Let F be a family of functions meromorphic in a domain D, let n ≥ 2 be a positive integer, and let a ≠ 0, b be two finite complex numbers. If, for each f ∈ F, all of whose zeros have multiplicity at least k ＋ 1, a...Let F be a family of functions meromorphic in a domain D, let n ≥ 2 be a positive integer, and let a ≠ 0, b be two finite complex numbers. If, for each f ∈ F, all of whose zeros have multiplicity at least k ＋ 1, and f ＋ a（f^（k））^n≠b in D, then F is normal in D.展开更多
Let κ 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 κ + 1. L...Let κ 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 κ + 1. Let α and b be two distinct finite complex numbers. If for each f ∈ F, all zeros of f;-α are of multiplicity at least 2,and for each pair of functions f, g ∈ F, f;and g;share b in D, then F is normal in D.展开更多
In this article,we use Zalcman Lemma to investigate the normal family of meromorphic functions concerning shared values,which improves some earlier related results.
In this paper, we study the normality of families of meromorohic functions related to a Hayman conjecture. We prove that the conditions in Hayman conjecture and in other criterions can be relaxed. The results in this ...In this paper, we study the normality of families of meromorohic functions related to a Hayman conjecture. We prove that the conditions in Hayman conjecture and in other criterions can be relaxed. The results in this paper improve some previous results.展开更多
In this paper, we use Pang-Zalcman lemma to investigate the normal family of meromorphic functions concerning shared analytic function, which improves some earlier related results.
Let F be a family of meromorphic functions in D, and let ψ（≠ 0） be a meromorphic function in D all of whose poles are simple. Suppose that, for each f ∈ F, f ≠0 in D. If for each pair of functions {f, g} ∩ F, f...Let F be a family of meromorphic functions in D, and let ψ（≠ 0） be a meromorphic function in D all of whose poles are simple. Suppose that, for each f ∈ F, f ≠0 in D. If for each pair of functions {f, g} ∩ F, f′ and g′ share ψ in D, then F is normal in D.展开更多
In this paper, we study the normality criteria of meromorphic functions concerning shared fixed-points, we obtain: Let F be a family of meromorphic functions defined in a domain D. Let n, k ≥ 2 be two positive intege...In this paper, we study the normality criteria of meromorphic functions concerning shared fixed-points, we obtain: Let F be a family of meromorphic functions defined in a domain D. Let n, k ≥ 2 be two positive integers. For every f ∈ F, all of whose zeros have multiplicity at least (nk+2)/(n-1). If f(f(k))nand g(g(k))nshare z in D for each pair of functions f and g, then F is normal.展开更多
In this paper, we consider normality criteria for a family of meromorphic functions concerning shared values. Let F be a family of meromorphic functions defined in a domain D, m, n, k and d be four positive integers s...In this paper, we consider normality criteria for a family of meromorphic functions concerning shared values. Let F be a family of meromorphic functions defined in a domain D, m, n, k and d be four positive integers satisfying m ≥ n ＋ 2 and d≥k＋1/m-n-1 and a（≠ 0）, b be two finite constants. Suppose that every f∈F has all its zeros and poles of multiplicity at least k and d, respectively. If （fn）（k）-afm and （gn）（k） -agm share the value b for every pair of functions （f, g） of ~, then is normal in D. Our results improve the related theorems of Schwick （Schwick W. Normality criteria for families of meromorphic function. J. Anal. Math., 1989, 52： 241-289）, Li and Gu （Li Y T, Gu Y X. On normal families of meromorphic functions. J. Math. Anal. Appl., 2009, 354： 421-425）.展开更多
Let k be a positive integer,let h(z)■0 be a holomorphic functions in a domain D,and let F be a family of zero-free meromorphic functions in D,all of whose poles have order at least l.If,for each f∈P(f)(z)-h(z) has a...Let k be a positive integer,let h(z)■0 be a holomorphic functions in a domain D,and let F be a family of zero-free meromorphic functions in D,all of whose poles have order at least l.If,for each f∈P(f)(z)-h(z) has at most k+l-1 distinct zeros(ignoring multiplicity) in D,where P(f)(z)=f(k)(z)+a1(z)f((k-1)(z)+…+ak(z)f(z) is a differential polynomial of f and aj(z)(j=1,2,···,k) are holomorphic functions in D,then F is normal in D.展开更多
We studied the normality criterion for families of meromorphic functions which related to One-way sharing set, and obtain two normal criterions, which improve the previous results.
In this paper, we investigate normal families of meromorphic functions, prove some theorems of normal families sharing a holomorphic function, and give a counterex- ample to the converse of the Bloch principle based o...In this paper, we investigate normal families of meromorphic functions, prove some theorems of normal families sharing a holomorphic function, and give a counterex- ample to the converse of the Bloch principle based on the theorems.展开更多
We obtain some normality criteria of families of meromorphic functions sharing values related to Hayman conjecture, which improves some earlier related results.
In 1996, C. C. Yang and P. C. Hu [8] showed that： Let f be a transcendental meromorphic function on the complex plane, and a ≠ 0 be a complex number; then assume that n 〉 2, n1,… , nk are nonnegative integers such...In 1996, C. C. Yang and P. C. Hu [8] showed that： Let f be a transcendental meromorphic function on the complex plane, and a ≠ 0 be a complex number; then assume that n 〉 2, n1,… , nk are nonnegative integers such that n1＋… ＋ nk ≥1; thus fn（f′）n1…（f（k））nk-a has infinitely zeros. The aim of this article is to study the value distribution of differential polynomial, which is an extension of the result of Yang and Hu for small function and all zeros of f having multiplicity at least k ≥2. Namely, we prove that fn（f′）n1…（f（k））nk-a（z） has infinitely zeros, where f is a transcendental meromorphic function on the complex plane whose all zeros have multiplicity at least k≥ 2, and a（z） 0 is a small function of f and n ≥ 2, n1,… ,nk are nonnegative integers satisfying n1＋ …＋ nk ≥1. Using it, we establish some normality criterias for a family of meromorphic functions under a condition where differential polynomials generated by the members of the family share a holomorphic function with zero points. The results of this article are supplement of some problems studied by d. Yunbo and G. Zongsheng [6], and extension of some problems studied X. Wu and Y. Xu [10]. The main result of this article also leads to a counterexample to the converse of Bloeh＇s principle.展开更多
In the paper,we prove the main result:Let k(≥2)be an integer,and a,b and c be three distinct complex numbers.Let F be a family of functions holomorphic in a domain D in complex plane,all of whose zeros have multiplic...In the paper,we prove the main result:Let k(≥2)be an integer,and a,b and c be three distinct complex numbers.Let F be a family of functions holomorphic in a domain D in complex plane,all of whose zeros have multiplicity at least k.Suppose that for each f∈F,f(z)and f(k)(z)share the set{a,b,c}.Then F is a normal family in D.展开更多
In this paper,we study the normal criterion of meromorphic functions concerning shared analytic function.We get some theorems concerning shared analytic function,which improves some earlier related results.
Let k, m be two positive integers with m ≤ k and let F be a family of zero-free meromorphic functions in a domain D, let h(z) ≡ 0 be a meromorphic function in D with all poles of h has multiplicity at most m. If, fo...Let k, m be two positive integers with m ≤ k and let F be a family of zero-free meromorphic functions in a domain D, let h(z) ≡ 0 be a meromorphic function in D with all poles of h has multiplicity at most m. If, for each f ∈ F, f(k)(z) = h(z) has at most k- m distinct roots(ignoring multiplicity) in D, then F is normal in D. This extends the results due to Chang[1], Gu[3], Yang[11]and Deng[1]etc.展开更多
We studied the normality conditions in families of meromorphic functions, improved the results of Fang and Zalcman [Fang ML, Zalcman L, Normal families and shared values of meromorphic functions, Computational Methods...We studied the normality conditions in families of meromorphic functions, improved the results of Fang and Zalcman [Fang ML, Zalcman L, Normal families and shared values of meromorphic functions, Computational Methods and Function Theory, 2001, 1 （1）： 289-299], and generalized two new normality criterions. Let F be a family of meromorphic functions in a domain D, a a non-zero finite complex number, B a positive real number, and k and m two positive integers satisfying m〉2k＋4. If every function denoted by f belonging to F has only zeros with multiplicity at least k and satisfies f^m（z）f^（k）（Z）=α→ ｜^f（k）（z）｜ ≤B or f^m（z）f^（k）（z）=α→｜f（z）｜ ≥, then F is normal in D.展开更多
We proved： Let F be a family of meromorphic functions in a domain D anda α ≠0, b ∈C. If f1（z） - α（f（z））^2 ≠ b, f≠ 0 and the poles of f（z） are of multiplicity ≥ 3 foreach f（z） ∈F, then F is normal in D.
In this paper,we study the normality criterion for families of meromorphic functions concerning shared set depending on f∈F.Let F be a family of meromorphic functions in the unit disc A.For each f∈F,all zeros of f h...In this paper,we study the normality criterion for families of meromorphic functions concerning shared set depending on f∈F.Let F be a family of meromorphic functions in the unit disc A.For each f∈F,all zeros of f have multiplicity at least 2 and there exist nonzero complex numbers b_f,c_f satisfying(i) b_f/c_f is a constant;(ii) min{σ(0,b_f),σ(0,c_f),σ(b_f,c_f)} ≥m for some m > 0;(iii) E_f'(S_f)■ E_f(S_f),where S_f = {b_f,c_f}.Then F is normal in A.At the same time,the corresponding results are also proved.The results in this paper improve and generalize the related results展开更多
Let F- be a family of zero-free meromorphic functions in a domain D, let h（z） ≠ be a holomorphic function in D and let k be a positive integer. If, for each f ∈ P（f）（z） - h（z） has at most k distinct zeros（i...Let F- be a family of zero-free meromorphic functions in a domain D, let h（z） ≠ be a holomorphic function in D and let k be a positive integer. If, for each f ∈ P（f）（z） - h（z） has at most k distinct zeros（ignoring multiplicity） in D, where P（f） = f（k）（z） ＋ al（z）f（k-1）（z） ＋… ＋ ak（z）f（z） is a differential polynomial of f and aj（z）（j=1, 2,…, k） are holomorphic functions in D, then F is normal in D. This extends the results due to Deng[2], Chang[1], Gu[3], Yang[2], etc.展开更多
基金Supported by the NNSF of China(11071083)the Tianyuan Foundation(11126267)
文摘Let F be a family of functions meromorphic in a domain D, let n ≥ 2 be a positive integer, and let a ≠ 0, b be two finite complex numbers. If, for each f ∈ F, all of whose zeros have multiplicity at least k ＋ 1, and f ＋ a（f^（k））^n≠b in D, then F is normal in D.
基金The NSF(11301076)of Chinathe NSF(2014J01004) of Fujian Province
文摘Let κ 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 κ + 1. Let α and b be two distinct finite complex numbers. If for each f ∈ F, all zeros of f;-α are of multiplicity at least 2,and for each pair of functions f, g ∈ F, f;and g;share b in D, then F is normal in D.
文摘In this article,we use Zalcman Lemma to investigate the normal family of meromorphic functions concerning shared values,which improves some earlier related results.
基金The NSF(11271090) of Chinathe NSF(S2012010010121) of Guangdong Provincethe Graduate Research and Innovation Projects(XJGRI2013131) of Xinjiang Province
文摘In this paper, we study the normality of families of meromorohic functions related to a Hayman conjecture. We prove that the conditions in Hayman conjecture and in other criterions can be relaxed. The results in this paper improve some previous results.
文摘In this paper, we use Pang-Zalcman lemma to investigate the normal family of meromorphic functions concerning shared analytic function, which improves some earlier related results.
基金the National Natural Science Foundation of China(Grant Nos.1137113911261029+1 种基金11001081)the Scientific Research Foundation of CUIT(Grant No.KYTZ201403)
文摘Let F be a family of meromorphic functions in D, and let ψ（≠ 0） be a meromorphic function in D all of whose poles are simple. Suppose that, for each f ∈ F, f ≠0 in D. If for each pair of functions {f, g} ∩ F, f′ and g′ share ψ in D, then F is normal in D.
文摘In this paper, we study the normality criteria of meromorphic functions concerning shared fixed-points, we obtain: Let F be a family of meromorphic functions defined in a domain D. Let n, k ≥ 2 be two positive integers. For every f ∈ F, all of whose zeros have multiplicity at least (nk+2)/(n-1). If f(f(k))nand g(g(k))nshare z in D for each pair of functions f and g, then F is normal.
文摘In this paper, we consider normality criteria for a family of meromorphic functions concerning shared values. Let F be a family of meromorphic functions defined in a domain D, m, n, k and d be four positive integers satisfying m ≥ n ＋ 2 and d≥k＋1/m-n-1 and a（≠ 0）, b be two finite constants. Suppose that every f∈F has all its zeros and poles of multiplicity at least k and d, respectively. If （fn）（k）-afm and （gn）（k） -agm share the value b for every pair of functions （f, g） of ~, then is normal in D. Our results improve the related theorems of Schwick （Schwick W. Normality criteria for families of meromorphic function. J. Anal. Math., 1989, 52： 241-289）, Li and Gu （Li Y T, Gu Y X. On normal families of meromorphic functions. J. Math. Anal. Appl., 2009, 354： 421-425）.
文摘Let k be a positive integer,let h(z)■0 be a holomorphic functions in a domain D,and let F be a family of zero-free meromorphic functions in D,all of whose poles have order at least l.If,for each f∈P(f)(z)-h(z) has at most k+l-1 distinct zeros(ignoring multiplicity) in D,where P(f)(z)=f(k)(z)+a1(z)f((k-1)(z)+…+ak(z)f(z) is a differential polynomial of f and aj(z)(j=1,2,···,k) are holomorphic functions in D,then F is normal in D.
文摘We studied the normality criterion for families of meromorphic functions which related to One-way sharing set, and obtain two normal criterions, which improve the previous results.
文摘In this paper, we investigate normal families of meromorphic functions, prove some theorems of normal families sharing a holomorphic function, and give a counterex- ample to the converse of the Bloch principle based on the theorems.
基金supported by Nature Science Foundation of China(11461070),supported by Nature Science Foundation of China(11271227)PCSIRT(IRT1264)
文摘We obtain some normality criteria of families of meromorphic functions sharing values related to Hayman conjecture, which improves some earlier related results.
基金funded by Vietnam National Foundation for Science and Technology Development(NAFOSTED)under grant number 101.04-2014.41the Vietnam Institute for Advanced Study in Mathematics for financial support
文摘In 1996, C. C. Yang and P. C. Hu [8] showed that： Let f be a transcendental meromorphic function on the complex plane, and a ≠ 0 be a complex number; then assume that n 〉 2, n1,… , nk are nonnegative integers such that n1＋… ＋ nk ≥1; thus fn（f′）n1…（f（k））nk-a has infinitely zeros. The aim of this article is to study the value distribution of differential polynomial, which is an extension of the result of Yang and Hu for small function and all zeros of f having multiplicity at least k ≥2. Namely, we prove that fn（f′）n1…（f（k））nk-a（z） has infinitely zeros, where f is a transcendental meromorphic function on the complex plane whose all zeros have multiplicity at least k≥ 2, and a（z） 0 is a small function of f and n ≥ 2, n1,… ,nk are nonnegative integers satisfying n1＋ …＋ nk ≥1. Using it, we establish some normality criterias for a family of meromorphic functions under a condition where differential polynomials generated by the members of the family share a holomorphic function with zero points. The results of this article are supplement of some problems studied by d. Yunbo and G. Zongsheng [6], and extension of some problems studied X. Wu and Y. Xu [10]. The main result of this article also leads to a counterexample to the converse of Bloeh＇s principle.
基金Supported by the NSF of China(10771220)Supported by the Doctorial Point Fund of National Education Ministry of China(200810780002)
文摘In the paper,we prove the main result:Let k(≥2)be an integer,and a,b and c be three distinct complex numbers.Let F be a family of functions holomorphic in a domain D in complex plane,all of whose zeros have multiplicity at least k.Suppose that for each f∈F,f(z)and f(k)(z)share the set{a,b,c}.Then F is a normal family in D.
基金Supported by the National Natural Science Foundation of China(Grant No.11961068).
文摘In this paper,we study the normal criterion of meromorphic functions concerning shared analytic function.We get some theorems concerning shared analytic function,which improves some earlier related results.
文摘Let k, m be two positive integers with m ≤ k and let F be a family of zero-free meromorphic functions in a domain D, let h(z) ≡ 0 be a meromorphic function in D with all poles of h has multiplicity at most m. If, for each f ∈ F, f(k)(z) = h(z) has at most k- m distinct roots(ignoring multiplicity) in D, then F is normal in D. This extends the results due to Chang[1], Gu[3], Yang[11]and Deng[1]etc.
文摘We studied the normality conditions in families of meromorphic functions, improved the results of Fang and Zalcman [Fang ML, Zalcman L, Normal families and shared values of meromorphic functions, Computational Methods and Function Theory, 2001, 1 （1）： 289-299], and generalized two new normality criterions. Let F be a family of meromorphic functions in a domain D, a a non-zero finite complex number, B a positive real number, and k and m two positive integers satisfying m〉2k＋4. If every function denoted by f belonging to F has only zeros with multiplicity at least k and satisfies f^m（z）f^（k）（Z）=α→ ｜^f（k）（z）｜ ≤B or f^m（z）f^（k）（z）=α→｜f（z）｜ ≥, then F is normal in D.
文摘We proved： Let F be a family of meromorphic functions in a domain D anda α ≠0, b ∈C. If f1（z） - α（f（z））^2 ≠ b, f≠ 0 and the poles of f（z） are of multiplicity ≥ 3 foreach f（z） ∈F, then F is normal in D.
基金Supported by the National Natural Science Foundation of China（l1461070, 11271090） Supported by the Natural Science Foundation of Guangdong Province（S2012010010121）
文摘In this paper,we study the normality criterion for families of meromorphic functions concerning shared set depending on f∈F.Let F be a family of meromorphic functions in the unit disc A.For each f∈F,all zeros of f have multiplicity at least 2 and there exist nonzero complex numbers b_f,c_f satisfying(i) b_f/c_f is a constant;(ii) min{σ(0,b_f),σ(0,c_f),σ(b_f,c_f)} ≥m for some m > 0;(iii) E_f'(S_f)■ E_f(S_f),where S_f = {b_f,c_f}.Then F is normal in A.At the same time,the corresponding results are also proved.The results in this paper improve and generalize the related results
文摘Let F- be a family of zero-free meromorphic functions in a domain D, let h（z） ≠ be a holomorphic function in D and let k be a positive integer. If, for each f ∈ P（f）（z） - h（z） has at most k distinct zeros（ignoring multiplicity） in D, where P（f） = f（k）（z） ＋ al（z）f（k-1）（z） ＋… ＋ ak（z）f（z） is a differential polynomial of f and aj（z）（j=1, 2,…, k） are holomorphic functions in D, then F is normal in D. This extends the results due to Deng[2], Chang[1], Gu[3], Yang[2], etc.
