In this paper, general modular theorems are obtained for meromorphic functions and their derivatives. The related criteria for normality of families of meromorphic functions are proved.
In this paper, the Bloch principle is discussed and a normal criterion is asserted. Let (?) be a family of meromorphic functions on a domain D, a≠0,∞; b≠∞, n≥4. If for any f ∈(?) there existsf’ - afn≠b, then (...In this paper, the Bloch principle is discussed and a normal criterion is asserted. Let (?) be a family of meromorphic functions on a domain D, a≠0,∞; b≠∞, n≥4. If for any f ∈(?) there existsf’ - afn≠b, then (?) is normal in D.展开更多
<正> Improvement of classical Marty’s criterion for normality of a family of meromorphic functions is given and Zalcman’s theorem for a family which is not normal is improved when the functions in the family h...<正> Improvement of classical Marty’s criterion for normality of a family of meromorphic functions is given and Zalcman’s theorem for a family which is not normal is improved when the functions in the family have only zeros of degree at least k. As an application, we prove that a family is normal if each function in the family has only poles of degree at least k+2 and satisfies f(k)+af^3≠b. This improves a theorem of D. Drasin.展开更多
Let k, K ∈1N and F be a family of zero-free meromorphic functions in a domain D such that for each f ∈ F, f（k） - 1 has at most K zeros, ignoring multiplicity. Then F is quasinormal of order at most v=k/k＋1,where ...Let k, K ∈1N and F be a family of zero-free meromorphic functions in a domain D such that for each f ∈ F, f（k） - 1 has at most K zeros, ignoring multiplicity. Then F is quasinormal of order at most v=k/k＋1,where v is equal to the largest integer not exceeding k/k＋1.In particular, if K = k, then F is normal. The results are sharp.展开更多
Let F be a family of holomorphic functions in a domain D, k be a positive integer, a, b（≠0）, c（≠0） and d be finite complex numbers. If, for each f∈F, all zeros of f-d have multiplicity at least k, f^（k） = a w...Let F be a family of holomorphic functions in a domain D, k be a positive integer, a, b（≠0）, c（≠0） and d be finite complex numbers. If, for each f∈F, all zeros of f-d have multiplicity at least k, f^（k） = a whenever f=0, and f=c whenever f^（k） = b, then F is normal in D. This result extends the well-known normality criterion of Miranda and improves some results due to Chen-Fang, Pang and Xu. Some examples are provided to show that our result is sharp.展开更多
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
In this paper, we mainly discuss the normality of two families of functions concerning shared values and proved： Let F and G be two families of functions meromorphic on a domain D C C, a1, a2, a3, a4 be four distinct...In this paper, we mainly discuss the normality of two families of functions concerning shared values and proved： Let F and G be two families of functions meromorphic on a domain D C C, a1, a2, a3, a4 be four distinct finite complex numbers. If G is normal, and for every f 9~, there exists g C G such that f（z） and g（z） share the values a1, a2, a3, a4, then F is normal on D.展开更多
文摘In this paper, general modular theorems are obtained for meromorphic functions and their derivatives. The related criteria for normality of families of meromorphic functions are proved.
文摘In this paper, the Bloch principle is discussed and a normal criterion is asserted. Let (?) be a family of meromorphic functions on a domain D, a≠0,∞; b≠∞, n≥4. If for any f ∈(?) there existsf’ - afn≠b, then (?) is normal in D.
基金Project supported by the National Natural Science Foundation of China.
文摘<正> Improvement of classical Marty’s criterion for normality of a family of meromorphic functions is given and Zalcman’s theorem for a family which is not normal is improved when the functions in the family have only zeros of degree at least k. As an application, we prove that a family is normal if each function in the family has only poles of degree at least k+2 and satisfies f(k)+af^3≠b. This improves a theorem of D. Drasin.
基金Supported by National Natural Science Foundation of China(Grant No.10871094)NSFU of Jiangsu,China(Grant No.08KJB110001)Qinglan Project of Jiangsu,China,and SRF for ROCS,SEM
文摘Let k, K ∈1N and F be a family of zero-free meromorphic functions in a domain D such that for each f ∈ F, f（k） - 1 has at most K zeros, ignoring multiplicity. Then F is quasinormal of order at most v=k/k＋1,where v is equal to the largest integer not exceeding k/k＋1.In particular, if K = k, then F is normal. The results are sharp.
基金The first author is supported in part by the Post Doctoral Fellowship at Shandong University.The second author is supported by the national Nature Science Foundation of China (10371065).
文摘Let F be a family of holomorphic functions in a domain D, k be a positive integer, a, b（≠0）, c（≠0） and d be finite complex numbers. If, for each f∈F, all zeros of f-d have multiplicity at least k, f^（k） = a whenever f=0, and f=c whenever f^（k） = b, then F is normal in D. This result extends the well-known normality criterion of Miranda and improves some results due to Chen-Fang, Pang and Xu. Some examples are provided to show that our result is sharp.
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 National Natural Science Foundation of China(Grant No.11071074)supported by Outstanding Youth Foundation of Shanghai(Grant No.slg10015)
文摘In this paper, we mainly discuss the normality of two families of functions concerning shared values and proved： Let F and G be two families of functions meromorphic on a domain D C C, a1, a2, a3, a4 be four distinct finite complex numbers. If G is normal, and for every f 9~, there exists g C G such that f（z） and g（z） share the values a1, a2, a3, a4, then F is normal on D.
