Normal Criteria of Function Families Related to a Result of Drasin

Abstract

We study the normality of families of meromorphic functions related to a result of Drasin. We consider whether a family meromorphic functions ℱ whose each function does not take zero is normal in 𝐷, if for every pair of functions 𝑓 and 𝑔 in ℱ, 𝑓(𝑧) and 𝑔(𝑧) share ∞ or 𝐻(𝑓)−1 and 𝐻(𝑔)−1 share 0, where 𝐻(𝑓)∶=𝑓^(𝑘)(𝑧)+𝑎_𝑘−1𝑓^(𝑘−1)(𝑧)+⋯,𝑎₀𝑓(𝑧). Some examples show that the conditions in our results are best possible.

1. Introduction and Main Result

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 (see [2–4] or [1]).

Influenced from Bloch’s principle [5], 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 [6]), many authors proved 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 in which every function shares three distinct finite complex numbers with its first derivative. And later, more results about normality criteria concerning shared values can be found, for instance, (see [9–13]) and so on. In recent years, this subject has attracted the attention of many researchers worldwide.

The following result is due to Schiff [14].

Theorem 1.1. Let ℱ be a family holomorphic functions in 𝐷, and 𝑎_𝑗∈ℂ,𝑗=0,1,…,𝑘−1,𝑘 a positive integer. If 𝑓(𝑧)≠0 and the zeros of 𝐻(𝑓)−1 are of multiplicity ≥𝑝>𝑘+1 for each 𝑓(𝑧)∈ℱ, where 𝐻(𝑓)∶=𝑓^(𝑘)(𝑧)+𝑎_𝑘−1𝑓^(𝑘−1)(𝑧)+⋯,𝑎₀𝑓(𝑧) and 𝑝 is a positive integer, then ℱ is normal in 𝐷.

In 2001, Fang and Yuan [15] extended Theorem 1.1 as follows.

Theorem 1.2. Let ℱ be a family holomorphic functions in 𝐷, and 𝑎_𝑗∈ℂ,𝑗=0,1,…,𝑘−1,𝑘 a positive integer. If 𝑓(𝑧)≠0 and the zeros of 𝐻(𝑓)−1 are of multiplicity ≥2 for each 𝑓(𝑧)∈ℱ, where 𝐻(𝑓)∶=𝑓^(𝑘)(𝑧)+𝑎_𝑘−1𝑓^(𝑘−1)(𝑧)+⋯,𝑎₀𝑓(𝑧), then ℱ is normal in 𝐷.

It is natural to ask whether or not Theorems 1.1 and 1.2 hold for meromorphic case or holomorphic case by the idea of shared values. In this paper, we answer above question and prove the following results.

Theorem 1.3. Let ℱ be a family meromorphic functions in 𝐷, and 𝑎_𝑗∈ℂ,𝑗=0,1,…,𝑘−1,𝑘 a positive integer. If (i) 𝑓(𝑧)≠0 and the zeros of 𝐻(𝑓)−1 are of multiplicity ≥2 for each 𝑓(𝑧)∈ℱ, (ii) 𝑓(𝑧) and 𝑔(𝑧) share ∞ for every pair of functions 𝑓(𝑧) and 𝑔(𝑧) in ℱ, where 𝐻(𝑓)∶=𝑓^(𝑘)(𝑧)+𝑎_𝑘−1𝑓^(𝑘−1)(𝑧)+⋯,𝑎₀𝑓(𝑧), then ℱ is normal in 𝐷.

Theorem 1.4. Let ℱ be a family holomorphic functions in 𝐷, and 𝑎_𝑗∈ℂ,𝑗=0,1,…,𝑘−1,𝑘 a positive integer. If (i) 𝑓(𝑧)≠0 for each 𝑓(𝑧)∈ℱ, (ii) 𝐻(𝑓)−1 and 𝐻(𝑔)−1 share the value 0 for every pair of functions 𝑓(𝑧) and 𝑔(𝑧) in ℱ, where 𝐻(𝑓)∶=𝑓^(𝑘)(𝑧)+𝑎_𝑘−1𝑓^(𝑘−1)(𝑧)+⋯,𝑎₀𝑓(𝑧), then ℱ is normal in 𝐷.

Example 1.5. The family of holomorphic functions ℱ={𝑓_𝑛(𝑧)=𝑛𝑧𝑒^𝑧−𝑛𝑒^𝑧∶𝑛=1,2,…} is not normal in 𝐷={𝑧∶|𝑧|<1}. Obviously 𝑓_𝑛^(𝑘)−𝑓_𝑛=𝑘𝑒^𝑧≠0. On the other hand, 𝑓_𝑛(0)=−𝑛,𝑓_𝑛(1/√𝑛)=−1/2+(1/2√𝑛)+𝑜(1/𝑛)→−1/2, as 𝑛→∞. This implies that the family ℱ fails to be equicontinuous at 0, and thus ℱ is not normal at 0.

Example 1.6. The family of holomorphic functions ℱ={𝑓_𝑛(𝑧)=𝑛(𝑧+1)−1∶𝑛=1,2,…} is normal in 𝐷={𝑧∶|𝑧|<1}. Obviously, 𝑓_𝑛(𝑧)≠0 and 𝑓^′_𝑛−𝑓_𝑛=−𝑛𝑧+1. So for each pair 𝑚,𝑛,𝑓^′_𝑛−𝑓_𝑛 and 𝑓^′_𝑚−𝑓_𝑚 share the value 1 in 𝐷. Theorem 1.4 implies that the family ℱ is normal in 𝐷.

Example 1.7. The family of meromorphic functions ℱ={𝑓_𝑛(𝑧)=𝑧/𝑛−1∶𝑛=1,2,…} is normal in 𝐷={𝑧∶|𝑧|<1}. The reason is the conditions of Theorems 1.3 and 1.4 hold, that is, 𝑓_𝑛(𝑧)≠0 and 𝑓^′_𝑛−𝑓_𝑛=(1−𝑧)/𝑛+1≠1 for each 𝑓_𝑛(𝑧)∈ℱ in 𝐷={𝑧∶|𝑧|<1}.

Example 1.8. The family of meromorphic functions ℱ={𝑓_𝑛(𝑧)=𝑛/𝑧−1∶𝑛=1,2,…} is normal in 𝐷={𝑧∶|𝑧|<1}. The reason is the conditions of Theorem 1.3 hold, that is, 𝑓_𝑛(𝑧)≠0 and has only one pole 0 for each 𝑓(𝑧)∈ℱ; 𝑓_𝑛^(𝑘)−𝑓_𝑛−1=𝑛((−1)𝑘!−𝑧^𝑘)/𝑧^𝑘+1≠0 for each 𝑓_𝑛(𝑧)∈ℱ in 𝐷={𝑧∶|𝑧|<1}.

Remark 1.9. Example 1.5 shows that 𝐻(𝑔)−1 is not replaced by 𝐻(𝑔) in Theorems 1.3 and 1.4. is not valid when 𝑛=1. All of Examples 1.6, 1.7 and 1.8 show that Theorems 1.3 and 1.4 occur.

2. Preliminary Lemmas

In order to prove our result, we need the following lemmas. The first one extends a famous result by Zalcman [16] concerning normal families.

Lemma 2.1 (see [17]). Let ℱ be a family of meromorphic functions on the unit disc satisfying all zeros of functions in ℱ having multiplicity ≥𝑝 and all poles of functions in ℱ have multiplicity ≥𝑞. Let 𝛼 be a real number satisfying −𝑝<𝛼<𝑞. Then, ℱ is not normal at 0 if and only if there exist(a)a number 0<𝑟<1,(b)points 𝑧_𝑛 with |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 of functions in ℱ have multiplicity ≥𝑝 and all poles of functions in ℱ 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 the Nevanlinna’s characteristic function 𝑇(𝑟,𝑔)𝜌𝑔=lim_𝑟→∞suplog𝑇𝑟,𝑔log𝑟. (2.1) Here, 𝑔^#(𝜉) denotes the spherical derivative 𝑔^#(𝜉)=𝑔^′(𝜉)1+𝑔(𝜉)². (2.2)

Lemma 2.3 (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 (2.3)(2.4)

Remark 2.4. Both (2.3) and (2.4) are called Hayman inequality and Milloux inequality, respectively.

2.1. Proof of the Results

Proof of Theorem 1.3. 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^−𝑘𝑓_𝑗𝑧_𝑗+𝜌_𝑗𝜉⟹𝑔(𝜉), (2.5) locally uniformly with respect to the spherical metric, where 𝑔 is a nonconstant meromorphic function in ℂ. Moreover, the order of 𝑔 is less than 2.
Since 𝑓_𝑗(𝑧)∈ℱ, by Hurwitz’s theorem, we see 𝑔≠0.
From (2.5), we know 𝑔_𝑗^(𝑘)(𝜉)=𝑓_𝑗^(𝑘)𝑧_𝑗+𝜌_𝑗𝜉⟹𝑔^(𝑘)(𝜉),𝑔_𝑗^(𝑘)(𝜉)−1+^𝑘−1_𝑖=1𝜌_𝑗^𝑘−𝑖𝑎_𝑖𝑔_𝑗^(𝑖)𝑧_𝑗+𝜌_𝑗𝜉=𝐻𝑓_𝑗𝑧_𝑗+𝜌_𝑗𝜉−1⟹𝑔^(𝑘)(𝜉)−1, (2.6)(2.7) also locally uniformly with respect to the spherical metric.
If 𝑔^(𝑘)(𝜉)−1≡0, then 𝑔 is a polynomial with degree 𝑘. This contradicts 𝑔≠0.
If 𝑔^(𝑘)(𝜉)−1≠0, then by Hayman inequality (2.3) of Lemma 2.3, we know that 𝑔 is a constant, which is impossible.
Hence, 𝑔^(𝑘)(𝜉)−1 is a nonconstant meromorphic function and has at least one zero. By (2.7) and Hurwitz’s theorem, we see that the zeros of 𝑔^(𝑘)(𝜉)−1 are multiple.
Next, we prove that 𝑔 has at most one distinct pole. By contraries, let 𝜉₀ and 𝜉^∗₀ be two distinct poles of 𝑔, 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 (2.5), by Hurwitz’s theorem, there exist points 𝜉_𝑗∈𝐷(𝜉₀,𝛿), 𝜉^∗_𝑗∈𝐷(𝜉^∗₀,𝛿) such that for sufficiently large 𝑗𝑓_𝑗⁻¹𝑧_𝑗+𝜌_𝑗𝜉_𝑗=0,𝑓_𝑗⁻¹𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗=0. (2.8)
By the hypothesis that for each pair of functions 𝑓 and 𝑔 in ℱ, 𝑓⁻¹ and 𝑔⁻¹ share 0 in 𝐷, we know that for any positive integer 𝑚𝑓_𝑚⁻¹𝑧_𝑗+𝜌_𝑗𝜉_𝑗=0,𝑓_𝑚⁻¹𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗=0. (2.9)
Fix 𝑚, take 𝑗→∞, and note 𝑧_𝑗+𝜌_𝑗𝜉_𝑗→0, 𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗→0, then 𝑓_𝑚⁻¹(0)=0. Since the zeros of 𝑓_𝑚⁻¹ has no accumulation point, so.𝑧_𝑗+𝜌_𝑗𝜉_𝑗=0,𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗=0. (2.10) 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 has at most one distinct pole.
For 𝑔 applying Milloux inequality (2.4) of Lemma 2.3, we deduce 𝑇𝑟,𝑔≤𝑁1𝑟,𝑔+𝑁𝑟,𝑔+𝑁1𝑟,𝑔^(𝑘)1−1+𝑆𝑟,𝑔≤log𝑟+2𝑁1𝑟,𝑔^(𝑘)≤1−1+𝑆𝑟,𝑔2𝑇𝑟,𝑔+log𝑟+𝑆𝑟,𝑔. (2.11) From (2.11), we know that 𝑔 is a rational function of degree at most 2. Noting that 𝑔≠0, we can write 𝑔(𝜉)=1/(𝑎𝜉+𝑏)^𝑚,1≤𝑚≤2 where 𝑎≠0 and 𝑏 are two finite complex numbers. Simple calculating shows that 𝑔^(𝑘)−1 has at least 𝑘+1 distinct simple zeros. This is impossible.
The proof of Theorem 1.3 is complete.

Proof of Theorem 1.4. Similarly with the proof of Theorem 1.3, we have that (2.5) and (2.7) also hold. Moreover, 𝑔(𝜉) is an entire function.
If 𝑔^(𝑘)(𝜉)−1≡0, then 𝑔 is a polynomial with degree 𝑘. This contradicts 𝑔≠0.
If 𝑔^(𝑘)(𝜉)−1≠0, then by Hayman inequality (2.3) of Lemma 2.3 we know that 𝑔 is a constant, which is impossible.
Hence, 𝑔^(𝑘)(𝜉)−1 is a nonconstant entire function and has at least one zero.
Next, we prove that 𝑔^(𝑘)(𝜉)−1 has one distinct zero. By contraries, let 𝜉₀ and 𝜉^∗₀ be two distinct zeros of 𝑔^(𝑘)(𝜉)−1, 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 (2.7), by Hurwitz’s theorem, there exist points 𝜉_𝑗∈𝐷(𝜉₀,𝛿), 𝜉^∗_𝑗∈𝐷(𝜉^∗₀,𝛿) such that for sufficiently large 𝑗𝐻𝑓_𝑗𝑧_𝑗+𝜌_𝑗𝜉_𝑗−1=0,𝐻𝑓_𝑗𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗−1=0. (2.12)
By the hypothesis that for each pair of functions 𝑓 and 𝑔 in ℱ, 𝐻(𝑓)−1 and 𝐻(𝑔)−1 share 0 in 𝐷, we know that for any positive integer 𝑚𝐻𝑓_𝑚𝑧_𝑗+𝜌_𝑗𝜉_𝑗−1=0,𝐻𝑓_𝑚𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗−1=0. (2.13)
Fix 𝑚, take 𝑗→∞, and note 𝑧_𝑗+𝜌_𝑗𝜉_𝑗→0, 𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗→0, then 𝑓_𝑚^(𝑘)(0)−1=0. Since the zeros of 𝑓_𝑚^(𝑘)−1 has no accumulation point, so𝑧_𝑗+𝜌_𝑗𝜉_𝑗=0,𝑧_𝑗+𝜌_𝑗𝜉^∗_𝑗=0. (2.14) 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^(𝑘)(𝜉)−1 has one distinct zero.
For 𝑔 applying Milloux inequality (2.4) of Lemma 2.3, we deduce𝑇𝑟,𝑔≤𝑁1𝑟,𝑔+𝑁𝑟,𝑔+𝑁1𝑟,𝑔^(𝑘).−1+𝑆𝑟,𝑔≤log𝑟+𝑆𝑟,𝑔 (2.15) From (2.15) and 𝑔≠0, we know that 𝑔 is a polynomial of degree 1. We can write 𝑔(𝜉)=(𝑐𝜉+𝑑), where 𝑐≠0 and 𝑑 are two finite complex numbers. Hence, 𝑔^(𝑘)−1 has no only one distinct zero. This is impossible.
The proof of Theorem 1.4 is complete.

Acknowlegment

This work was completed with the support of the NSF of China (no. 10771220) and Doctorial Point Fund of National Education Ministry of China (no. 200810780002).