)². (1.1)

Theorem 1.3. Let ℱ be a family of meromorphic functions in a domain 𝐷 and 𝑎≠0,𝑏∈ℂ. If 𝑓^′(𝑧)+𝑎(𝑓(𝑧))²−𝑏≠0 and the poles of 𝑓(𝑧) are of multiplicity ≥3 for each 𝑓(𝑧)∈ℱ, then ℱ is normal in 𝐷.

In 2008, by the ideas of shared values, Zhang [11] proved the following.

Theorem 1.4. Let ℱ be a family of meromorphic (holomorphic) functions in 𝐷, 𝑛 a positive integer, and 𝑎, 𝑏 two finite complex numbers such that 𝑎≠0. If 𝑛≥4(𝑛≥2) and, for every pair of functions 𝑓 and 𝑔 in ℱ, 𝑓′−𝑎𝑓^𝑛 and 𝑔′−𝑎𝑔^𝑛 share the value 𝑏, then ℱ is normal in 𝐷.

Example 1.5 (see [11]). The family of meromorphic functions ℱ={𝑓_𝑗(𝑧)=1/(𝑗(𝑧−1/𝑗))∶𝑗=1,2,…,} is not normal in 𝐷={𝑧∶|𝑧|<1}. Obviously 𝑓^′_𝑗−𝑓³_𝑗=−𝑧/(𝑗(𝑧−1/𝑗)³). So, for each pair 𝑚,𝑗,𝑓^′_𝑗−𝑓³_𝑗 and 𝑓^′_𝑚−𝑓³_𝑚 share the value 0 in 𝐷, but ℱ is not normal at the point 𝑧=0 since 𝑓^#_𝑗(0)=2(𝑗)³/1+𝑗→∞, as 𝑗→∞.

Remark 1.6. Example 1.5 shows that Theorem 1.4 is not valid when 𝑛=3, and the condition 𝑛=4 is the best possible for the meromorphic case.

It is natural to ask under what conditions Theorem 1.4 holds for 𝑛≤3. In this paper, we answer the above question and prove the following results.

Theorem 1.7. Let ℱ be a family of meromorphic functions in 𝐷 and 𝑎 and 𝑏 two finite complex numbers such that 𝑎≠0. Suppose that each 𝑓(𝑧)∈ℱ has no simple pole. If 𝑓′−𝑎𝑓³ and 𝑔′−𝑎𝑔³ share the value 𝑏 for every pair of functions 𝑓 and 𝑔 in ℱ, then ℱ is normal in 𝐷.

Remark 1.8. Example 1.5 shows that the condition added in Theorem 1.7 about the multiplicity of poles of 𝑓(𝑧) is the best possible.

Theorem 1.9. Let ℱ be a family of meromorphic functions in 𝐷 and 𝑎 and 𝑏 two finite complex numbers such that 𝑎≠0. Suppose that 𝑓(𝑧) admits the zeros of multiple and the poles of multiplicity ≥3 for each 𝑓(𝑧)∈ℱ. If 𝑓′−𝑎𝑓² and 𝑔′−𝑎𝑔² share the value 𝑏 for every pair of functions 𝑓 and 𝑔 in ℱ, then ℱ is normal in 𝐷.

Remark 1.10. Example 1.2 shows that the condition added in Theorem 1.9 about the multiplicity of poles and zeros of 𝑓(𝑧) is the best possible.

Theorem 1.11. Let ℱ be a family of meromorphic functions in 𝐷 and 𝑎 and 𝑏 two nonzero finite complex numbers. Suppose that 𝑓(𝑧)≠0 and that its poles are multiple for each 𝑓(𝑧)∈ℱ. If 𝑓′−𝑎𝑓 and 𝑔′−𝑎𝑔 share the value 𝑏 for every pair of functions 𝑓 and 𝑔 in ℱ, then ℱ is normal in 𝐷.

Corollary 1.12. Let ℱ be a family of holomorphic functions in 𝐷 and 𝑎 and 𝑏 two finite complex numbers such that 𝑏≠0. Suppose that 𝑓(𝑧)≠0 for each 𝑓(𝑧)∈ℱ. If 𝑓′−𝑎𝑓 and 𝑔′−𝑎𝑔 share the value 𝑏 for every pair of functions 𝑓 and 𝑔 in ℱ, then ℱ is normal in 𝐷.

Example 1.13. The family of holomorphic functions ℱ={𝑓_𝑗(𝑧)=𝑗𝑧𝑒^𝑧−𝑗𝑒^𝑧+𝑗−𝑏∶𝑗=1,2,…,} is not normal in 𝐷={𝑧∶|𝑧|<1}. Obviously 𝑓^′_𝑗−𝑓_𝑗=𝑗(𝑒^𝑧−1)+𝑏. So, for each pair 𝑚,𝑗,𝑓^′_𝑗−𝑓_𝑗 and 𝑓^′_𝑚−𝑓_𝑚 share the value 𝑏 in 𝐷. On the other hand, 𝑓_𝑗(0)=−𝑏,𝑓_𝑗(1/𝑗)=𝑗(1+(1/𝑗)+𝑜(1))→∞, as 𝑗→∞. This implies that the family ℱ fails to be equicontinuous at 0, and thus ℱ is not normal at 0.

Theorem 1.14. Let ℱ be a family of meromorphic functions in 𝐷 and 𝑎 and 𝑏 two finite complex numbers such that 𝑏≠0. Suppose that 𝑓(𝑧)≠0 and 𝑓^′(𝑧)−𝑎𝑓(𝑧)≠𝑏 for each 𝑓(𝑧)∈ℱ. Then ℱ is normal in 𝐷.

Example 1.15. The family of holomorphic functions ℱ={𝑓_𝑗(𝑧)=𝑗(𝑧+1)−1∶𝑗=1,2,…,} is normal in 𝐷={𝑧∶|𝑧|<1}. Obviously 𝑓_𝑗(𝑧)≠0 and 𝑓^′_𝑗−𝑓_𝑗=−𝑗𝑧+1. So, for each pair 𝑚,𝑗,𝑓^′_𝑗−𝑓_𝑗 and 𝑓^′_𝑚−𝑓_𝑚 share the value 1 in 𝐷. Corollary 1.12 implies that the family ℱ is normal in 𝐷.

Example 1.16. The family of meromorphic functions ℱ={𝑓_𝑗(𝑧)=(𝑧/𝑗)−1∶𝑗=1,2,…,} is normal in 𝐷={𝑧∶|𝑧|<1}. The reason is that the conditions of Theorem 1.14 hold that 𝑓_𝑗(𝑧)≠0 and 𝑓^′_𝑗−𝑓_𝑗=((1−𝑧)/𝑗)+1≠1 in 𝐷={𝑧∶|𝑧|<1}.

Remark 1.17. Example 1.13 shows that Theorem 1.4 is not valid when 𝑛=1 and in the holomorphic case and the condition 𝑓(𝑧)≠0 is necessary in Theorem 1.11, Corollary 1.12. Both Examples 1.15 and 1.16 tell us that Corollary 1.12 and Theorem 1.14 occur.

2. Preliminary Lemmas

In order to prove our result, we need the following lemmas. The first one extends a famous result by Zalcman [22] concerning normal families.

Lemma 2.1 (see [23]). Let ℱ be a family of meromorphic functions on the unit disc satisfying all zeros of functions in ℱ  which have multiplicity ≥𝑝 and all poles of functions in ℱ which have multiplicity ≥𝑞. Let 𝛼 be a real number satisfying −𝑞<𝛼<𝑝. Then ℱ is not normal at 0 if and only if there exist (a)a number 0<𝑟<1,(b)points 𝑧_𝑛with |𝑧_𝑛|<𝑟,(c)functions 𝑓_𝑛∈ℱ,(d)positive numbers 𝜌_𝑛→0such that 𝑔_𝑛(𝜁)∶=𝜌^−𝛼𝑓_𝑛(𝑧_𝑛+𝜌_𝑛𝜁) converges spherically uniformly on each compact subset of ℂ to a nonconstant meromorphic function 𝑔(𝜁), whose all zeros have multiplicity ≥𝑝 and all poles have multiplicity ≥𝑞 and order is at most 2.

Remark 2.2. If ℱ is a family of holomorphic functions on the unit disc in Lemma 2.1, then 𝑔(𝜁) is a nonconstant entire function whose order is at most 1.

The order of 𝑔 is defined by using Nevanlinna’s characteristic function 𝑇(𝑟,𝑔): 𝜌𝑔=lim_𝑟→∞suplog𝑇𝑟,𝑔log𝑟. (2.1)

Lemma 2.3 (see [24] or [25]). Let 𝑓(𝑧) be a meromorphic function and 𝑐∈ℂ⧵{0}. If 𝑓(𝑧) has neither simple zero nor simple pole and 𝑓^′(𝑧)≠𝑐, then 𝑓(𝑧) is constant.

Lemma 2.4 (see [26]). Let 𝑓(𝑧) be a transcendental meromorphic function of finite order in ℂ, with no simple zero; then 𝑓^′(𝑧) assumes every nonzero finite value infinitely often.

Lemma 2.5 (see [3]). Let 𝑓(𝑧) be a meromorphic function in ℂ; then 𝑇≤1𝑟,𝑓2+𝑘𝑁1𝑟,𝑓+22+𝑘𝑁1𝑟,𝑓^(𝑘),𝑇≤−1+𝑆𝑟,𝑓𝑟,𝑓𝑁1𝑟,𝑓+𝑁𝑟,𝑓+𝑁1𝑟,𝑓^(𝑘).−1+𝑆𝑟,𝑓 (2.2)(2.3)

Remark 2.6. Both (2.2) and (2.3) are called the Hayman inequality and Milloux inequality, respectively.

3. Proof of the Results

Proof of Theorem 1.7. Suppose that ℱ is not normal in 𝐷. Then there exists at least one point 𝑧₀ such that ℱ is not normal at the point 𝑧₀. Without loss of generality we assume that 𝑧₀=0. By Lemma 2.1, there exist points 𝑧_𝑗→0, positive numbers 𝜌_𝑗→0, and functions 𝑓_𝑗∈ℱ such that 𝑔_𝑗(𝜉)=𝜌_𝑗^1/2𝑓_𝑗𝑧_𝑗+𝜌_𝑗𝜉⟹𝑔(𝜉) (3.1) locally uniformly with respect to the spherical metric, where 𝑔 is a nonconstant meromorphic function in ℂ. Moreover, the order of 𝑔 is no greater than 2 and the poles of 𝑔 are of multiplicity ≥2.
From (3.1) we know that𝑔^′_𝑗(𝜉)=𝜌_𝑗^3/2𝑓^′_𝑗𝑧_𝑗+𝜌_𝑗𝜉⟹𝑔^′(𝜉),𝜌_𝑗^3/2𝑓^′_𝑗𝑧_𝑗+𝜌_𝑗𝜉−𝑎𝑓³_𝑗𝑧_𝑗+𝜌_𝑗𝜉−𝑏=𝑔^′_𝑗(𝜉)−𝑎𝑔³_𝑗(𝜉)−𝜌_𝑗^3/2𝑏⟹𝑔^′(𝜉)−𝑎𝑔³(𝜉) (3.2)(3.3) in ℂ⧵𝐒 locally uniformly with respect to the spherical metric, where 𝐒 is the set of all poles of 𝑔(𝜉).
If 𝑔′−𝑎𝑔³≡0, then −(1/2)(1/𝑔²)≡𝑎𝜉+𝑐, where 𝑐 is a constant. This contradicts with the idea that the poles of 𝑔 are of multiplicity ≥2. So 𝑔′−𝑎𝑔³≢0.
If 𝑔′−𝑎𝑔³≠0, then 𝑔′/𝑔³≠𝑎. Set 𝑔=1/𝜑; then 𝜑𝜑′≠−𝑎. By Lemma 2.3, 𝜑 is a constant, so 𝑔 is also a constant which is a contradiction with 𝑔 being a nonconstant. Hence, 𝑔′−𝑎𝑔³ is a nonconstant meromorphic function and has at least one zero.
Next we prove that 𝑔′−𝑎𝑔³ has just a unique zero. By contraries, let 𝜉₀ and 𝜉^∗₀ be two distinct zeros of 𝑔′−𝑎𝑔³, and choose 𝛿(>0) small enough such that 𝐷(𝜉₀,𝛿)∩𝐷(𝜉^∗₀,𝛿)=𝜙, where 𝐷(𝜉₀,𝛿)={𝜉∶|𝜉−𝜉₀|<𝛿} and 𝐷(𝜉^∗₀,𝛿)={𝜉∶|𝜉−𝜉^∗₀|<𝛿}. From (3.3), by Hurwitz^'s theorem, there exist points 𝜉_𝑗∈𝐷(𝜉₀,𝛿), 𝜉^∗_𝑗∈𝐷(𝜉^∗₀,𝛿) such that for sufficiently large𝑗𝑓^′_𝑗𝑧_𝑗+𝜌_𝑗𝜉_𝑗−𝑎𝑓³_𝑗𝑧_𝑗+𝜌_𝑗𝜉_𝑗𝑓−𝑏=0,_′𝑗𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗−𝑎𝑓³_𝑗𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗−𝑏=0. (3.4)
By the hypothesis that, for each pair of functions 𝑓 and 𝑔 in ℱ, 𝑓′−𝑎𝑓³ and 𝑔′−𝑎𝑔³ share 𝑏 in 𝐷, we know that, for any positive integer 𝑚𝑓^′_𝑚𝑧_𝑗+𝜌_𝑗𝜉_𝑗−𝑎𝑓³_𝑚𝑧_𝑗+𝜌_𝑗𝜉_𝑗𝑓−𝑏=0,_′𝑚𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗−𝑎𝑓³_𝑚𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗−𝑏=0. (3.5)
Fix 𝑚, take 𝑗→∞, and note that 𝑧_𝑗+𝜌_𝑗𝜉_𝑗→0, 𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗→0; then 𝑓^′_𝑚(0)−𝑎𝑓³_𝑚(0)−𝑏=0. Since the zeros of 𝑓_𝑚′−𝑎𝑓³_𝑚−𝑏 have no accumulation point,𝑧_𝑗+𝜌_𝑗𝜉_𝑗=0,𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗=0. (3.6) Hence, 𝜉_𝑗=−(𝑧_𝑗/𝜌_𝑗),𝜉^∗_𝑗=−(𝑧_𝑗/𝜌_𝑗). This contradicts with 𝜉_𝑗∈𝐷(𝜉₀,𝛿),𝜉^∗_𝑗∈𝐷(𝜉^∗₀,𝛿), and 𝐷(𝜉₀,𝛿)∩𝐷(𝜉^∗₀,𝛿)=𝜙. So 𝑔′−𝑎𝑔³ has just a unique zero, which can be denoted by 𝜉₀.
Set 𝑔=1/𝜑 again; then 𝑔′−𝑎𝑔³=−(𝜑′𝜑+𝑎)/𝜑³. So(𝜑′𝜑+𝑎)/𝜑³ has only a unique zero 𝜉₀. Therefore, 𝜉₀ is a multiple pole of 𝜑, or else a zero of 𝜑′𝜑+𝑎. If 𝜉₀ is a multiple pole of 𝜑, since (𝜑′𝜑+𝑎)/𝜑³ has only one zero 𝜉₀, then 𝜑′𝜑+𝑎≠0. By Lemma 2.3 again, 𝜑 is a constant which contradicts with the idea that 𝑔 is a nonconstant.
So 𝜑 has no multiple pole and 𝜑′𝜑+𝑎 has just a unique zero 𝜉₀. By Lemma 2.3, 𝜑 is not any transcendental function.
If 𝜑 is a nonconstant polynomial, then 𝜑′𝜑+𝑎=𝐴(𝜉−𝜉₀)^𝑙, where A is a nonzero constant and 𝑙 is a positive integer because the poles of 𝑔 are of multiplicity ≥2. So the zeros of 𝜑 are of multiplicity ≥2, and thus, 𝑙≥3. Set 𝜓=(1/2)𝜑²; then 𝜓′=𝐴(𝜉−𝜉₀)^𝑙−𝑎 and 𝜓^′′=𝐴𝑙(𝜉−𝜉₀)^𝑙−1. Note that the zeros of 𝜓 are of multiplicity ≥4. But 𝜓^′′ has only one zero 𝜉₀, and so 𝜓 has only the same zero 𝜉₀. Hence 𝜓^′(𝜉₀)=0 which contradicts with 𝜓^′(𝜉₀)=𝑎≠0. Therefore 𝜑 and 𝜓 are rational functions which are not polynomials and 𝜓′+𝑎 has just a unique zero 𝜉₀.
Next we prove that there exists no rational function such as 𝜓. Noting that 𝜓=(1/2)𝜑², 𝜑 has no multiple pole, and the zeros of 𝜑 are of multiplicity ≥2, we can set𝜓(𝜉)=𝐴𝜉−𝜉₁^𝑚1𝜉−𝜉₂^𝑚2⋯𝜉−𝜉_𝑠^𝑚𝑠𝜉−𝜂₁²𝜉−𝜂₂²⋯𝜉−𝜂_𝑡², (3.7) where 𝐴 is a nonzero constant, 𝑠≥1,𝑡≥1,and𝑚_𝑖≥4(𝑖=1,2,…,𝑠). For stating briefly, denote 𝑚=𝑚₁+𝑚₂+⋯+𝑚_𝑠≥4𝑠. (3.8) From (3.7), 𝜓^′(𝜉)=𝐴𝜉−𝜉₁^𝑚1−1𝜉−𝜉₂^𝑚2−1⋯𝜉−𝜉_𝑠^𝑚𝑠−1ℎ(𝜉)𝜉−𝜂₁³𝜉−𝜂₂³⋯𝜉−𝜂_𝑡³=𝑝₁(𝜉)𝑞₁(𝜉), (3.9) where ℎ(𝜉)=(𝑚−2𝑡)𝜉^{𝑠+𝑡−1}+𝑎_{𝑠+𝑡−2}𝜉^{𝑠+𝑡−2}+⋯+𝑎₀,𝑝₁(𝜉)=𝐴𝜉−𝜉₁^𝑚1−1𝜉−𝜉₂^𝑚2−1⋯𝜉−𝜉_𝑠^𝑚𝑠−1ℎ(𝜉),𝑞₁(𝜉)=𝜉−𝜂₁³𝜉−𝜂₂³⋯𝜉−𝜂_𝑡³ (3.10) are polynomials. Since 𝜓^′(𝜉)+𝑎 has only a unique zero 𝜉₀, set 𝜓^′(𝜉)𝐵+𝑎=𝜉−𝜉₀^𝑙𝜉−𝜂₁³𝜉−𝜂₂³⋯𝜉−𝜂_𝑡³, (3.11) where 𝐵 is a nonzero constant, and so 𝜓^′′(𝜉)=𝜉−𝜉₀^𝑙−1𝑝₂(𝜉)𝜉−𝜂₁⁴𝜉−𝜂₂⁴⋯𝜉−𝜂_𝑡⁴, (3.12) where 𝑝₂(𝜉)=𝐵(𝑙−3𝑡)𝜉^𝑡+𝑏_𝑡−1𝜉^𝑡−1+⋯+𝑏₀ is a polynomial. From (3.9), we also have 𝜓^′′(𝜉)=𝜉−𝜉₁^𝑚1−2𝜉−𝜉₂^𝑚2−2⋯𝜉−𝜉_𝑠^𝑚𝑠−2𝑝₃(𝜉)𝜉−𝜂₁⁴𝜉−𝜂₂⁴⋯𝜉−𝜂_𝑡⁴, (3.13) where 𝑝₃(𝜉) is also a polynomial.
Let deg(𝑝) denote the degree of a polynomial 𝑝(𝜉).
From (3.9) and (3.10),deg(ℎ)≤𝑠+𝑡−1,deg𝑝₁≤𝑚+𝑡−1,deg𝑞₁=3𝑡. (3.14) Similarly from (3.12) and (3.13) and noting (3.14), deg𝑝₂𝑝≤𝑡,deg₃≤deg𝑝₁+𝑡−1−(𝑚−2𝑠)≤2𝑡+2𝑠−2. (3.15)(3.16)
Note that 𝑚_𝑖≥4(𝑖=1,2,…,𝑠). It follows from (3.9) and (3.11) that 𝜓^′(𝜉_𝑖)=0(𝑖=1,2,…,𝑠) and 𝜓^′(𝜉₀)=𝑎≠0. Thus 𝜉₀≠𝜉_𝑖(𝑖=1,2,…,𝑠), and then (𝜉−𝜉₀)^𝑙−1 is a factor of 𝑝₃(𝜉). Hence we get that 𝑙−1≤deg(𝑝₃). Combining (3.12) and (3.13) we also have 𝑚−2𝑠=deg(𝑝₂)+𝑙−1−deg(𝑝₃)≤deg(𝑝₂). By (3.15) we obtain𝑚−2𝑠≤deg𝑝₂≤𝑡. (3.17)
Since 𝑚≥4𝑠, we know by (3.17) that2𝑠≤𝑡. (3.18)
If 𝑙≥3𝑡, by (3.16), then 3𝑡−1≤𝑙−1≤deg(𝑝₃)≤2𝑡+2𝑠−2. Noting (3.18), we obtain 1≤0, a contradiction.
If 𝑙<3𝑡, from (3.9) and (3.11), deg(𝑝₁)=deg(𝑞₁). Noting that deg(𝑝₁)=𝑚+𝑡−𝑖,1≤𝑖≤𝑠+𝑡,deg(𝑞₁)=3𝑡, so 𝑚+𝑡−𝑖=3𝑡,𝑚=2𝑡+𝑖≠2𝑡. From (3.10), then deg(ℎ)=𝑠+𝑡−1, and then deg(𝑝₁)=𝑚+𝑡−1. Noting that deg(𝑞₁)=3𝑡, 𝑚=2𝑡+1. By (3.17), 𝑡≤2𝑠−1. From (3.18), we obtain 1≤0, a contradiction.
The proof of Theorem 1.7 is complete.

Proof of Theorem 1.9. Similarly with the proof of Theorem 1.7, 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_𝑗𝑓_𝑗(𝑧_𝑗+𝜌_𝑗𝜉) converges uniformly with respect to the spherical metric to a nonconstant mermorphic function 𝑔(𝜉) and all zeros and poles of 𝑔(𝜉) are multiple. Moreover, 𝑔(𝜉) is of order 2 at most.
Thus𝑔^′_𝑗(𝜉)=𝜌²_𝑗𝑓^′_𝑗𝑧_𝑗+𝜌_𝑗𝜉⟹𝑔^′(𝜉),𝜌²_𝑗𝑓^′_𝑗𝑧_𝑗+𝜌_𝑗𝜉−𝑎𝑓²_𝑗𝑧_𝑗+𝜌_𝑗𝜉−𝑏=𝑔^′_𝑗(𝜉)−𝑎𝑔²_𝑗(𝜉)−𝜌²_𝑗𝑏⟹𝑔^′(𝜉)−𝑎𝑔²(𝜉) (3.19)(3.20) also locally uniformly with respect to the spherical metric.
If 𝑔′−𝑎𝑔²≡0, then −(1/𝑔)≡𝑎𝜉+𝑐 where 𝑐 is a constant. This contradicts with the idea that the poles of 𝑔 are multiple. So 𝑔′−𝑎𝑔²≢0.
If 𝑔′−𝑎𝑔²≠0, then (1/𝑔)′≠−𝑎. By Lemma 2.3, 𝑔 is a constant which contradicts with our conclusion. Hence, 𝑔′−𝑎𝑔² is a nonconstant meromorphic function and has at least one zero.
As the same argument in the proof of Theorem 1.7, we obtain that 𝑔′−𝑎𝑔² has only one distinct zero denoted by 𝜉₀.
Set 𝑔=1/𝜑; then 𝑔′−𝑎𝑔²=−(𝜑′+𝑎)/𝜑². So (𝜑′+𝑎)/𝜑² has only a unique zero 𝜉₀. Therefore 𝜉₀ is a multiple pole of 𝜑 or a zero of 𝜑′+𝑎. If 𝜉₀ is a multiple pole of 𝜑, since (𝜑′+𝑎)/𝜑² has only one zero 𝜉₀, then 𝜑′+𝑎≠0. By Lemma 2.3 again, 𝜑 is a constant, which is a contradiction.
Hence 𝜑 is an entire function with no simple zero and growth order at most 2 and 𝜑′+𝑎 has just a unique zero 𝜉₀. By Lemma 2.4, 𝜑 is not any transcendental function. Therefore 𝜑 is a nonconstant polynomial and has the form that 𝜑′+𝑎=𝐶(𝜉−𝜉₀)^𝑙, where 𝐶 is a nonzero constant and 𝑙 is a positive integer because the poles of 𝑔 are of multiplicity ≥3. So the zeros of 𝜑 are of multiplicity ≥3; thus, 𝑙≥2,𝜑^′′=𝐶𝑙(𝜉−𝜉₀)^𝑙−1. Note that 𝜑^′′ has only one zero 𝜉₀, and so 𝜑 has only the same zero 𝜉₀ too. Hence 𝜑^′(𝜉₀)=0 which contradicts with 𝜑^′(𝜉₀)=−𝑎≠0.
The proof of Theorem 1.9 is complete.

Proof of Theorem 1.11. Similarly with the proof of Theorem 1.7, 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 meromorphic function 𝑔(𝜉) whose poles are multiple and 𝑔(𝜉)≠0. Moreover, 𝑔(𝜉) is of order 2 at most.
Thus𝑔^′_𝑗(𝜉)=𝑓^′_𝑗𝑧_𝑗+𝜌_𝑗𝜉⟹𝑔^′(𝜉),𝑔^′_𝑗(𝜉)−𝑎𝜌_𝑗𝑔_𝑗(𝜉)−𝑏=𝑓^′_𝑗𝑧_𝑗+𝜌_𝑗𝜉−𝑎𝑓_𝑗𝑧_𝑗+𝜌_𝑗𝜉−𝑏⟹𝑔^′(𝜉)−𝑏 (3.21)(3.22) also locally uniformly with respect to the spherical metric.
If 𝑔′−𝑏≡0, then 𝑔=𝑏𝜉+𝑐, where 𝑐 is a constant. This contradicts with 𝑔(𝜉)≠0. So 𝑔′−𝑏≢0.
If 𝑔′−𝑏≠0, then by Milloux inequality (2.3) of Lemma 2.5 we have 𝑇𝑟,𝑔≤𝑁1𝑟,𝑔+𝑁𝑟,𝑔+𝑁1𝑟,𝑔^′≤1−𝑏+𝑆𝑟,𝑔2𝑁≤1𝑟,𝑔+𝑆𝑟,𝑔2𝑇𝑟,𝑔+𝑆𝑟,𝑔. (3.23) From (3.23) we know that 𝑔 is a constant which contradicts with our conclusion. Hence, 𝑔′−𝑏 is a nonconstant meromorphic function and has at least one zero.
As the same argument in the proof of Theorem 1.7, we obtain that 𝑔′−𝑏 has only one distinct zero denoted by 𝜉₀. Thus the Hayman inequality (2.2) of Lemma 2.5 implies that 𝑔 is a rational function of degree 4 at most. Noting that 𝑔≠0 and has no simple pole, we obtain that 𝑔 has at most two distinct poles. Using the Milloux inequality (2.3) of Lemma 2.5 again we get that 𝑔 has at most one distinct pole. Hence we can write 𝑔(𝜉)=1/(𝑑𝜉+𝑒)^𝑚,2≤𝑚≤3, where 𝑑≠0 and 𝑒 are two finite complex numbers. Simple calculation shows that 𝑔′−𝑏 has at least three distinct zeros. This is impossible.
The proof of Theorem 1.11 is complete.

Proof of Theorem 1.14. Similarly with the proof of Theorem 1.11, we have that (3.22) also holds. Moreover, 𝑔(𝜉)≠0 and 𝑔(𝜉) is of order 2 at most.
If 𝑔′−𝑏≡0, then 𝑔=𝑏𝜉+𝑐, where 𝑐 is a constant. This contradicts with 𝑔(𝜉)≠0.
If 𝑔′−𝑏≠0, then by the Hayman inequality (2.2) of Lemma 2.5 we have 𝑇1𝑟,𝑔≤3𝑁𝑟,𝑔+4𝑁1𝑟,𝑔^′.−𝑏+𝑆𝑟,𝑔≤𝑆𝑟,𝑔 (3.24) From (3.24) we know that 𝑔 is a constant which contradicts with our conclusion.
The proof of Theorem 1.8 is complete.

Acknowledgments

The authors would like to express their hearty thanks to Professor Qingcai Zhang for supplying them with his helpful reprint. The authors wish to thank the managing editor and referees for their very helpful comments and useful suggestions. This work was completed with the support of the NSF of China (10771220) and Doctorial Point Fund of National Education Ministry of China (200810780002).