Normal Families Of Meromorphic Functions And Its Applications

The theory of normal families is an important and classical branch of the complex analysis. At the beginning of the20th century, Montel introduced the definition of the normal families of meromorphic functions. A family of meromorphic functions F on a domain D(?)Cis said to be normal on D if each sequence of functions{fn}(?)F has a subsequence{fn,i} which converges locally uniformly with spherical distance to a meromorphic function or oo(cf.[18,37,47]). Up to now, the theory of normal families has achieved great development. Many domestic and foreign scholars such as Q. L. Xiong, C. T. Chuang, L. Yang, G. H. Zhang, Y. X. Gu, H. H. Chen, X. C. Pang, Hayman, Drasin, Zalcman, Bergweiler, et al., have obtained many profound results.The research on normal families of meromorphic functions has not only impor-tant theoretical significance but also a wide range of application. The theory of normal families has intimate connection with many theories such as the modulus distribution and argument distribution of meromorphic functions, the complex differential equa-tions, uniqueness of meromorphic functions, the complex dynamical systems and so on. Besides, there are some new perspectives such as:replacing the exceptional con-stant by the exceptional function; normal families on shared values or shard func-tions; the applications of (quasi)normal families in value distribution theory and so on. Therefore, the theory of normal families of meromorphic functions is still an active branch of complex analysis.In this thesis, we do some research on normal families of meromorphic functions and its applications, and obtained some significant results. The thesis is divided into five chapters.In Chapter1, we mainly give the notations, definitions, fundamental facts and important theorems in value distribution theory and normal family.Chapter2mainly deals with the families of meromorphic functions concerning exceptional functions. In1979, Gu[17] proved:Let F be a family of meromorphic functions defined on a domain D and k be a positive integer. If for every function f∈F,f≠0,f(k)≠1, then F is normal on D.Remarkl:The case for holomorphic functions is proved by Miranda[26]. The above theorem is called Gu-Miranda normality criterion.Recently, Chang[6] generalized the well-known Gu-Miranda normality criterion by allowing f(k)-1to have zeros but restricting their numbers. He proved:Theorem A2Let k be a positive integer and F be a family of zero-free mero-morphic functions on a domain D. If for each f∈F,f(k)-1has at most k distinct zeros(ignoring multiplicity) on D, then F is normal on D.Naturally, we consider whether we can replace the constant1in Theorem A2by an analytic function φ(z)((?)0). In [13], Deng-Fang-Liu proved:Theorem A4Let k be a positive integer,φ(z)((?)0) be an analytic function on a domain D, and F be a family of zero-free meromorphic functions on D. If, for each f∈F, f(k)(z)-φ(z) has at most k distinct zeros (ignoring multiplicity) on D, then F is normal on D.We continue to study this problem and obtain the following result.Theorem2.1.1Let k be a positive integer,φ(z)((?)0) be an analytic function on a domain D, and F be a family of zero-free meromorphic functions on D, all of whose poles are multiple. If for each f∈F, f(k)(z)-φ(z) has at most k+1distinct zeros(ignoring multiplicity)on D, then F is normal on D.Remark2:Theorem A4is correct but the proof is incomplete ([13]p320, they ignored the possibility that G is a nonzero constant). We complete it during the proof of Theorem2.1.1Besides, Theorem2.1.1extends and improves, in some sense, the related results of Chang[6], Schwick[39] and Yang[48]. As an application, we obtain the following two quasinormality criteria.Theorem2.4.1Let k, K be positive integers,φ(z)((?)0) be an analytic function, and F be a family of zero-free meromorphic functions on a domain D. If for each f∈F, f(k)(z)-φ(z) has at most K distinct zeros(ignoring multiplicity)on D, then F is quasinormal of order at most v on D, where v=[K/k+1] is equal to the largest integer not exceedingK/k+1. Theorem2.4.2Let k, K be positive integers,φ(z)((?)0) be an analytic func-tion, and F be a family of zero-free meromorphic functions on a domain D, all of whose poles are multiple. If for each f∈F, f(k)(z)-φ(z) has at most K distinct zeros(ignoring multiplicity) on D, then F is quasinormal of order at most v on D, where v=[K/k+2] is equal to the largest integer not exceeding K/k+2.In Chapter3, we consider normal families concerning shared functions.The connection between normal families and shared values was first discovered by Schwick [38]. In2002, Fang-Zalcman[16] proved:Theorem B1Let k be a positive integer and let a and b be two non-zero finite values, F be a family of meromorphic functions defined on a domain D, all of whose zeros have multiplicity at least k+1. If for each f∈F,=a(?)f(k)=b, then F is normal on D.By considering holomorphic functions a(z) and b(z) instead of the values a and b, Lei-Yang-Fang[23] extended the above theorem for the case k≥2and obtained the following result.Theorem B2Let k≥2be an integer, and let a(z)(≠0), b(z)((?)0) be two holomorphic functions on a domain D, F be a family of meromorphic functions on D, all of whose zeros have multiplicity at least k+1. If for each f∈F, f(z)=a(z)(?)f(k)(z)=b(z), then F is normal on D.A natural problem arises:Whether the theorem is true when k=1? In this chapter, we obtainTheorem3.1.1Let a(z)(≠0) and b(z)((?)0) be two holomorphic functions on a domain D such that a’(z) does not take the value0at the zeros of b(z). Let F be a family of meromorphic functions on D, all of whose zeros are multiple. If for each f∈F, f(z)=a(z)(?)f’(z)=b(z), then F is normal on D.Remark3:There is an example in this thesis (see P22, Example3.1.1) which shows that the condition "a’(z) does not take the value0at the zeros of b(z)" in Theorem3.1.1is necessary.In Chapter4, by weakening some assumption of the existing normality criteria, we discuss non-normal families and the uniqueness of counterexamples. In2004, Xu[43](cf.[34,36,39,42,48] etc.) generalized and improved Gu-Miranda normality criterion as follows.Theorem C1Let φ(z)((?)0) be a function holomorphic on a domain D and k be a positive integer. Let F be a family of meromorphic functions defined on D, all of whose poles are multiple and whose zeros have multiplicity at least k+2. If for every function f∈F, f(k)(z)≠φ(z), then F is normal on D.Theorem C2Let φ(z)((?)0) be a function holomorphic on a domain D and k be a positive integer. Let F be a family of meromorphic functions defined on D, all of whose zeros have multiplicity at least k+3. If for every function f∈F, f(k)(z)≠φ(z), then F is normal on D.Theorem C3Let φ(z)((?)0) be a function holomorphic on a domain D, and k be a positive integer. Let F be a family of meromorphic functions defined on D, all of whose zeros have multiplicity at least k+2. If for every function f∈F, f(k)(z)≠φ(z), and φ(z) has no simple zeros in D, then F is normal on D.We first construct an example(see P37, Example4.1.1) to show that the condition "all of whose poles are multiple" in Theorem C1is necessary; the number k+3in Theorem C2is best possible; the hypothesis "φ(z) has no simple zeros" in Theorem C3cannot be omitted. Then we study the property of non-normal families and obtain the following result.Theorem4.1.1Let k be a positive integer,φ(z)((?)0) be a function holomorphic on a domain D, and F be a family of meromorphic functions defined on D, all of whose zeros have multiplicity at least k+2such that for every function f∈F, f(k)(z)≠φ(z). If F is not normal at z0∈D, then z0must be the simple zero of φ(z) and there exist δ>0and {fn}(?)F such that fn(z)=(z-ζn)k+2/(z-ηn)fn(z), on Δ(z0,δ)={z:|z-z0|<δ}, where(ζn-z0)/ρn�'-c,(ηn-z0)/ρn�'-(k+2)c for some sequence of positive numbers ρn�'0and some constant c≠0. Moreover, fn(z) is holomorphic and non-vanishing on Δ(z0,δ) such that fn(z)�'f(z) locally uniformly on Δ(z0,δ), where f(z) satisfies [(z-z0)k+1f(z)](k)=φ(z).Remark4:Theorem4.1.1shows the counterexample about Theorem C1-C3(see P37,Example4.1.1)is unique in some sense.In2006,Huang-Gu[21](see Yuan-Fang[15])proved:Theorem C6Let F be a family of meromorphic functions on a domain D,and a(z)(≠0),b(z)be two analytic functions on D.For every function f∈F,if (1)f(z)≠∞when a(z)=0,(2)f’(z)-a(z)f2(z)≠b(z),and (3)all poles of f(z)are of multuokucuty at least4, then F is normal on D.Remark5:There is an example[21]which shows that Condition(1)in Theorem C6is necessary.Here,we give another example(see P46,Example4.2.1),which shows that Con-dition(3)in Theorem C6is sharp.Meanwhile,we proved:Theorem4.2.1Let F be a farnily of meromorphic functions on a domain D, all of whose poles have multiplicity at least3,and a(z)(≠0),b(z)be two analytic functions on D such that for each function f∈F,’(z)-a(z)f2(z)≠b(z)and f(z)≠∞when a(z)=0.If F is not normal at z0∈D,then z0must be a simple zero of a(z)and there exist δ>0and{fn)(?)such that fn(z)=z-ηn/(z-ζn)3fn(z), on△(z0,δ)={z:|z-zol<δ},where(ζn-z0)/ρn�'-c,(ηn-z0)/ρn�'-3c for some sequence of positive numbers ρn�'0and some constant c≠0.Moreover, fn(z)is holomorphic and non-vanishing on△(z0,δ)Such that,fn(z)�'f(z)locally uniformly on△(z0,δ),where f(z)satisfies the Riccati differential equations ω’(z)=(z-z0)2b(z)+2/z-z0ω(z)+a(z)/(z-z0)2ω2(z).Remark6:Theorem4.2.1shows the counterexample about Theorem C6(see P46,Example4.2.1)is unique in some sense.On the other hand,we get two new normality criteria,which can be viewed as the complement of Theorem C6.Theorem4.2.2Let F be a family of meromorphic functions on a domain D,and a(z)(≠0),b(z)be two analytic functions on D.For every function f∈F,if (1) f(z)≠∞when a(z)=0,(2) f’(z)-a(z)f2(z)b(z),and(3) all poles of f(z) are of multiplicity at least3and all zeros of f(z) are multiple, then F is normal on D.Theorem4.2.3Let F be a family of meromorphic functions on a domain D, and a(z)((?)0),b(z) be two analytic functions on D such that a(z) has no simple zeros. For every function f∈F, if(1) f(z)≠∞when a(z)=0,(2) f’(z)-a(z)f2(z)=≠b(z),and(3) all poles of f(z) are of multiplicity at least3, then F is normal on D.Chapter5is devoted to the theory of quasinormality criterion.Recently, Chang[5] proved the following quasinormality criterion, which ex-tended the result of Bergweiler[2].Theorem D4Let F be a family of meromorphic functions on a domain D. If, for each f∈F, f’(z)≠1, and there exists a constant K>0such that|f’(z)|≤K whenever f(z)=0in D, then F is quasinormal of order1on D.We extend the result to the case f(k) and obtain the following result.Theorem5.1.1Let k≥2be a positive integer and F be a family of meromor-phic functions on a domain D, all of whose zeros are of multiplicity at least k. If, for each f∈F, f(k)(z)≠1, and there exists a constant K>0such that|ftk)(z)|≤K whenever f(z)=0in D, then F is quasinormal of order1on D.Remark7:Theorem5.1.1also generalizes the related results due to Xu-Fang[46] and Nevo-Pang-Zalcman[29].The well-known Hayman’s alternative [18](cf.[19,47]) asserts that:for a tran-scendental meromorphic function f and k∈N, either f takes zero infinitely many times or f(k) takes each finite nonzero value infinitely many times.By using an argument from Nevanlinna theory, Wang-Fang [42] proved:Theorem D5Let f be a transcendental meromorphic function and k∈N. If all but finitely many zeros of f have multiplicity at least3, then f(k) takes each finite nonzero value infinitely many times. For k=1, Nevo-Pang-Zalcman [29] proved the following result, in which they used an argument involving quasinormal families.Theorem D6Let f be a transcendental meromorphic function with finitely many simple zeros. Then f’ takes each finite nonzero value infinitely many times.Chang[5] improved Theorem D6as follows, which also confirms a conjecture due to Bergweiler [2].Theorem D7Let f be a transcendental meromorphic function. If there exists a constant K>0such that|f’(z)|<K whenever f(z)=0in D, then f’ takes each finite nonzero value infinitely many times.As an application of Theorem5.1.1, we prove the following result, which im-proves Theorem D5for the case k=2and extends Theorem D7.Theorem5.4.1Let f be a transcendental meromorphic function with finitely many simple zeros. If there exists a constant K>0such that|f"(z)|<K whenever f(z)=0, then f" takes each finite nonzero value infinitely many times.