Normality Criteria of Meromorphic Functions That Share a Holomorphic Function

Abstract

Let ℱ be a family of meromorphic functions defined in 𝐷, let 𝜓(≢0), 𝑎₀,𝑎₁,...,𝑎_𝑘−1 be holomorphic functions in 𝐷, and let 𝑘 be a positive integer. Suppose that, for every function 𝑓∈ℱ, 𝑓≠0, 𝑃(𝑓)=𝑓^(𝑘)+𝑎_𝑘−1𝑓^(𝑘−1)+⋯+𝑎₁𝑓′+𝑎₀𝑓≠0 and, for every pair functions (𝑓,𝑔)∈ℱ, 𝑃(𝑓), 𝑃(𝑔) share 𝜓, then ℱ is normal in 𝐷.

1. Introduction and Main Results

Let ℂ be complex plane. Let 𝐷 be a domain in ℂ. Let ℱ be a family meromorphic functions defined in the domain 𝐷. ℱ is said to be normal in 𝐷, 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 uniformly in 𝐷, to a meromorphic function or ∞.

Let 𝑓(𝑧) and 𝑔(𝑧) be two meromorphic functions, let 𝑎 be a finite complex number. If 𝑓(𝑧)−𝑎 and 𝑔(𝑧)−𝑎 have the same zeros, then we say they share 𝑎 or share 𝑎 IM (ignoring multiplicity) (see [1–3]).

Definition 1.1. Let 𝑎_𝑖(𝑧),(𝑖=1,2,…,𝑞−1),𝑏_𝑗(𝑧),(𝑗=1,2,…,𝑛) be analytic in 𝐷, let 𝑛₀,𝑛₁,…,𝑛_𝑘 be nonnegative integers, set 𝑃(𝜔)=𝜔^𝑞+𝑎_𝑞−1(𝑧)𝜔^𝑞−1+⋯+𝑎₁(𝑧)𝑀𝜔,𝑓,𝑓^′,…,𝑓^(𝑘)=𝑓^𝑛0𝑓^′𝑛1⋯𝑓^{(𝑘)𝑛𝑘},𝛾_𝑀=𝑛₀+𝑛₁+⋯+𝑛_𝑘,Γ_𝑀=𝑛₀+2𝑛₁+⋯+(𝑘+1)𝑛_𝑘, (1.1) where 𝑀(𝑓,𝑓′,…,𝑓^(𝑘)) is called a differential monomial of 𝑓, 𝛾_𝑀 the degree of 𝑀(𝑓,𝑓^′,…,𝑓^(𝑘)), and Γ_𝑀 the weight of 𝑀(𝑓,𝑓′,…,𝑓^(𝑘)).
From Definition 1.1, we give Definition 1.2.

Definition 1.2. Let 𝑀_𝑗(𝑓,𝑓^′,…,𝑓^(𝑘)),(𝑗=1,2,…,𝑛) be differential monomials of 𝑓. Set 𝐻𝑓,𝑓^′,…,𝑓^(𝑘)=𝑏₁(𝑧)𝑀₁𝑓,𝑓^′,…,𝑓^(𝑘)+⋯+𝑏_𝑛(𝑧)𝑀_𝑛𝑓,𝑓^′,…,𝑓^(𝑘),𝛾_𝐻=max𝛾_𝑀1,𝛾_𝑀2,…,𝛾_𝑀𝑛,Γ_𝐻=maxΓ_𝑀1,Γ_𝑀2,…,Γ_𝑀𝑛, (1.2) where 𝐻(𝑓,𝑓′,…,𝑓^(𝑘)) is called the differential polynomial of 𝑓, 𝛾_𝐻 the degree of 𝐻(𝑓,𝑓^′,…,𝑓^(𝑘)), and Γ_𝐻 the weight of 𝐻(𝑓,𝑓′,…,𝑓^(𝑘)),Γ𝛾_𝐻Γ=max^𝑀1𝛾_𝑀1,Γ_𝑀2𝛾_𝑀2Γ,…,^𝑀𝑛𝛾_𝑀𝑛,𝐺𝑓=𝑃𝑓^(𝑘)+𝐻𝑓,𝑓^′,…,𝑓^(𝑘). (1.3) In 1979, Gu [4] proved the following result.

Theorem A. Let ℱ be a family of meromorphic functions defined in 𝐷, let 𝑘 be a positive integer, and let 𝑎 be a nonzero constant. If, for each function 𝑓∈ℱ, 𝑓≠0, 𝑓^(𝑘)≠𝑎 in 𝐷, then ℱ is normal in 𝐷.

Yang [5] and Schwick [6] proved that Theorem A still holds if 𝑎 is replaced by a holomorphic function 𝜓(≢0) in Theorem A.

Xu [7] improved Theorem A by the ideas of shared values and obtained the following result.

Theorem B. Let ℱ be a family of meromorphic functions defined in 𝐷, let 𝜓(≢0) be a holomorphic functions and with only simple zeros in 𝐷, and let 𝑘 be a positive integer. Suppose that, for every function 𝑓∈ℱ, 𝑓 has all multiple poles and 𝑓≠0. If, for every pair of functions 𝑓 and 𝑔, 𝑓^(𝑘) and 𝑔^(𝑘) share 𝜓 in 𝐷, then ℱ is normal in 𝐷.

Recently, Xu [7] did not know whether the condition 𝜓 has only simple zero in 𝐷 and 𝑓 has all multiple poles are necessary or not in Theorem B.

In 2007, Fang and Chang considered the case 𝑎=0 in Theorem A. In this note, Fang and Chang [8] proved the following result.

Theorem C. Let ℱ be a family of meromorphic functions defined in 𝐷, and let 𝑘 be a positive integer, and let 𝑏 be a nonzero complex number. If, for each 𝑓∈ℱ, 𝑓≠0, 𝑓^(𝑘)≠0 and the zeros of 𝑓^(𝑘)−𝑏 have multiplicity at least (𝑘+2)/𝑘, then ℱ is normal in 𝐷.

Remark 1.3. The number (𝑘+2)/𝑘 is sharp, as is shown by the examples in [8].
In 2009, Xia and Xu [9] replaced the constant 1 by a function 𝜓(𝑧)≢0 in Theorem C. They obtained the following result.

Theorem D. Let ℱ be a family of meromorphic functions defined in 𝐷, let 𝜓(≢0), 𝑎₀,𝑎₁,…,𝑎_𝑘−1 be holomorphic functions in 𝐷, and let 𝑘 be a positive integer. Suppose that, for every function 𝑓∈ℱ, 𝑓≠0, 𝑓^(𝑘)+𝑎_𝑘−1𝑓^(𝑘−1)+⋯+𝑎₁𝑓^′+𝑎₀𝑓≠0 and all zeros of 𝑓^(𝑘)+𝑎_𝑘−1𝑓^(𝑘−1)+⋯+𝑎₁𝑓^′+𝑎₀𝑓−𝜓(𝑧) have multiplicity at least (𝑘+2)/𝑘. If, for 𝑘=1, 𝜓 has only zeros with multiplicities at most 2 and, for 𝑘≥2, 𝜓 has only simple zeros, then ℱ is normal in 𝐷.

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

Theorem 1.4. Let ℱ be a family of meromorphic functions defined in 𝐷, let 𝜓(≢0), 𝑎₀,𝑎₁,…,𝑎_𝑘−1 be holomorphic functions in 𝐷, and let 𝑘 be a positive integer. Suppose that, for every function 𝑓∈ℱ, 𝑓≠0, 𝑃(𝑓)=𝑓^(𝑘)+𝑎_𝑘−1𝑓^(𝑘−1)+⋯+𝑎₁𝑓^′+𝑎₀𝑓≠0 and, for every pair functions (𝑓,𝑔)∈ℱ, 𝑃(𝑓) and 𝑃(𝑔) share 𝜓, then ℱ is normal in 𝐷.

By Theorem 1.4, we immediately deduce.

Corollary 1.5. Let ℱ be a family of meromorphic functions defined in 𝐷, let 𝜓(≢0), 𝑎₀,𝑎₁,,…,𝑎_𝑘−1 be holomorphic functions in 𝐷, and 𝑘 be a positive integer. Suppose that, for every function 𝑓∈ℱ, 𝑓≠0, 𝑓^(𝑘)≠0 and for every pair functions (𝑓,𝑔)∈ℱ, 𝑓^(𝑘) and 𝑔^(𝑘) share 𝜓, then ℱ is normal in 𝐷.

Remark 1.6. By the ideas of sharing values, Theorem 1.4 and Corollary 1.5 yield the number (𝑘+2)/𝑘 can be omitted.

Remark 1.7. Obviously, Corollary 1.5 omitted the conditions 𝜓 with only simple zeros, and, for every function 𝑓∈ℱ, 𝑓 has all multiple poles in Theorem D. But the condition for every function 𝑓∈ℱ, 𝑓^(𝑘)≠0 is additional. Hence, Corollary 1.5 improves Theorem B in some sense.

The condition 𝜓≢0 in Theorem 1.4 is necessary. For example, we consider the following families.

Example 1.8. ℱ={𝑓_𝑚(𝑧)=𝑒^𝑚𝑧,𝑚=1,2,…}, obviously, any 𝑓∈𝐹 satisfies 𝑓≠0, 𝑓^(𝑘)≠0. For distinct positive integers 𝑚, 𝑙, 𝑓_𝑚^(𝑘), and 𝑓_𝑙^(𝑘) share 0 IM. However, the families ℱ are not normal at 𝑧=0.

Remark 1.9. Some ideas of this paper are based on [7, 9, 10].

2. Preliminary Lemmas

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

The well-known Zalcman’s lemma is a very important tool in the study of normal families. It has also undergone various extensions and improvements. The following is one up-to-date local version, which is due to Pang and Zaclman [11].

Lemma 2.1 (see [11, 12]). Let ℱ be a family of meromorphic functions in the unit disc Δ with the property that, for each 𝑓∈ℱ, all zeros are of multiplicity at least 𝑘. Suppose that there exists a number 𝐴≥1 such that |𝑓^(𝑘)(𝑧)|≤𝐴 whenever 𝑓∈ℱ and 𝑓=0. If ℱ is not normal in Δ, then, for 0≤𝛼≤𝑘, there exist (1)a number 𝑟∈(0,1);(2)a sequence of complex numbers 𝑧_𝑛, |𝑧_𝑛|<𝑟;(3)a sequence of functions 𝑓_𝑛∈ℱ;(4)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.

Here, as usual, 𝑔^♯(𝜉)=|𝑔^′(𝜉)|/(1+|𝑔(𝜉)|²) is the spherical derivative.

Lemma 2.2 (see [1]). Let 𝑓(𝑧) be a transcendental meromorphic function in ℂ, let 𝑘(≥1) be a integer, and let 𝑏 be a nonzero finite value, then 𝑓 or 𝑓^(𝑘)−𝑏 has infinite zeros.

Lemma 2.3 (see [7]). Let 𝑓(𝑧) be a nonconstant rational function. Let 𝑘≥1 be an integer, and let 𝑏 be a non-zero finite value. If 𝑓≠0, then 𝑓^(𝑘)(𝑧)−𝑏 has at least two distinct zeros in the plane.

Lemma 2.4. Let 𝑓(𝑧) be a nonconstant rational function. Let 𝑘≥1 be an integer, and let 𝑙 be a positive integer. If 𝑓≠0, 𝑓^(𝑘)≠0, then 𝑓^(𝑘)(𝑧)−𝑧^𝑙 has at least two distinct zeros in the plane.

Proof. Since 𝑓≠0 and 𝑓^(𝑘)≠0, then 𝑓 is a nonpolynomial rational function and has the form 𝑓(𝑧)=𝐴𝑧−𝑧₁^𝑚1𝑧−𝑧₂^𝑚2⋯𝑧−𝑧_𝑡^𝑚𝑡, (2.1) where 𝐴≠0 is a constant, and 𝑚₁,𝑚₂,…,𝑚_𝑡 are positive integers. Set 𝑚=𝑚₁+𝑚₂+⋯+𝑚_𝑡. Then, 𝑓^′(𝑧)=−𝐴𝑚𝑧^𝑡−1+𝑏_𝑡−2𝑧^𝑡−2+⋯+𝑏₀𝑧−𝑧₁^𝑚1+1𝑧−𝑧₂^𝑚2+1⋯𝑧−𝑧_𝑡^𝑚𝑡+1, (2.2) where 𝑏_𝑡−2,…,𝑏₀ are constants. For 𝑘≥2, by mathematical induction, we have 𝑓^(𝑘)(𝑧)=𝐵𝑧^{𝑘𝑡−𝑘}+𝑐_{𝑘𝑡−𝑘−1}𝑧^{𝑘𝑡−𝑘−1}+⋯+𝑐₀𝑧−𝑧₁^𝑚1+𝑘𝑧−𝑧₂^𝑚2+𝑘⋯𝑧−𝑧_𝑡^{𝑚𝑡+𝑘}, (2.3) where 𝐵=(−1)^𝑘𝑚(𝑚+1)(𝑚+2)⋯(𝑚+𝑘−1)𝐴≠0,𝑐_{𝑘𝑡−𝑘−1},…,𝑐₀ are constants. Since 𝑓^(𝑘)≠0, we deduce that 𝑡=1, and thus 𝑓(𝑧)=𝐴𝑧−𝑧₁^𝑚1,𝑓^(𝑘)(𝑧)=𝐵𝑧−𝑧₁^𝑚1+𝑘. (2.4)(2.5)Case 1 (if 𝑓^(𝑘)−𝑧^𝑙 has exactly one zero 𝑧₀). From (2.5), we set 𝑓^(𝑘)(𝑧)−𝑧^𝑙=𝐵𝑧−𝑧₁^𝑚1+𝑘−𝑧^𝑙=𝐵^′𝑧−𝑧₀^{𝑚1+𝑘+𝑙}𝑧−𝑧₁^𝑚1+𝑘. (2.6) Obviously, 𝐵^′ is a nonzero constant and 𝑙≥1.
From (2.6), we obtain 𝑓^{(𝑘+𝑙+1)}(𝑧)=𝑧−𝑧₀^{𝑚1+𝑘−1}𝑃₁(𝑧)𝑧−𝑧₁^{𝑚1+𝑘+𝑙+1}, (2.7) where 𝑃₁(𝑧)≢0. By (2.4), we deduce 𝑓^{(𝑘+𝑙+1)}(𝑧)=𝐴^′𝑧−𝑧₁^{𝑚1+𝑘+1+1}, (2.8) where 𝐴^′ is nonzero constant.
Comparing (2.7) and (2.8), we obtain that deg𝐴^′=0≥𝑚₁+𝑘−1 is impossible.Case 2 (if 𝑓^(𝑘)(𝑧)−𝑧^𝑙≠0). By (2.5), clearly Case 2 is impossible.
Lemma 2.4 is proved.

Lemma 2.5 (see [7]). Let ℱ be a family of meromorphic functions defined in 𝐷, let 𝑘 be a positive integer, and let 𝜓(≠0) be a holomorphic function in 𝐷. If, for any 𝑓∈ℱ satisfying 𝑓≠0 and if 𝑓^(𝑘),𝑔^(𝑘) share 𝜓 IM for every pair of functions 𝑓,𝑔∈ℱ, then ℱ is normal in 𝐷.

In this paper, by the same method of [7], we consider the differential polynomial in Lemma 2.5 and prove a more general result.

Lemma 2.6. Let ℱ be a family of meromorphic functions defined in 𝐷, let 𝑘 be a positive integer, and let 𝜓(≠0) be a holomorphic function in 𝐷. If, for any 𝑓∈ℱ satisfying 𝑓≠0 and if 𝐺(𝑓),𝐺(𝑔) share 𝜓 IM for every pair of functions 𝑓,𝑔∈ℱ, where 𝐺(𝑓) is a differential polynomial of 𝑓 as the definition 1 satisfying 𝑞≥𝛾_𝐻, and Γ/𝛾|_𝐻<𝑘+1, then ℱ is normal in 𝐷, where 𝑞, Γ/𝛾|_𝐻 are as in Definitions 1.1 and 1.2.

Proof. We may assume that 𝐷=Δ={|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 exists a number 𝑟∈(0,1); a sequence of complex numbers 𝑧_𝑗, 𝑧_𝑗→0(𝑗→∞); a sequence of functions 𝑓_𝑗∈ℱ; 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 meromorphic functions 𝑔(𝜉) in 𝐶. Moreover, 𝑔(𝜉) is of order at most 2. Hurwitz’s theorem implies that g(𝜉)≠0.
We have 𝐺𝑓_𝑗𝑧_𝑗+𝜌_𝑗𝜁=𝑃𝑓_𝑗^(𝑘)𝑧_𝑗+𝜌_𝑗𝜁+𝐻𝑓_𝑗,𝑓^′_𝑗,…,𝑓_𝑗^(𝑘)𝑧_𝑗+𝜌_𝑗𝜁,𝐻𝑓_𝑗,𝑓^′_𝑗,…,𝑓_𝑗^(𝑘)𝑧_𝑗+𝜌_𝑗𝜁=^𝑛_𝑖=1𝑏_𝑖𝑧_𝑗+𝜌_𝑗𝜁𝜌^{(𝑘+1)𝛾𝑀𝑖−Γ𝑀𝑖}_𝑗𝑀_𝑖𝑔_𝑗,𝑔^′_𝑗,…,𝑔_𝑗^(𝑘)(𝜁). (2.9) Considering 𝑏_𝑖(𝑧) is analytic on 𝐷(𝑖=1,2,…,𝑛), we have 𝑏_𝑖𝑧_𝑗+𝜌_𝑗𝜁≤𝑀1+𝑟2,𝑏_𝑖(𝑧)<∞,(𝑖=1,2,…,𝑛) (2.10) for sufficiently large 𝑗.
Hence, we deduce from Γ/𝛾|_𝐻<𝑘+1 that ^𝑛_𝑖=1𝑏_𝑖𝑧_𝑗+𝜌_𝑗𝜁𝜌^{(𝑘+1)𝛾𝑀𝑖−Γ𝑀𝑖}_𝑗𝑀_𝑖𝑔_𝑗,𝑔^′_𝑗,…,𝑔_𝑗^(𝑘)(𝜁) (2.11) converges uniformly to 0 on every compact subset of ℂ which contains no poles of 𝑔(𝜉).
Thus, we have 𝐺𝑓_𝑗𝑧_𝑗+𝜌_𝑗𝜁⟶𝑃𝑔^(𝑘)(𝜁),𝐺𝑓_𝑗𝑧_𝑗+𝜌_𝑗𝜁−𝜓𝑧_𝑗+𝜌_𝑗𝜁⟶𝑃𝑔^(𝑘)(𝜁)−𝜓𝑧₀ (2.12) on every compact subset of ℂ which contains no poles of 𝑔(𝜁).
Next, we will prove that 𝐺(𝑓_𝑗)(𝜁)−𝜓(𝑧₀) has just a unique zero. By way of contradiction, let 𝜁₀ and 𝜁^∗₀ be two distinct solutions of 𝐺(𝑓_𝑗)(𝜁)−𝜓(𝑧₀), and choose 𝛿(>0) small enough such that 𝐷(𝜁₀,𝛿)∩𝐷(𝜁^∗₀,𝛿)=∅ where 𝐷(𝜁₀,𝛿)={𝜁∶|𝜁−𝜁₀|<𝛿} and 𝐷(𝜁^∗₀,𝛿)={𝜁∶|𝜁−𝜁^∗₀|<𝛿}. By Hurwitz’s theorem, there exist points 𝜁_𝑗∈𝐷(𝜁₀,𝛿), 𝜁^∗_𝑗∈𝐷(𝜁^∗₀,𝛿) such that, for sufficiently large 𝑗, 𝐺𝑓_𝑗𝑧_𝑗+𝜌_𝑗𝜁_𝑗−𝜓𝑧₀𝐺𝑓=0,_𝑗𝑧_𝑗+𝜌_𝑗𝜁^∗_𝑗−𝜓𝑧₀=0. (2.13)
By the hypothesis that for each pair of functions 𝑓 and 𝑔 in ℱ, 𝐺(𝑓) and 𝐺(𝑔) share 𝜓(𝑧₀) in 𝐷, we know that, for any positive integer 𝑚, 𝐺𝑓_𝑚𝑧_𝑗+𝜌_𝑗𝜁_𝑗−𝜓𝑧₀𝐺𝑓=0,_𝑚𝑧_𝑗+𝜌_𝑗𝜁^∗_𝑗−𝜓𝑧₀=0. (2.14) Fix 𝑚, take 𝑗→∞, and note 𝑧_𝑗+𝜌_𝑗𝜁_𝑗→0,𝑧_𝑗+𝜌_𝑗𝜁^∗_𝑗→0, then 𝐺𝑓_𝑚(0)−𝜓𝑧₀=0. (2.15) Since the zeros of 𝐺(𝑓_𝑚)(0)−𝜓(𝑧₀)=0 have no accumulation point, so 𝑧_𝑗+𝜌_𝑗𝜁_𝑗=0, 𝑧_𝑗+𝜌_𝑗𝜁^∗_𝑗=0.
Hence, 𝜁_𝑗𝑧=−^𝑗𝜌_𝑗,𝜁^∗_𝑗𝑧=−^𝑗𝜌_𝑗. (2.16) This contradicts with 𝜁_𝑗∈𝐷(𝜁₀,𝛿), 𝜁^∗_𝑗∈𝐷(𝜁^∗₀,𝛿), and 𝐷(𝜁₀,𝛿)∩𝐷(𝜁^∗₀,𝛿)=∅. So 𝐺(𝑓_𝑗)−𝜓(𝑧₀) has just a unique zero. By Hurwitz’s theorem, we know 𝑃(𝑔^(𝑘))(𝜁)−𝜓(𝑧₀) has just a unique zero.
By Lemmas 2.2 and 2.3, we know 𝑔^(𝑘)(𝜁)−𝜓(𝑧₀) has at least two distinct zeros. From the definition of 𝑃(𝑤), we deduce that 𝑃(𝑔^(𝑘)(𝜁))−𝜓(𝑧₀) has more than two distinct zeros, a contradiction.
So ℱ is normal in 𝐷. Lemma 2.6 is proved.

By Lemma 2.6, we immediately deduce the following lemma.

Lemma 2.7. Let ℱ be a family of meromorphic functions defined in 𝐷, let 𝜓(≠0), 𝑎₀,𝑎₁,…,𝑎_𝑘−1 be holomorphic functions in 𝐷, and let 𝑘 be a positive integer. Suppose that, for every function 𝑓∈ℱ, 𝑓≠0, 𝑓^(𝑘)+𝑎_𝑘−1𝑓^(𝑘−1)+⋯+𝑎₁𝑓^′+𝑎₀𝑓≠0 and, for every pair functions (𝑓,𝑔)∈ℱ, 𝑓^(𝑘)+𝑎_𝑘−1𝑓^(𝑘−1)+⋯+𝑎₁𝑓^′+𝑎₀𝑓, 𝑔^(𝑘)+𝑎_𝑘−1𝑔^(𝑘−1)+⋯+𝑎₁𝑔^′+𝑎₀𝑔 share 𝜓, then ℱ is normal in 𝐷.

Lemma 2.8 (see [1]). Let 𝑓(𝑧) be a meromorphic function. Let 𝑘 be a positive integer. If 𝑓(𝑧)≠0, then 𝑓^(𝑘)(𝑧)≠1, then 𝑓 is a constant.

Lemma 2.9 (see [13, 14]). Let 𝑓(𝑧) be a transcendental meromorphic function in ℂ, and let 𝑃(≢0) be a polynomial. Let 𝑘 be a positive integer. If all zeros (except at most finite zeros) of 𝑓(𝑧) have the multiplicity at least 3, then 𝑓^(𝑘)(𝑧)−𝑃(𝑧) has infinite zeros.

3. Proof of Theorem 1.4

Proof. Since normality is a local property, without loss of generality, we may assume 𝐷=Δ={𝑧∶|𝑧|<1}, and 𝜓(𝑧)=𝑧^𝑙𝜑(𝑧)(𝑧∈Δ), (3.1) where 𝑙 is a positive integer, 𝜑(0)=1, 𝜑(𝑧)≠0 on Δ^′={𝑧∶0<|𝑧|<1}. By Lemma 2.6, we only need to prove that ℱ is normal at 𝑧=0.
If 𝑓∈ℱ, 𝑃(𝑓)(0)≠𝜓(0), then there exists 𝛿>0 such that 𝑃(𝑓)(𝑧)≠𝜓(𝑧) on Δ_𝛿. By condition of Theorem, for every 𝑔∈ℱ, we know 𝑃(𝑔)(𝑧)≠𝜓(𝑧) on Δ_𝛿. By theorem D, ℱ is normal on Δ_𝛿, so ℱ is normal on 𝑧=0.
Now, we consider 𝑃(𝑓)(0)=𝜓(0). Suppose 𝑃(𝑓)(𝑧)≢𝜓(𝑧) on the neighborhood |𝑧|<𝛿 (where 𝛿 is a small positive number) (otherwise, 𝑃(𝑓)(𝑧)≡𝜓(𝑧) on the neighborhood |𝑧|<𝛿, by condition of theorem, for every 𝑔∈ℱ, we also obtain 𝑃(𝑔)(𝑧)≡𝜓(𝑧). So 𝑃(𝑔)(𝑧)≠𝜓(𝑧)+1. By Theorem D, ℱ is normal at 𝑧=0. So Theorem 1.4 is proved), there exists 𝛿>0 such that 𝑃(𝑓)(𝑧)≠𝜓(𝑧) on (𝑧∈Δ^′_𝛿). So, for every 𝑔∈ℱ, we obtain 𝑃𝑔(𝑧)≠𝜓(𝑧)𝑧∈Δ^′_𝛿. (3.2) By Theorem D, ℱ is normal on Δ^′.
Next, we will prove ℱ is normal at 𝑧=0. Suppose, on the contrary, that ℱ is not normal at 𝑧=0∈Δ, then there exists a sequence functions (we also denote ℱ) that has no any normal subsequence on 𝑧=0.
Consider the family ℑ={𝑔(𝑧)=(𝑓(𝑧)/𝜓(𝑧))∶𝑓∈ℱ,𝑧∈Δ}. Since 𝑓≠0 for 𝑓∈ℱ, we have that 𝑔(0)=∞ for each 𝑔∈ℑ.
We first prove that ℑ is normal in Δ. Suppose, on the contrary, that ℑ is not normal at 𝑧₀∈Δ. By Lemma 2.1, there exist a sequence of functions 𝑔_𝑛∈ℑ, a sequence of complex numbers 𝑧_𝑛→𝑧₀, and a sequence of positive numbers 𝜌_𝑛→0, such that 𝐺_𝑛(𝜉)=𝑔_𝑛𝑧_𝑛+𝜌_𝑛𝜉𝜌^𝑘_𝑛⟶𝐺(𝜉) (3.3) converges spherically uniformly on compact subsets of ℂ where 𝐺(𝜉) is a nonconstant meromorphic function on ℂ, and 𝐺(𝜉)≠0.
We distinguish two cases.
Case 1 (𝑧_𝑛/𝜌_𝑛→∞). By a simple calculation, for 0≤𝑖≤𝑘, we have 𝑔_𝑛^(𝑖)(𝑧)=𝑓_𝑛^(𝑖)(𝑧)𝜓(𝑧)−^𝑖_𝑗=1𝐶^𝑗_𝑖𝑔_𝑛^{(𝑖−𝑗)}(𝑧)𝜓^(𝑗)(𝑧)𝜓(𝑧)=𝑓_𝑛^(𝑖)(𝑧)𝜓(𝑧)−^𝑖_𝑗=1𝐶^𝑗_𝑖𝑔_𝑛^{(𝑖−𝑗)}(𝑧)^𝑗_𝑡=0𝐴_𝑗𝑡1𝑧^𝑗−𝑡𝜑^(𝑡)(𝑧)𝜑(𝑧), (3.4) where 𝐴_𝑗𝑡=𝑙(𝑙−1)⋯(𝑙−𝑗+𝑡+1)𝐶^𝑡_𝑗 if 𝑙<𝑗, for 𝑡=0,1,…,𝑗−1 and 𝐴_𝑗𝑗=1.
Thus, from (3.4), we have 𝜌_𝑛^𝑘−𝑖𝐺_𝑛^(𝑖)(𝜉)=𝑔_𝑛^(𝑖)𝑧_𝑛+𝜌_𝑛𝜉=𝑓_𝑛^(𝑖)𝑧_𝑛+𝜌_𝑛𝜉𝜓𝑧_𝑛+𝜌_𝑛𝜉−^𝑖_𝑗=1𝐶^𝑗_𝑖𝑔_𝑛^{(𝑖−𝑗)}𝑧_𝑛+𝜌_𝑛𝜉^𝑗_𝑡=0𝐴_𝑗𝑡1𝑧_𝑛+𝜌_𝑛𝜉^𝑗−𝑡𝜑^(𝑡)𝑧_𝑛+𝜌_𝑛𝜉𝜑𝑧_𝑛+𝜌_𝑛𝜉=𝑓_𝑛^(𝑖)𝑧_𝑛+𝜌_𝑛𝜉𝜓𝑧_𝑛+𝜌_𝑛𝜉−^𝑖_𝑗=1𝐶^𝑗_𝑖𝑔_𝑛^{(𝑖−𝑗)}𝜌^𝑗_𝑛𝑧_𝑛+𝜌_𝑛𝜉^𝑗_𝑡=0𝐴_𝑗𝑡1𝑧_𝑛+𝜌_𝑛𝜉^𝑗−𝑡𝜌^𝑡_𝑛𝜑^(𝑡)𝑧_𝑛+𝜌_𝑛𝜉𝜑𝑧_𝑛+𝜌_𝑛𝜉. (3.5) On the other hand, we have lim_𝑛→∞1𝑧_𝑛/𝜌_𝑛+𝜉=0,lim_𝑛→∞𝜌^𝑡_𝑛𝜑^(𝑡)𝑧_𝑛+𝜌_𝑛𝜉𝜑𝑧_𝑛+𝜌_𝑛𝜉=0, (3.6) for 𝑡≥1. Noting that 𝑔_𝑛^{(𝑖−𝑗)}(𝑧_𝑛+𝜌_𝑛𝜉)/𝜌^𝑗_𝑛 is locally bounded on ℂ minus the set of poles of 𝐺(𝜉) since 𝑔_𝑛(𝑧_𝑛+𝜌_𝑛𝜉)/𝜌^𝑘_𝑛→𝐺(𝜉). Therefore, on every subset of ℂ which contains no poles of 𝐺(𝜉), we have 𝑓_𝑛^(𝑘)𝑧_𝑛+𝜌_𝑛𝜉𝜓𝑧_𝑛+𝜌_𝑛𝜉⟶𝐺^(𝑘)(𝜉),𝑓_𝑛^(𝑖)𝑧_𝑛+𝜌_𝑛𝜉𝜓𝑧_𝑛+𝜌_𝑛𝜉⟶0, (3.7) for 𝑖=0,1,…,𝑘−1, and thus 𝑓_𝑛^(𝑘)𝑧_𝑛+𝜌_𝑛𝜉+Σ^𝑘−1_𝑖=0𝑎_𝑖𝑧_𝑛+𝜌_𝑛𝜉𝑓_𝑛^(𝑖)𝑧_𝑛+𝜌_𝑛𝜉𝜓𝑧_𝑛+𝜌_𝑛𝜉⟶𝐺^(𝑘)(𝜉),𝑓_𝑛^(𝑘)𝑧_𝑛+𝜌_𝑛𝜉+Σ^𝑘−1_𝑖=0𝑎_𝑖𝑧_𝑛+𝜌_𝑛𝜉𝑓_𝑛^(𝑖)𝑧_𝑛+𝜌_𝑛𝜉−𝜓𝑧_𝑛+𝜌_𝑛𝜉𝜓𝑧_𝑛+𝜌_𝑛𝜉⟶𝐺^(𝑘)(𝜉)−1, (3.8) since 𝑎₀,…,𝑎_𝑘−1 are analytic in 𝐷.
By 𝐺(𝜉)≠0, we know 𝐺^(𝑘)(𝜉)≢1. In fact, if 𝐺^(𝑘)(𝜉₀)=1, by Hurwitz’s theorem, then exists 𝜉_𝑛→𝜉₀, for 𝑛 sufficiently large, 𝑃𝑓𝑧_𝑛+𝜌_𝑛𝜉_𝑛=𝜓𝑧_𝑛+𝜌_𝑛𝜉_𝑛. (3.9) By the condition of theorem, for every positive number 𝑚, we obtain 𝑃(𝑓_𝑚)(𝑧_𝑛+𝜌_𝑛𝜉_𝑛)=𝜓(𝑧_𝑛+𝜌_𝑛𝜉_𝑛). We know 𝑧_𝑛+𝜌_𝑛𝜉_𝑛→𝑧₀∈Δ_𝛿, and, for sufficiently large 𝑛, 𝑧_𝑛+𝜌_𝑛𝜉_𝑛∈Δ_𝛿. However, 𝑧_𝑛+𝜌_𝑛𝜉_𝑛≠0 (otherwise, 𝑧_𝑛+𝜌_𝑛𝜉_𝑛=0, so 𝜉_𝑛=−(𝑧_𝑛/𝜌_𝑛)→∞, a contradiction), so for sufficiently large 𝑛, 𝑧_𝑛+𝜌_𝑛𝜉_𝑛∈Δ^′_𝛿. This contradicts with (3.2).
So 𝐺(𝜉)≠0 and 𝐺^(𝑘)(𝜉)≠1, by Lemma 2.8, we obtain 𝐺 is a constant, a contradiction.
Case 2. 𝑧_𝑛/𝜌_𝑛→𝛼 is a finite complex number. Then, 𝑔_𝑛𝜌_𝑛𝜉𝜌^𝑘_𝑛=𝑔_𝑛𝑧_𝑛+𝜌_𝑛𝜉−𝑧_𝑛/𝜌_𝑛𝜌^𝑘_𝑛=𝐺_𝑛𝑧𝜉−^𝑛𝜌_𝑛⟶𝐺(𝜉−𝛼)=𝔾(𝜉). (3.10) Obviously, 𝔾(𝜉)≠0, and 𝜉=0 is a pole of 𝔾 with order at least 𝑙.
Set 𝐻_𝑛(𝜉)=𝑓_𝑛𝜌_𝑛𝜉𝜌_𝑛^𝑘+𝑙. (3.11) Then, 𝐻_𝑛(𝜉)=𝜓𝜌_𝑛𝜉𝜌^𝑙_𝑛𝑓_𝑛𝜌_𝑛𝜉𝜌^𝑘_𝑛𝜓𝜌_𝑛𝜉=𝜓𝜌_𝑛𝜉𝜌^𝑙_𝑛𝑔_𝑛𝜌_𝑛𝜉𝜌^𝑘_𝑛. (3.12) Noting that 𝜓(𝜌_𝑛𝜉)/𝜌^𝑙_𝑛→𝜉^𝑙, thus 𝐻_𝑛(𝜉)⟶𝜉^𝑙𝔾(𝜉)=𝐻(𝜉), (3.13) uniformly on compact subsets of ℂ. Since 𝔾 has a pole of order at least at 𝜉=0, we have 𝐻(0)≠0, so that 𝐻(𝜉)≠0.
From (3.11), we get 𝐻_𝑛^(𝑖)=𝑓_𝑛^(𝑖)𝜌_𝑛𝜉𝜌_𝑛^{𝑘+𝑙−𝑖}⟶𝐻^(𝑖)(𝜉), (3.14) spherically uniformly on compact subsets of ℂ minus the set of poles of 𝔾(𝜉). As the above, on every compact subset of ℂ minus the set of poles of 𝐺(𝜉), we have 𝑓_𝑛^(𝑘)𝜌_𝑛𝜉+Σ^𝑘−1_𝑖=0𝑎_𝑖𝜌_𝑛𝜉𝑓_𝑛^(𝑖)𝜌_𝑛𝜉𝜌^𝑙_𝑛⟶𝐻^(𝑘)(𝜉), (3.15)𝑓_𝑛^(𝑘)𝜌_𝑛𝜉+Σ^𝑘−1_𝑖=0𝑎_𝑖𝜌_𝑛𝜉𝑓_𝑛^(𝑖)𝜌_𝑛𝜉−𝜓𝜌_𝑛𝜉𝜌^𝑙_𝑛⟶𝐻^(𝑘)(𝜉)−𝜉^𝑙, (3.16) locally uniformly on ℂ.
By the assumption of Theorem and (3.16), Hurwitz’s theorem implies 𝐻^(𝑘)(𝜉)≠0.
Next, we proof that if 𝜉∈ℂ/{0}, then 𝐻^(𝑘)(𝜉)≠𝜉^𝑙.
First, 𝐻^(𝑘)(𝜉)≢𝜉^𝑙, otherwise 𝐻^(𝑘)(𝜉)≡𝜉^𝑙, which contradicts with 𝐻(𝜉)≠0. If there exists a 𝜉₀≠0 such that 𝐻^(𝑘)(𝜉₀)=𝜉^𝑙₀, by Hurwitz’s theorem and (3.16), there exists 𝜉_𝑛→𝜉₀ such that 𝑓_𝑛^(𝑘)(𝜌_𝑛𝜉_𝑛)+Σ^𝑘−1_𝑖=0𝑎_𝑖(𝜌_𝑛𝜉_𝑛)𝑓_𝑛^(𝑖)(𝜌_𝑛𝜉_𝑛)=𝜓(𝜌_𝑛𝜉_𝑛). By the assumption of Theorem 1.4, for every positive 𝑚 such that 𝑃(𝑓_𝑚)(𝜌_𝑛𝜉_𝑛)=𝜓(𝜌_𝑛𝜉_𝑛). However, for 𝑛 sufficiently large, 𝜌_𝑛𝜉_𝑛∈Δ^′_𝛿, all of these contradict with (3.2). So if 𝜉∈ℂ/{0}, then 𝐻^(𝑘)(𝜉)≠𝜉^𝑙.
Noting 𝐻(𝜉)≠0, By Lemma 2.9, we know 𝐻 must be a rational function. If 𝐻 is not a constant, By Lemma 2.4, we know 𝐻^(𝑘)(𝜉)−𝜉^𝑙 has at least two distinct zeros, a contradiction. So 𝐻 must be a nonzero constant, also contradicts with 𝐻^(𝑘)(𝜉)≠0. Now, we have proved the ℑ is normal on Δ_𝛿.
It remains to show that ℱ is normal at 𝑧=0. Since ℑ is normal in Δ, then the family ℑ is equicontinuous on Δ with respect to the spherical distance. On the other hand, 𝑔(0)=∞ for each 𝑔∈ℑ, so there exists 𝛿>0 such that |𝑔(𝑧)|≥1 for all 𝑔∈ℑ and each 𝑧∈Δ_𝛿={𝑧∶|𝑧|<𝛿}. Suppose that ℱ is not normal at 𝑧=0. Since ℱ is normal in 0<|𝑧|<1, the family ℱ₁={1/𝑓∶𝑓∈ℑ} is normal in Δ={𝑧∶0<|𝑧|<1}, but it is not normal at 𝑧=0. Then, there exists a sequence {1/𝑓_𝑛}⊂ℱ₁ which converges locally uniformly in Δ^′, but not in Δ. Noting that 𝑓_𝑛≠0 in Δ, 1/𝑓_𝑛 is holomorphic in Δ for each 𝑛. The maximum modulus principle implies that 1/𝑓_𝑛→∞ in Δ^′. Thus, 𝑓_𝑛→0 converges locally uniformly in Δ^′, and hence so does {𝑔_𝑛}⊂ℑ, where 𝑔_𝑛=𝑓_𝑛/𝜑. But |𝑔_𝑛(𝑧)|≥1 for each 𝑧∈Δ_𝛿, a contradiction. This finally completes the proof of Theorem 1.4.

Acknowledgments

This paper is supported by Leading Academic Discipline Project 10XKJ01, by Key Development Project 12C102 of Shanghai Dianji University, by the National Natural Science Foundation of China (11171184), and by National Natural Science Youth Fund Project (51008190).