Meromorphic functions that share a set with their derivatives

There exists a set S with three elements such that if a meromorphic function f, having at most finitely many simple poles, shares the set S CM with its derivative f^′, then f^′≡f.     

Entire functions that share a set with their derivatives

Let f be a nonconstant entire function and let S = {a, b, c}, where a, b and c are distinct complex numbers. If E(S, f) = E(S, f′), then either (i) f = Ce ^z ; or (ii) ; or (iii) , where C is a nonzero constant. Received: 22 May 2007     

与其导数具有分担值的亚纯函数唯一性

为进一步丰富亚纯函数唯一性理论,寻求更佳的唯一性条件,利用亚纯函数Nevanlinna理论更精确地估计亚纯函数的n重值点的计数函数,得到两个亚纯函数与其导数具有某些分担值时的唯一性定理,推广和改进了相关文献的相关结果.     

Entire functions that share a polynomial with their derivatives

Let f be a nonconstant entire function, k and q be positive integers satisfying k>q, and let Q be a polynomial of degree q. This paper studies the uniqueness problem on entire functions that share a polynomial with their derivatives and proves that if the polynomial Q is shared by f and f^′ CM, and if f^(k)(z)−Q(z)=0 whenever f(z)−Q(z)=0, then f≡f^′. We give two examples to show that the hypothesis k>q is necessary.     

Entire functions that share a small function with their derivatives

Let f be an entire function of finite order and a an entire function of order less than f's. If f−a and f^′−a have the same zeros with the same multiplicities, then f^′−a≡c(f−a) for some non-zero constant c.     

Entire functions that share one value CM with their derivatives

This paper studies the uniqueness problem on entire function that share a finite, nonzero value CM with their derivatives and proves two main theorems which generalize some results given by Jank, Mues and Volkmann, P. Li and C.C. Yang, H.L. Zhong etc. An example shows that the condition of one of our theorems is necessary.     

Uniqueness of meromorphic functions whose derivatives share four small functions

In this paper, we will prove some uniqueness theorems of meromorphic functions whose derivatives share four distinct small functions. The results in this paper improve those given by R. Nevanlinna, L. Yang, G.D. Qiu, and other authors. An example is provided to show that the results in this paper are best possible.     

具三个公共值的亚纯函数的唯一性

主要讨论具有三个公共值的两个亚纯函数的唯一性问题,推广并改进了H,Ueda与Brosch等人的有关结果。     

分担四个小函数的亚纯函数的唯一性定理

设亚纯函数f和g分担四个小函数,如果(N)≤uT(r,f)+S(r,f)且(N) ≤(μ)T≤(μ)T(r,f)+S(r,f),(u, (μ))∈[0,1/16×[0,1/16),那么f=g.     

Uniqueness results of meromorphic functions whose derivatives share four small functions

We prove an oscillation theorem of two meromorphic functions whose derivatives share four values IM. From this we obtain some uniqueness theorems, which improve the corresponding results given by Yang [16] and Qiu [10], and supplement results given by Nevanlinna [9] and Gundersen [3, 4]. Some examples are provided to show that the results in this paper are best possible.     

On meromorphic functions that share three values of finite weights

A uniqueness theorem for two distinct non-constant meromorphic functions that share three values of finite weights is proved, which generalizes two previous results by H.X. Yi, and X.M. Li and H.X. Yi. As applications of it, many known results by H.X. Yi and P. Li, etc. could be improved. Furthermore, with the concept of finite-weight sharing, extensions on Osgood-Yang's conjecture and Mues' conjecture, and a generalization of some prevenient results by M. Ozawa and H. Ueda, ect. could be obtained.     

Unicity of meromorphic functions and their derivatives

In this paper, we prove that if a transcendental meromorphic function f shares two distinct small functions CM with its kth derivative f^(k) (k>1), then f=f^(k). We also resolve the same question for the case k=1. These results generalize a result due to Frank and Weissenborn.     

Uniqueness and set sharing of derivatives of meromorphic functions

We prove some uniqueness theorems concerning the derivatives of meromorphic functions when they share two or three sets which will improve some existing results.     

Uniqueness of meromorphic functions concerning sharing two small functions with their derivatives

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 $f\equiv f"$.     

Julia directions of meromorphic functions and their derivatives

We prove that if a transcendental meromorphic function has no Julia direction and is bounded on a path to ￥ \infty then there is a common Julia direction for all derivatives. Related statements are obtained under the assumption that f is o(?{ | z | }) o(\sqrt{\mid z \mid}) or O(?{ | z | }) O(\sqrt{\mid z \mid}) on a path to ￥ \infty . Further we disprove a conjecture of Frank and Wang by means of a counterexample.     

Value sharing of meromorphic functions and their derivatives

In this paper, we estimate the size of ρn's in the famous L. Zalcman's lemma. With it, we obtain some uniqueness theorems for meromorphic functions f and f^′ when they share two transcendental meromorphic functions.     

具有两个权分担的公共值集的亚纯函数

讨论了亚纯函数分担两个集合的唯一性问题,证明了一个定理.所得结果改进了先前的一些定理,例子表明结果是精确的.