1 Introduction and Main Results

First of all we recall that a family F of functions meromorphic in a plane domain D is called to be normal in D,in the sense of Montel,if every sequence{fn}⊂F contains a subsequence{fnj}which converges spherically locally uniformly in D,to a meromorphic function or the constant ∞ (see[1]–[3]).

在只有气水两相存在的气藏中，由于重力分异气水分布常有一定的规律：气在上，水在下，形成气水界面［1］。气水界面对于储量评价和井位设计等工作意义重大。在石油天然气行业标准中，提供了很多确定垂向上气水界面的方法［2］，但如何将垂向上的气水界面在平面构造等深图上表示出来，目前未见相关报道。

Let f and g be meromorphic in a domain D,b∈ C∪{∞}.If f(z)−b and g(z)−b assume the same zeros ignoring multiplicity,we say that f and g share b in D.

Inspired by heuristic Bloch’s principle(see[4]–[5])that there is an analogue in normal family theory corresponding to every Liouville-Picard type theorem,Gu[6]proved the following famous normality criterion related to the well-known Hayman’s alternative(see[7]).

Theorem A[6]Let F be a family of meromorphic functions in a domain D,k be a positive integer,and b be a nonzero finite complex number.If for each f∈F,f0 and f(k)b in D,then F is normal in D.

游泳过程中，上臂旋转外展时，水的阻力会对肩部造成很大的压力。一旦长时间保持不正确的泳姿，会让肩袖处于过度运动状态。几种泳姿比较起来，自由泳、蝶泳和仰泳比较容易出现游泳肩。比如自由泳时，在“拉水阶段”多使用胸肌、阔背肌等内旋转肌，而在“恢复阶段”则多使用三角肌、冈下肌等外旋转肌。通常来说，外旋转肌较薄弱，极容易损伤。而蛙泳的基本姿势确定了其肩部活动的范围较小，因此会好一些。

Recently,by the idea of shared values,Fang and Zalcman[8],[9]extended Theorem A as follows.

Theorem B[8],[9]Let k be a positive integer and F be a family of meromorphic functions in a domain D such that for each f∈F,all zeros of f are of multiplicity at least k+2.Let a and b0 be two finite complex numbers.If for each pair of functions f,g∈F,f and g share a,f(k)and g(k)share b in D,then F is normal in D.

In 1989,Schwick[10]obtained the following theorem.

a contradiction.

In 2009,Li and Gu[11]improved Theorem C and proved the following result with the idea of shared values.

（2）IoT设备自身的资源受限。IoT节点主要由一些嵌入式的传感设备组成，这类设备的计算能力、存储空间和通信效率极其有限。由于这种限制，当前互联网的诸多安全解决方案（例如：漏洞检测、流量审计、访问控制等）不能很好地迁移到IoT系统中，导致IoT设备在面对形如Mirai病毒时却无能为力，这种因设备资源受限而导致安全检测能力的降低（甚至丧失）给IoT系统的安全造成了严重的威胁。

Theorem D[11]Let F be a family of meromorphic functions in a domain D,n,k be positive integers with n≥k+2,and b be a nonzero finite complex number.If for each pair of functions f,g∈F,(fn)(k)and(gn)(k)share b in D,then F is normal in D.

In 1998,Wang and Fang[12]proved the following theorem.

Theorem E[12]Let k be a positive integer and F be a family of meromorphic functions in a domain D such that for each f∈F,all poles of f are of multiplicity at least 2,and all zeros of f are of multiplicity at least k+1.Let b be a nonzero finite complex number.If for each f∈F,f(k)b in D,then F is normal in D.

It is natural to ask whether Theorem E can be extended in the same way that Theorem B extends Theorem A or Theorem D extends Theorem C.In this paper,we offer such an extension.

Theorem 1.1 Let k be a positive integer and F be a family of meromorphic functions in a domain D such that for each f∈F,all poles of f are of multiplicity at least 2,and all zeros of f are of multiplicity at least k+1.Let a and b be two distinct finite complex numbers.If for each f∈F,all zeros of f(k)−a are of multiplicity at least 2,and for each pair of functions f,g∈F,f(k)and g(k)share b in D,then F is normal in D.

Corollary 1.1 Let k be a positive integer and F be a family of holomorphic functions in a domain D such that for each f∈F,all zeros of f are of multiplicity at least k+1.Let a and b be two distinct finite complex numbers.If for each f∈F,all zeros of f(k)−a are of multiplicity at least 2,and for each pair of functions f,g∈F,f(k)and g(k)share b in D,then F is normal in D.

Moreover,we can prove the following result by restricting the numbers of the zeros of f(k)−b.

Theorem 1.2 Let k be a positive integer and F be a family of meromorphic functions in a domain D such that for each f∈F,all poles of f are of multiplicity at least 2,and all zeros of f are of multiplicity at least k+1.Let a and b be two distinct finite complex numbers.If for each f∈F,all zeros of f(k)−a are of multiplicity at least 2,f(k)−b has at most 1 zero in D ignoring multiplicity,then F is normal in D.

Brother精心研发的家用标签产品分为两大阵营：智能手机蓝牙标签机一族的PT-P710BT和PTP300BT以及卡通标签机一族的PT-D200 (KT)和PT-D200 (SN)，方便用户在做个人物品识别或家庭收纳时有更多的选择。Brother家用标签产品独特的覆膜特性使其具备防水、防油污等六大特性，即使反复擦拭都能保证标签字迹清晰、长久保存。配合PT-P710BT同时上市的还有6款不同宽度的珠光标签，满足用户的各种巧思。

Corollary 1.2 Let k be a positive integer and F be a family of holomorphic functions in a domain D such that for each f∈F,all zeros of f are of multiplicity at least k+1.Let a and b be two distinct finite complex numbers.If for each f∈F,all zeros of f(k)−a are of multiplicity at least 2,f(k)−b has at most 1 zero in D ignoring multiplicity,then F is normal in D.

Example 1.1 Let D={z:|z| ＜ 1}and F={fm:m=1,2,3,···},where fm(z)=mzk+2.Then,for each fm∈F,all zeros of fmare of multiplicity at least k+1 in D,and

Thus we see that for each pair of functions fm,fn∈F,andshare 0 in D,and for each fm∈F,has only one zero z=0 of exact multiplicity 2 in D.But F fails to be normal in D.This shows that the condition in Theorems 1.1 and 1.2 that a and b be two distinct finite complex numbers is necessary.

2Some Lemmas

Lemma 2.1[13] Let k be a positive integer and F be a family of meromorphic functions in a domain D on C,all of whose zeros have multiplicity at least k,and suppose that there exists A≥1 such that|f(k)(z)|≤A whenever f(z)=0,f∈F.Then,if F is not normal at some point z0∈D,there exist,for each 0≤α≤k,

(a)points zn∈D,zn→z0;

(b)functions fn∈F,and

(c)positive numbers ρn → 0

such that

locally uniformly with respect to the spherical metric,where g is a nonconstant meromorphic function on C,all of whose zeros have multiplicity at least k,such that g#(ζ) ≤ g#(0)=kA+1.In particular,g is of order at most two.

Lemma 2.2[12] Let k be a positive integer and f be a transcendental meromorphic function in C such that all poles of f are of multiplicity at least 2,and all zeros of f are of multiplicity at least k+1.Then f(k)assumes every nonzero finite complex number in finitely often.

Lemma 2.3 Let k be a positive integer and f be a nonconstant rational function such that all poles of f are of multiplicity at least 2,and all zeros of f are of multiplicity at least k+1.Let a and b be two distinct finite complex numbers.If all zeros of f(k)−a are of multiplicity at least 2,then f(k)−b has at least two distinct zeros.Proof.We consider two cases.

Case 1. f(k)−b has exactly one zero z0.

从这个意义上讲，给村社集体以实权，让他们具有依据当地实际来进行公共事业建设的权力，而不是限制村社集体权力，约束村社集体权力，甚至直接虚化和架空村社集体，国家与农民之间才会有一个有力的对接平台与工作抓手。

where B1is a nonzero constant,z0 ξi(i=1,2,···,s)since ab.Differentiating(2.4)we have

提高大学生课外活动的参与度，应当首先了解大学生的兴趣所在，倾听大学生的实际需求，根据高校的人才培养目标，结合大学生的心理需求与兴趣爱好，创新活动的内容。例如实行“点餐式”的课外活动项目：读书学习活动、科技创新活动、社会实践活动、户外运动等，让学生选择自己喜欢的内容。

Clearly,f(k)(z)a and f(k)(z)b,for otherwise f would be a polynomial of degree at most k,which contradicts the fact that all zeros of f are of multiplicity at least k+1.Since f(k)−b has exactly one zero z0and all zeros of f(k)−a are of multiplicity at least 2,then we can put

where A is a nonzero constant,n ≥ 2,and lp ≥ 2,β1 αp(p=1,2,···,q).Thus,by(2.1)and(2.2),we get

Differentiating(2.1)and(2.2)it follows that

Since all poles of f are of multiplicity at least 2,all zeros of f are of multiplicity at least k+1,and all zeros of f(k)− a are of multiplicity at least 2,it follows by the Nevanlinna’s first and second fundamental theorem that

i.e.,

This contradicts with(2.3).

3.时间方面。申请者在申请时必须承诺每年至少出庭26个半天[6]13，一个半天一般是3-4个小时。英格兰和威尔士的许多地区申请者要承诺每周出庭半天或每两周出庭一整天。宪法事务部秘书和大法官则希望治安法官每年参加审理35个半天。[6]4治安法官出庭时间一般会在年初有个时间表，以便各个治安法官尽早做好出庭安排，遇到紧急情况也可以申请调剂。除了必需的出庭时间外，治安法官还需接受培训和实践教育不断提升自我和跟上时代的需要。如果治安法官时间比较充裕，那么可以出庭更多的时间服务治安法官工作。

Subcase 1.2. f is a nonpolynomial rational function.

From the conditions we can set

where B is a nonzero constant,P and Q are two polynomials having no common factors,mi≥ 2(i=1,2,···,s),and nj ≥ k+2(j=1,2,···,t).For the sake of convenience,we put

Since f(k)−b has exactly one zero z0,from(2.4)it follows that

For this case we consider two subcases.

where degg ≤ s+t−1.Differentiating(2.7)we get

where

with some constants d0,d1,···,dt−1.

Next we distinguish two subcases again.

初中生的情感体验不够丰富，对未来世界充满了好奇心和探究心理。在美术中渗透情感教育直接会对初中生的人生价值观产生很大的影响，关系到他们将来看待事物和问题的心态、思路以及角度。因此在渗透情感教育的时候老师需要以身作则，为学生树立一个正面的榜样，并且加强在课上的师生互动，以便随时了解每个学生的情感动态，促进师生关系和谐，帮助学生健康发展。

Subcase 1.2.1. lN.

By(2.4)–(2.7)we see that degP ≥ degQ and so M ≥ N.Since z0 ξi(i=1,2,···,s),it follows from(2.8)and(2.9)that M−s≤degh=t and so M≤s+t.This together with(2.5)and(2.6)gives

采用SPSS 21.0统计学软件对数据进行处理，计数资料以例数（n）、百分数（%）表示，采用x2检验；计量资料以“±s”表示，采用t检验，以P＜0.05为差异有统计学意义。

Theorem C[10]Let F be a family of meromorphic functions in a domain D,n,k be positive integers with n≥k+3,and b be a nonzero finite complex number.If for each f∈F,(fn)(k)b in D,then F is normal in D.

Subcase 1.2.2.l=N.

If M≥N,then proceeding as in the argument in Subcase 1.2.1,from(2.8)and(2.9)we also have M＜M,a contradiction.

If M ＜N,then(2.8)and(2.9)means that l−1≤s+t−1 and so by(2.5)and(2.6)it follows that

a contradiction.

Case 2. f(k)−b has no zeros,i.e.,f(k)b.

where ϕ is a polynomial.Now from above two equalities and the fact that β1 αp(p=1,2,···,q),we have

which implies that T(r,f(k))=S(r,f(k)),a contradiction.

This completes the proof of Lemma 2.3.

Example 2.1 Let Then by a simple calculation it follows that

and

2.3 孕妇焦虑抑郁的影响因素分析 孕妇焦虑抑郁的发生在工作待遇上差异没有统计学意义(P>0.05)；但是在身体状况、家庭环境、家庭支持程度及家庭经济状况等方面差异显示出统计学意义(P<0.05)。

Clearly,not all zeros of f′(z)are of multiplicity at least 2,and f′(z)− 1 has only one zeroThis shows the condition in Lemma 2.3 that all zeros of f(k)−a are of multiplicity at least 2 cannot be omitted.

3 Proof of Theorem 1.1

Let z0∈D.We show that F is normal at z0.Let f∈F.We consider two cases.

Case 1. f(k)(z0)b.

There exists a δ＞ 0 such that f(k)(z)b in Dδ={z:|z−z0|＜ δ}⊂ D.Thus by the assumptions,for each h∈F,all poles of h are of multiplicity at least 2,all zeros of h are of multiplicity at least k+1,and h(k)(z)b in Dδ.By Theorem E,F is normal in Dδ.Hence,F is normal at z0.

一是防汛抗旱取得积极成效。2013年，河南省因持续高温少雨，发生了较为严重的春旱、伏旱和秋旱。全省累计投入抗旱资金22.2亿元，累计抗旱浇地13 349万亩次，解决21.94万人临时性饮水困难，挽回经济作物损失22.53亿元、粮食作物损失88.58亿元。完成了56个平原县计算机网络、视频会商系统和264个山丘乡（镇）视频会商系统建设，完善了防汛抗旱指挥体系。

Case 2. f(k)(z0)=b.

There exists a δ＞0 such that f(k)(z)b in D0δ={z:0＜|z−z0|＜δ}⊂D.Thus by the assumptions,for each h∈F,all poles of h are of multiplicity at least 2,all zeros of h are of multiplicity at least k+1,and h(k)(z)b in D0δ.By Theorem E,F is normal in D0δ.

Next we prove that F is normal at z0.Without loss of generality,we may assume that z0=0.Suppose,on the contrary,that F is not normal at z0.Then from Lemma 2.1 there exist

(i)points zj∈D,zj→z0;

uniformly on compact subsets of C disjoint from the poles of g.

(iii)positive numbers ρj → 0 such that

locally uniformly with respect to the spherical metric,where g is a nonconstant meromorphic function on C.Moreover,all poles of g are of multiplicity at least 2,and all zeros of g are of multiplicity at least k+1.

Wnt/β-catenin通路的活性决定了MPCs是分化为成骨细胞还是软骨细胞。实验表明［23］，敲除βcatenin的间充质细胞，会向软骨方向分化，而使骨生成受到抑制，生成异位的软骨。

Now,from(3.1)we have

(ii)functions fj∈F,and

Subcase 1.1. f is a nonconstant polynomial.

We claim that all zeros of g(k)(ζ)− a are of multiplicity at least 2.

Suppose that g(k)(ζ0)=a.Clearly,g(k)(ζ) a,for otherwise g would be a polynomial of degree at most k,which contradicts the fact that all zeros of g are of multiplicity at least k+1.Then by(3.2)and Hurwitz’s theorem there exist ζj, ζj → ζ0,such that,for j sufficiently large,

Thus

since all zeros of f(k)(z)−a are of multiplicity at least 2.It now follows that

根据层次分析法和熵权法分别计算出一级指标权重，再通过公式(10)，充分利用样本熵值对主客观权重进行加权组合，确定综合权重。二级指标涵盖内容较多，无法获取统计数据，因此二级权重仅采用层次分析法确定，主要参数及结果见表5。

which implies that all zeros of g(k)(ζ)−a are of multiplicity at least 2.The claim is proved.

Obviously,g(k)(ζ) b.Next we consider further two subcases.

Subcase 2.1.

By Hurwitz’s theorem we see that g(k)(ζ)b.This contradicts with Lemmas 2.2 and 2.3.

Subcase 2.2.a finite complex number.

By Hurwitz’s theorem we deduce that g(k)(ζ)b for ζ α,and g(k)(α)=b.This implies that g(k)(ζ)− b has only one zero ζ= α,contradicting with Lemmas 2.2 and 2.3 again.

Hence F is normal at z0and so F is normal in D.The proof of Theorem 1.1 is completed.

4 Proof of Theorem 1.2

Suppose that F is not normal in D.Then there exists at least one z0∈D such that F is not normal at the point z0.Without loss of generality,we can assume that z0=0.Thus from Lemma 2.1 we can find

(i)points zj∈D,zj→z0;

(ii)functions fj∈F,and

(iii) positive numbers ρj → 0 such that

locally uniformly with respect to the spherical metric,where g is a nonconstant meromorphic function on C.Moreover,all poles of g are of multiplicity at least 2,and all zeros of g are of multiplicity at least k+1.

Now,from(4.1)we have

uniformly on compact subsets of C disjoint from the poles of g.

Proceeding as in the proof in Theorem 1.1,we can prove that all zeros of g(k)(ζ)−a are of multiplicity at least 2.

We claim that g(k)(ζ)− b has at most 1 zero ignoring multiplicity.

Suppose that g(k)(ζ)− b has at least 2 distinct zero ζ1and ζ2.Clearly,g(k)(ζ) b,for otherwise g would be a polynomial of degree at most k,which contradicts the fact that all zeros of g are of multiplicity at least k+1.Then by(4.2)and Hurwitz’s theorem there exist ζj,1,ζj,2,ζj,1 → ζ1,ζj,2 → ζ2,such that,for j sufficiently large,

Since f(k)j(z)− b has at most 1 zero in D ignoring multiplicity and zj+ ρjζj,1 → z0,zj+ρjζj,2 → z0,it follows that

Then ζj,1= ζj,2and so ζ1= ζ2.This is a contradiction and the claim thus is proved.

However,from Lemmas 2.2 and 2.3,we see that there do not exist nonconstant meromorphic functions that have the above properties.This contradiction shows that F is normal in D,and therefore the proof of Theorem 1.2 is completed.

References

[1]Hayman W K.Meromorphic Functions.Oxford:Clarendon Press,1964.

[2]SchiffJ.Normal Families.Berlin:Springer-Verlag,1993.

[3]Yang L.Value Distribution Theory.Berlin:Springer-Verlag,1993.

[4]Bergweiler W.Bloch’s principle.Comput.Methods Funct.Theory,2006,6:77–108.

[5]Zalcman L.Normal families:new perspectives,Bull.Amer.Math.Soc.,1998,35:215-230.

[6]Gu Y X.Un crit`ere de normalité des familles de fonctions méromorphes.Sci.Sinica,1979,Special Issue I:267–274.

[7]Hayman W K.Picard values of meromorphic functions and their derivatives.Math.Ann.,1959,70:9–42.

[8]Fang M L,Zalcman L.Normality and shared sets.J.Aust.Math.Soc.,2009,86:339–354.

[9]Fang M L,Zalcman L.A note on normality and shared values.J.Aust.Math.Soc.,2004,76:141–150.

[10]Schwick W.Normality criteria for families of meromorphic functions.J.d’Anal.Math.,1989,52:241–289.

[11]Li Y T,Gu Y X.On normal families of meromorphic functions.J.Math.Anal.Appl.,2009,354:421–425.

[12]Wang Y F,Fang M L.Picard values and normal families of meromorphic functions with multiple zeros.Acta Math.Sinica(N.S.),1998,14(1):17–26.

[13]Pang X C,Zalcman L.Normal families and shared values.Bull.London Math.Soc.,2000,32:325–331.