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1.
Normality and quasinormality of zero-free meromorphic functions 总被引:1,自引:0,他引:1
Jian Ming Chang 《数学学报(英文版)》2012,28(4):707-716
Let k, K ∈ N 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 ν = K k+1 , where ν is equal to the largest integer not exceeding K/k+1 . In particular, if K = k, then F is normal. The results are sharp. 相似文献
2.
Pei-Chu Hu 《Journal of Mathematical Analysis and Applications》2009,357(2):323-731
Take positive integers n,k?2. Let F be a family of meromorphic functions in a domain D⊂C such that each f∈F has only zeros of multiplicity at least k. If, for each pair (f,g) in F, fn(f(k)) and gn(g(k)) share a non-zero complex number a ignoring multiplicity, then F is normal in D. 相似文献
3.
Let k be a positive integer and F be a family of meromorphic functions in a domain D⊂C such that each f∈F has only zeros of multiplicity at least k+1. If for each pair (f, g) in F, ff(k) and gg(k) share a non-zero complex number a ignoring multiplicity, then F is normal in D. 相似文献
4.
Uniqueness of meromorphic functions concerning sharing two small functions with their derivatives 下载免费PDF全文
In this paper, we study the uniqueness of meromorphic functions that share two small functions with their derivatives. We prove the following result: Let $f$ be a nonconstant meromorphic function such that $\mathop {\overline{\lim}}\limits_{r\to\infty} \frac{\bar{N}(r,f)}{T(r,f)}<\frac{3}{128}$, and let $a$, $b$ be two distinct small functions of $f$ with $a\not\equiv\infty$ and $b\not\equiv\infty$. If $f$ and $f"$ share $a$ and $b$ IM, then $f\equiv f"$. 相似文献
5.
Let F be a family of functions meromorphic in a domain D, let P be a polynomial with either deg P≥3 or deg P = 2 and P having only one distinct zero, and let b be a finite nonzero complex number. If, each pair of functions f and g in F, P (f)f and P (g)g share b in D, then F is normal in D. 相似文献
6.
JianMing Chang 《中国科学A辑(英文版)》2009,52(8):1717-1722
Let be a family of meromorphic functions in a plane domain D, and a and b be finite non-zero complex values such that . If for and , then is normal. We also construct a non-normal family of meromorphic functions in the unit disk Δ={|z|<1} such that for every and in Δ, where m is a given positive integer. This answers Problem 5.1 in the works of Gu, Pang and Fang.
This work was supported by National Natural Science Foundation of China (Grant Nos. 10671093, 10871094) and the Natural Science
Foundation of Universities of Jiangsu Province of China (Grant No. 08KJB110001), the Qing Lan Project of Jiangsu, China and
the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry 相似文献
7.
Qingcai Zhang 《Journal of Mathematical Analysis and Applications》2006,318(2):707-725
In this paper we study the problem of meromorphic functions sharing three values with weight and obtain some theorems which improve the results given by Lahiri and others. 相似文献
8.
Hong-Xun Yi 《Proceedings of the American Mathematical Society》2002,130(6):1689-1697
In this paper, we show that if two non-constant meromorphic functions and satisfy for , where are five distinct small functions with respect to and , and is a positive integer or with , then . As a special case this also answers the long-standing problem on uniqueness of meromorphic functions concerning small functions.
9.
Indrajit Lahiri 《Journal of Mathematical Analysis and Applications》2002,271(1):206-216
In this paper we prove a uniqueness theorem for meromorphic functions sharing three values with some weight which improves some known results. 相似文献
10.
设k为正整数,M为正数;F为区域D内的亚纯函数族,且其零点重级至少为k;h为D内的亚纯函数(h(z)≠0,∞),且h(z)的极点重级至多为k.若对任意给定的函数f∈F,f与f~((k))分担0,且f~((k))(z)-h(z)=0?|f(z)|≥M,则F在D内正规. 相似文献
11.
The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of functions holomorphic on a domain D ? ?, all of whose zeros have multiplicity at least k, where k ?? 2 is an integer. Let h(z) ? 0 and ?? be a meromorphic function on D. Assume that the following two conditions hold for every f ?? F: $$ \begin{gathered} (a)f(z) = 0 \Rightarrow |f^{(k)} (z)| < |h(z)|. \hfill \\ (b)f^{(k)} (z) \ne h(z). \hfill \\ \end{gathered} $$ Then F is normal on D. 相似文献
12.
Xiaojun Huang 《Journal of Mathematical Analysis and Applications》2003,277(1):190-198
In this paper, we study the normality of a family of meromorphic functions and general criteria for normality of families of meromorphic functions with multiple zeros concerning shared values are obtained. 相似文献
13.
Qingcai Zhang 《Journal of Mathematical Analysis and Applications》2008,338(1):545-551
In this paper we study the problem of normal families of meromorphic functions concerning shared values and prove that a family F of meromorphic functions in a domain D is normal if for each pair of functions f and g in F, f′−afn and g′−agn share a value b in D where n is a positive integer and a,b are two finite constants such that n?4 and a≠0. This result is not true when n?3. 相似文献
14.
Let f(z) be a meromorphic function in the complex plane, whose zeros have multiplicity at least k + 1(k ≥ 2). If sin z is a small function with respect to f(z), then f~(k)(z)-P(z) sin z has infinitely many zeros in the complex plane, where P(z) is a nonzero polynomial of deg(P(z)) ≠ 1. 相似文献
15.
Yan Xu 《Journal of Mathematical Analysis and Applications》2007,329(2):1343-1354
In this paper, we obtain some normality criteria for families of meromorphic functions that concern the exceptional functions of derivatives. 相似文献
16.
Ning Cui 《Journal of Difference Equations and Applications》2016,22(10):1452-1471
We mainly study the uniqueness of meromorphic functions sharing three distinct values CM with their difference operators, and the related result confirms the conjecture of Chen and Yi. We also obtain a uniqueness theorem on entire functions sharing two sets CM with their difference operators. 相似文献
17.
Chunlin Lei Degui Yang Xueqin Wang 《Journal of Mathematical Analysis and Applications》2008,341(1):224-234
Let k be a positive integer with k?2; let h(?0) be a holomorphic function which has no simple zeros in D; and let F be a family of meromorphic functions defined in D, all of whose poles are multiple, and all of whose zeros have multiplicity at least k+1. If, for each function f∈F, f(k)(z)≠h(z), then F is normal in D. 相似文献
18.
本文研究了亚纯函数及其 k 阶导数分担两个不同集合的亚纯函数族的正规性问题.证明了如下结论: 设 F 是平面区域 D上的亚纯函数族, 其中函数的零点重数至少为 k+1. 设S1, S2是两个集合,且|S1|=m, |S2|=n, S2 ≠ 0, 这里m, n是正整数. 如果任意f(z) ∈ F,满足f(z) ∈ S1?f(k)(z) ∈ S2, z ∈ D, 则 F 在区域 D 上正规.本文的研究结果是对刘晓俊和庞学诚[刘晓俊, 庞学诚. 分担值与正规族 [J].数学学报(中文版),2007, 50(2):409--412] 2007年研究结果的改进. 相似文献
19.
《复变函数与椭圆型方程》2012,57(11):793-806
In this article, we deal with the uniqueness problems on meromorphic functions concerning differential polynomials that share fixed-points. Moreover, we greatly improve a former result. 相似文献
20.
Normal families of meromorphic functions with multiple values 总被引:1,自引:0,他引:1
Jiying Xia 《Journal of Mathematical Analysis and Applications》2009,354(1):387-393
Let F be a family of meromorphic functions defined in a domain D, let ψ(?0) be a holomorphic function in D, and k be a positive integer. Suppose that, for every function f∈F, f≠0, f(k)≠0, and all zeros of f(k)−ψ(z) have multiplicities at least (k+2)/k. If, for k=1, ψ has only zeros with multiplicities at most 2, and for k?2, ψ has only simple zeros, then F is normal in D. This improves and generalizes the related results of Gu, Fang and Chang, Yang, Schwick, et al. 相似文献