Composite Holomorphic Functions and Normal Families
We study the normality of families of holomorphic functions. We prove the following result. Let α(z), ai(z), i=1,2,…,p, be holomorphic functions and F a family of holomorphic functions in a domain D, P(z,w):=(w-a1(z))(w-a2(z))⋯(w-ap(z)), p≥2. If Pw∘f(z) and Pw∘g(z) share α(z) IM for each pair f(z...
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| Language: | English |
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Wiley
2011-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2011/373910 |
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| author | Xiao Bing Wu Qifeng Yuan Wenjun |
| author_facet | Xiao Bing Wu Qifeng Yuan Wenjun |
| author_sort | Xiao Bing |
| collection | DOAJ |
| description | We study the normality of families of holomorphic functions. We prove the following result. Let α(z), ai(z), i=1,2,…,p, be holomorphic functions and F a family of holomorphic functions in a domain D, P(z,w):=(w-a1(z))(w-a2(z))⋯(w-ap(z)), p≥2. If Pw∘f(z) and Pw∘g(z) share α(z) IM for each pair f(z), g(z)∈F and one of the following conditions holds: (1) P(z0,z)-α(z0) has at least two distinct zeros for any z0∈D; (2) there exists z0∈D such that P(z0,z)-α(z0) has only one distinct zero and α(z) is nonconstant. Assume that β0 is the zero of P(z0,z)-α(z0) and that the multiplicities l and k of zeros of f(z)-β0 and α(z)-α(z0) at z0, respectively, satisfy k≠lp, for all f(z)∈F, then F is normal in D. In particular, the result is a kind of generalization of the famous Montel's criterion. At the same time we fill a gap in the proof of Theorem 1.1 in our original paper (Wu et al., 2010). |
| format | Article |
| id | doaj-art-9b8f1d94e6df41349eec37dfd2ca81f0 |
| institution | OA Journals |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-9b8f1d94e6df41349eec37dfd2ca81f02025-08-20T02:18:25ZengWileyAbstract and Applied Analysis1085-33751687-04092011-01-01201110.1155/2011/373910373910Composite Holomorphic Functions and Normal FamiliesXiao Bing0Wu Qifeng1Yuan Wenjun2Department of Mathematics, Xinjiang Normal University, Urumqi 830054, ChinaShaozhou Normal College, Shaoguan University, Shaoguan 512009, ChinaSchool of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, ChinaWe study the normality of families of holomorphic functions. We prove the following result. Let α(z), ai(z), i=1,2,…,p, be holomorphic functions and F a family of holomorphic functions in a domain D, P(z,w):=(w-a1(z))(w-a2(z))⋯(w-ap(z)), p≥2. If Pw∘f(z) and Pw∘g(z) share α(z) IM for each pair f(z), g(z)∈F and one of the following conditions holds: (1) P(z0,z)-α(z0) has at least two distinct zeros for any z0∈D; (2) there exists z0∈D such that P(z0,z)-α(z0) has only one distinct zero and α(z) is nonconstant. Assume that β0 is the zero of P(z0,z)-α(z0) and that the multiplicities l and k of zeros of f(z)-β0 and α(z)-α(z0) at z0, respectively, satisfy k≠lp, for all f(z)∈F, then F is normal in D. In particular, the result is a kind of generalization of the famous Montel's criterion. At the same time we fill a gap in the proof of Theorem 1.1 in our original paper (Wu et al., 2010).http://dx.doi.org/10.1155/2011/373910 |
| spellingShingle | Xiao Bing Wu Qifeng Yuan Wenjun Composite Holomorphic Functions and Normal Families Abstract and Applied Analysis |
| title | Composite Holomorphic Functions and Normal Families |
| title_full | Composite Holomorphic Functions and Normal Families |
| title_fullStr | Composite Holomorphic Functions and Normal Families |
| title_full_unstemmed | Composite Holomorphic Functions and Normal Families |
| title_short | Composite Holomorphic Functions and Normal Families |
| title_sort | composite holomorphic functions and normal families |
| url | http://dx.doi.org/10.1155/2011/373910 |
| work_keys_str_mv | AT xiaobing compositeholomorphicfunctionsandnormalfamilies AT wuqifeng compositeholomorphicfunctionsandnormalfamilies AT yuanwenjun compositeholomorphicfunctionsandnormalfamilies |