MEROMORPHIC FUNCTIONS THAT SHARE TWO SETS, II

This paper deals with the problem of uniqueness of meromorphic functions,and gets the following result: There exists a set S with 13 elements such that any two nonconstant meromorphic functions f and g satisfying E^-(S, f) = E^-(S, g) and E^-({oo}, f) =E^-({oo}, g) must be identical. This is the best result on this question until now.     

On meromorphic solutions of a certain type of nonlinear differential equations

We consider transcendental meromorphic solutions with N(r,f) = S(r,f) of the following type of nonlinear differential equations:f~n + Pn-2(f) = p1(z)e~(α1(z)) +p2(z)e~(α2(z)),where n≥ 2 is an integer, Pn-2(f) is a differential polynomial in f of degree not greater than n-2 with small functions of f as its coefficients, p1(z), p2(z) are nonzero small functions of f, and α1(z), α2(z)are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of meromorphic solutions and their possible forms of the above equation. Our results extend and improve some known results obtained most recently.     

Complex and p-Adic Meromorphic Functions f′P′( f ),g′P′(g) Sharing a Small Function

Let K be a complete algebraically closed p-adic field of characteristic zero.We apply results in algebraic geometry and a new Nevanlinna theorem for p-adic meromorphic functions in order to prove results of uniqueness in value sharing problems, both on K and on C. Let P be a polynomial of uniqueness for meromorphic functions in K or C or in an open disk. Let f, g be two transcendental meromorphic functions in the whole field K or in C or meromorphic functions in an open disk of K that are not quotients of bounded analytic functions. We show that if f′P′( f) and g′P′(g) share a small function α counting multiplicity, then f = g, provided that the multiplicity order of zeros of P′satisfy certain inequalities. A breakthrough in this paper consists of replacing inequalities n ≥ k+2 or n ≥ k+3 used in previous papers by Hypothesis(G). In the p-adic context, another consists of giving a lower bound for a sum of q counting functions of zeros with(q-1) times the characteristic function of the considered meromorphic function.     

The Uniqueness of Entire Functions with Their Derivatives

§ 1.Introduction　 By a“meromorphic function” we mean a function that is meromorphic in the wholecomplex plane.It is assumed that the reader is familiar with notations of Nevanlinnatheory such as T(r,f) ,m(r,f) ,N (r,1f) ,S(r,f ) and so on that can be found,forinstance,in[1 ] or[2 ] .Let f and g be two meromorphic functions and a be a complexnumber. We say that f and g share the value a CM (counting multiplicity) if f -a andg-a have the same zeros with the same multiplicity,and denote th…     

The Expression of Slow-growing Meromorphic Functions Sharing One Value CM with Their Derivatives

In this paper, we study the relations between meromorphic functions and their derivatives with one shared values, and obtain a concrete expression of meromorphic functions of zero order that share one value CM with their derivatives by a new method. Our main result is the supplementary of a related result due to Li and Yi(Li X M, Yi H X. Uniqueness of meromorphic functions sharing a meromorphic function of a small order with their derivatives. Ann. Polon. Math., 2010, 98(3):201–219).     

亚纯函数与其导数具有一个分担值（英）

1. IntroductionIh this paper a "meromorphic fUllction" will mean that is meromorphic in the wholecomplex plane. We say that two non-constant meromorphic functions f and g share avale c in the extended complex plane provided that f(4) = c if and only if g(to) = c.We will state weather a share vale is by CM (counting multiplicitics) or by iM (ignoringmultiplicities). We denote Ek)(c, f) the set of zeros of f(z) -- c with multiplicities less thenor equal to k (ignoring multiplicity), Nk)(h) de…     

MEROMORPHIC FUNCTIONS SHARING TWO FINITE SETS

Let S1 ＝ {∞} and S2 ＝ {w: Ps(w)＝ 0}, Ps(w) being a uniqueness polynomial under some restricted conditions. Then, for any given nonconstant meromorphic function f, there exist at most finitely many nonconstant meromorphic functions g such that f-1(Si) ＝ g-1(Si)(i ＝ 1,2), where f-1(Si) and g-1(Si) denote the pull-backs of Si considered as a divisor, namely, the inverse images of Si counted with multiplicities, by f and g respectively.     

Picard values and the growth of meromorphic functions

The purpose of this paper is to investigate the growth of meromorphic functions concern- ing Picard values with a radially distributed value. This generalizes a classic result due to Hayman [Hayman, W. K.： Picard values of meromorphic functions and their derivatives. Ann. of Math., 70, 9-42 （1959）].     

TWO ENTIRE FUNCTIONS SHARING FOUR SMALL FUNCTIONS IN THE SENSE OF _(k))(β,f)=_(k))(β,g)

This paper proves a result that if two entire functions f(z) and g(z) share four small functions aj(z) (j = 1,2,3,4) in the sense of Ek)(aj, f) = Ek)(aj,g), (j = 1,2,3,4) (k ≥ 11), then there exists f(z) = g(z).     

AN INEQUALITY ON THE ZEROS AND POLES OF VECTOR VALUED MEROMORPHIC FUNCTIONS

The author proves that if f : C→Cn is a transcendental vector valued meromorphic function of finite order and assume Σa∈Cn ∪{∞}δ(a) = 2, then,where .This result extends the related results for meromorphic function by Singh and Kulkarni.     

具有两个公共值集的亚纯函数

本文讨论了亚纯函数的唯一性问题，　证明了存在一个具有８个元素的集合Ｓ，使得对任何两个非常数亚纯函数　ｆ与ｇ，只要满足　Ｅ（Ｓ，ｆ）＝Ｅ（Ｓ，ｇ）　　和Ｅ（｛∞｝，ｆ）＝Ｅ（｛∞｝，ｇ）　　，必有ｆ　　ｇ　　．     

具有四个有穷的IM公共小函数的整函数

本文证明了两个非常数整函数若具有四个有穷的ＩＭ公共小函数，则它们必恒等．从而在整函数的情形下，Ｎｅｖａｎｌｉｎｎａ五值定理中的五个ＩＭ公共值能否更换为五个ＩＭ公共小函数（含∞）的问题的回答是肯定的．     

涉及导数与差分的亚纯函数唯一性

设f,g是两个非常数亚纯函数,a是一个非零有穷复数,n≥5是一个正整数.若[f(z)]~n与[g(z)]~n CM分担a,f(z)与g(z) CM分担∞,且N_(1))(r,f)=S(r,f),则或者f(z)三tg(z),其中t~n=1;或者f(z)g(z)≡t,其中t~n=a~2.由此改进了涉及导数与差分的一些亚纯函数唯一性的结果.     

具有亏值的亚纯函数的唯一性

改进了Ozawa的一个关于整函数的唯一性定理,得到了∞为亏值的亚纯函数唯一性的相应的几个结论.设亚纯函数f(z)与g(z)的级(或者下级)为有穷的非整数,满足.f=0→g=0,f=1g=1,f=∞9=∞,若∞为f(z)的Borel例外值,则f≡g.以及设f(z)与g(z)为C中非常数的亚纯函数,它们的级λ为有穷且非整数,再设它们满足f=0→g=0,f=1g=1,f=∞g=∞,若δ(∞,f)=1,f(z)为正规增长函数,则f≡g.     

关于CM分担四个公共小函数的亚纯函数结论的改进

关于CM分担四个公共小函数的亚纯函数结论，我们在考虑重值的条件下，改进了李平和杨重骏^[2]的结论：设f(z)与g(z)为非常数亚纯函数，aj(z)(j=1,2,….4)为f(z)与g(z)的四个判别的小函数，若f(z)与g(z)满足Ek)(aj,f)=Ek)(aj,g),(j=1,2,3,4)且k(≥15）是正整数，则f(z)是g(z)的拟分式线性变换。即:存在f(z)与g(z)的四个小函数a(z),b(z),c(z),d(z),使得f=ag b/cg d(ad-bc≠0），（亦称Quasi-Mobuys变换）。     

关于Nevanlinna四值定理的改进

在本文中,我们证明了如果两个亚纯函数分担两个值CM,并且在E_k))(β,f)=E_k))(β,g)(k≥5),意义下分担另外两个值,则这两个亚纯函数一个是另一个的分式线性变换。     

关于亚纯函数增长级的一个注记

研究了无穷级亚纯函数f与亚纯函数g在某个角域上具有分担集S = {a1,a2,a3}的增长级关系.     

关于亚纯函数据有三个公共值以及它们的导函数据有一个公共值的问题

这篇论文研究的是亚纯函数的唯一性。这篇论文的主要结论是：如果两个非常数亚纯函数ｆ和ｇ有三个ＣＭ公共值或四个ＩＭ公共值,并且ｆ’和ｇ’以零为ＩＭ公共值,则ｆ和ｇｒＭｏｂｉｕｓ变换。其中一个结论是Ｋ．Ｔｏｈｇｅ有一个结果的改进,而且我们采用了与Ｋ．Ｔｏｈｇｅ不同的方法。『     

关于全纯函数的正规定则

研究涉及微分多项式与公共值的全纯函数族的正规性问题.设F为区域D内的全纯函数族,n 1为任一正整数,b为有限常数,如果对F中任意两个函数f与g,fn(f-1)f′与gn(g-1)g′在D内都以b为公共值,则F在D内正规.此结果对亚纯函数不成立.     

亚纯函数及其n阶导数权分担两个值

研究亚纯函数及其n阶导数权分担两个值的唯一性问题.得到了：如果两个非常数亚纯函数f,g分担（∞,∞）,f（n）与g（n）分担（1,0）,n（≥0）为一整数,且满足C0：=（4n＋6）λ＋δn＋1（0,f）＋δn＋1（0,g）＋δn＋2（0,f）＋δn＋2（0,g）＋δn（0,f）〉4n＋10,其中λ=max{min{Θ（∞,f）,Θ（0,f）},min{Θ（∞,g）,Θ（0,g）}},那么f（n）·g（n）≡1,或者f≡g.该结果改进了前人的有关定理.