Nonconstant Meromorphic Functions Research Articles

Unicity of Meromorphic Function and its Derivative

Abstract. In this paper, we deal with the uniqueness problems of meromorphic functionsthat share a small function with its derivative and improve some results of Yang, Yu, Lahiri,and Zhang, also answer some questions of T. D. Zhang and W. R. Lu.¨ 1. Introduction and main resultsIn this article, by meromorphic functions we shall always mean meromorphicfunctions in the complex plane. we are going to mainly use the basic notationof Nevanlinna Theory, (see [1], [3], [2]) such as T(r,f), N(r,f), m(r,f), N(r,f)and S(r,f) = o(T(r,f)). Let f(z) and g(z) denote two non-constant meromorphicfunctions, and let a(z) be a meromorphic function. If f(z) − a(z) and g(z) − a(z)have the same zeros with the same multiplicities(ignoring multiplicities), then wesay that f(z) and g(z) share a(z) CM(IM). Let k be a positive integer. We denote byN k) (r,1/f−a) the counting function for the zeros of f−a with multiplicity ≤ k, andby N k) (r,1/f −a) the corresponding one for which the multiplicity is not counted.Let N

Meromorphic Functions That Share One Finite Value CM or IM with Their First Derivative

In this paper we shall prove that if a non-constant meromorphic function and its derivative share the value CM (IM) and if ( ), then either or ( ), where and are constants. These results give improvement and extension of the following result of Gundersen: if a non-constant meromorphic function and its derivative share two distinct values CM, then .

UNIQUENESS THEOREMS OF MEROMORPHIC FUNCTIONS OF A CERTAIN FORM

In this paper, we shall show that for any entire function f, the function of the form <TEX>$f^m(f^n$</TEX> - 1)f' has no non-zero finite Picard value for all positive integers m, n <TEX>${\in}\;{\mathbb{N}}$</TEX> possibly except for the special case m = n = 1. Furthermore, we shall also show that for any two nonconstant meromorphic functions f and g, if <TEX>$f^m(f^n$</TEX>-1)f' and <TEX>$g^m(g^n$</TEX>-1)g' share the value 1 weakly, then f <TEX>$\equiv$</TEX> g provided that m and n satisfy some conditions. In particular, if f and g are entire, then the restrictions on m and n could be greatly reduced.

TWO MEROMORPHIC FUNCTIONS SHARING SETS CONCERNING SMALL FUNCTIONS

The main purpose of this paper is to deal with the unique- ness of meromorphic functions sharing sets concerning small functions. We obtain two main theorems which improve and extend strongly some results due to R. Nevanlinna, Li-Qiao, Yao, Yi, Thai-Tan, and Cao-Yi. It is well known that two nonconstant polynomials f and g over an algebraic closed field of characteristic zero are identical if there exist two distinct values a and b such that f(x) = a if and only if g(x) = a and f(x) = b if and only if g(x) = b. In 1926, R. Nevanlinna (3) proved his famous five-value theorem that for two nonconstant meromorphic functions f and g on the whole complex plane C, if they have the same inverse images (ignoring multiplicities) for five distinct values, then f(z) · g(z). After this very work, the uniqueness of meromorphic functions with shared values onC attracted many investigations (for references, see (10)). It is very interesting to consider distinct small functions instead of distinct complex numbers on C. Over the last few years, there were several generaliza- tions of Nevanlinna's five-value theorem. To state some of these results, we must introduce some notions. Let h be a nonzero holomorphic function on C, expanding h as h(z) = P1 i=0 bi(ziz0) i around z0, then we define h(z0) := min{i : bi 6 0}. Let k and M be positive integers or +1. We set M h (z) = min{M,h(z)},

Entire Functions and Their Derivatives Share Two Finite Sets

Abstract. In this paper, we study the uniqueness of entire functions and prove thefollowing theorem. Let n(5), k be positive integers, and let S 1 = fz : z n = 1g,S 2 = fa 1 ;a 2 ; ;a m g, where a 1 ;a 2 ; ;a m are distinct nonzero constants. If two non-constant entire functions f and g satisfy E f (S 1 ;2) = E g (S 1 ;2) and E f (k) (S 2 ;1) =E g (k) (S 2 ;1), then one of the following cases must occur: (1) f = tg, fa 1 ;a 2 ; ;a m g=tfa 1 ;a 2 ; cz;a m g, where t is a constant satisfying t n = 1; (2) f(z) = de , g(z) = td e cz ,fa 1 ;a 2 ; 2;a m g= ( 1) k c k tf 1a 1 ; ; 1a m g, where t, c, d are nonzero constants and t n = 1.The results in this paper improve the result given by Fang (M.L. Fang, Entire functionsand their derivatives share two nite sets, Bull. Malaysian Math. Sc. Soc. 24(2001),7-16). 1. Introduction, de nitions and resultsLet f and gbe two nonconstant meromorphic functions de ned in the opencomplex plane C. If for some a2C[f1g, fand ghave the same set of a-pointswith the same multiplicities then we say that fand gshare the value aCM (count-ing multiplicities). If we do not take the multiplicities into account, f and garesaid to share the value aIM (ignoring multiplicities). We assume that the readeris familiar with the notations of Nevanlinna theory that can be found, for instance,in [5] or [9].Let Sbe a set of distinct elements of C[f1gand E

Meromorphic functions sharing four small functions

It is well known that if two nonconstant meromorphic functions f and g on the complex plane ℂ have the same inverse images counted with multiplicities for four distinct values, then g is a Mobius transformation of f. In this paper, we will show that the above result remains valid if f and g share four distinct small functions counted with multiplicities truncated by 2. This is the best possible truncation level.

Meromorphic Function Sharing Two Small Functions with Its Derivative

In this paper, we deal with the problem of uniqueness of meromorphic func- tions that share two small functions with their derivatives, and obtain the following result which improves a result of Yao and Li: Let f(z) be a nonconstant meromorphic function, k > 5 be an integer. If f(z) and g(z) = a1(z)f(z) + a2(z)f (k) (z) share the value 0 CM, and share b(z) IM, NE(r,f = 0 = f (k) ) = S(r), then f g, where a1(z), a2(z) and b(z) are small functions of f(z). 1. Introduction and main results In this paper, a meromorphic function will mean meromorphic in the whole complex plane. We say that two meromorphic functions f and g share a finite value a IM (ignoring multiplicities) when f a and g a have the same zeros. If f a and g a have the same zeros with the same multiplicities, then we say that f and g share the value a CM (counting multiplicities). Denote by N(r,f = b = g) the reduced counting function of the common zeros of f b and g b ignoring the multiplicities, and NE(r,f = b = g) the reduced counting function of the common zeros of f b and g b with the same multiplicities. We say that f and g share b IM provided that

Small Functions of Meromorphic Functions that Share Three Values GCM

Abstract. In this paper, we deal with the problem of uniqueness of meromorphic func-tions that share three values, and obtain some theorems which improve some results ofBrosch, Yi and other authors. 1. Introduction and de nitionsLet fand gbe two nonconstant meromorphic functions on the open complexplane C, and let abe a nite value in the complex plane. We say that fand gsharethe value aCM ( IM ) provided that f aand g ahave the same zeros countingmultiplicities ( ignoring multiplicities ), and f;gshare 1CM ( IM ) provided that1=f; 1=gshare 0 CM ( IM ). We do not explain the standard notations of valuedistribution theory as those are available in Hayman [4] or Yang and Yi [11].We denote by S(r;f) any function satisfying S(r;f) = o(T(r;f)) as r!+1possibly outside a set Eof nite Lebesgue measure. A meromorphic function a(z)is said to be a small function of f, if T(r;a) = S(r;f):Let fand gbe nonconstant meromorphic functions and abe a small meromor-phic function of fand g. We denote by N(r;a;f;g)( and N

Some Further Results on Weighted Sharing of Values for Meromorphic Functions Concerning a Result of Terglane

Abstract. In this paper, we deal with the problem of meromorphic functions that havethree weighted sharing values, and obtain some uniqueness theorems which improve thosegiven by N. Terglane, Hong-Xun Yi & Xiao-Min Li, and others. Some examples are pro-vided to show that the results in this paper are best possible. 1. Introduction and main resultsIn this paper, by meromorphic functions we will always mean meromorphicfunctions in the complex plane. We adopt the standard notations in the Nevan-linna theory of meromorphic functions as explained in [3]. It will be convenient tolet E denote any set of positive real numbers of finite linear measure, not necessarilythe same at each occurrence. For any nonconstant meromorphic function h(z), wedenote by S(r,h) any quantity satisfying S(r,h) = o(T(r,h)) (r → ∞,r 6∈E).Let f(z) and g(z) be two nonconstant meromorphic functions, and let a ∈C ∪ {∞}, where C ∪ {∞} denotes the extended complex plane. We denote byN 0 (r,a,f,g) the counting function of the common zeros of f(z) − a and g(z) − a,and each point is counted only once, where f(z)−∞ means 1/f(z) (see [10]). We saythat f and g share the value a CM, provided that f and g have the same a−pointswith the same multiplicities. Similarly, we say that f and g share the value a IM,provided that f and g have the same a−points ignoring multiplicities (see [12]).Throughout this paper, we denote by N

A REMARK ON MEROMORPHIC FUNCTIONS SHARING FOUR VALUES

In this paper, we prove that if two distinct non-constant meromorphic functions $f$ and $g$ share four distinct values $a_1, a_2, a_3, a_4$ \DM such that each $a_i$-point is either a $(p,q)$-fold or $(q,p)$-fold point of $f$ and $g$, then $(p,q)$ is either $(1,2)$ or $(1,3)$ and $f, g$ are in some particular forms.

Unicity of meromorphic functions related to their derivatives

AbstractIn this paper, we shall study the unicity of meromorphic functions definedover non-Archimedean fields of characteristic zero such that their valence func-tions of poles grow slower than their characteristic functions. If f is such afunction, and f and a linear differential polynomial P(f) of f, whose coeffi-cients are meromorphic functions growing slower than f, share one finite valuea CM, and share another finite value b (6= a) IM, then P(f) = f. 1 Introduction. In 1929, R. Nevanlinna studied the unicity of meromorphic functions in C. Thefive value theorem due to R. Nevanlinna states that if two non-constant meromor-phic functions f and gin C share five distinct complex numbers a j IM (ignoringmultiplicity), which meansf −1 (a j ) = g −1 (a j ), j= 1,2,...,5in the sense of sets, then it follows that f = g. The four value theorem of R.Nevanlinna states that if two non-constant meromorphic functions f and gin Cshare four distinct complex numbers a j CM (counting multiplicity), which meansf

On results of G. Brosch and T.C. Alzahary

We prove a uniqueness theorem for two non-constant meromorphic functions sharing three values which improves a recent result of T.C. Alzahary. As a consequence of our main result we also improve a theorem of G. Brosch.

Meromorphic functions sharing two finite sets

Let S 1 = {∞} and S 2 = { w: P s ( w) = 0}, P s( 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( S i ) = g −1( S i ), ( i = 1,2), where f −1( S i ) and g −1( S i) denote the pull-backs of S i considered as a divisor, namely, the inverse images of S i counted with multiplicities, by f and g respectively.

Uniqueness and value-sharing of meromorphic functions

In this paper, we study the uniqueness of meromorphic functions and prove the following theorem: Let f ( z ) and g ( z ) be two non-constant meromorphic functions, n , k two positive integers with n > 3 k + 8 . If [ f n ( z ) ] ( k ) and [ g n ( z ) ] ( k ) share 1 CM, then either f ( z ) = c 1 e c z , g ( z ) = c 2 e − c z , where c 1 , c 2 and c are three constants satisfying ( − 1 ) k ( c 1 c 2 ) n ( n c ) 2 k = 1 or f ( z ) = t g ( z ) for a constant t such that t n = 1 . Our results improves the results of Fang [M.L. Fang, Uniqueness and value-sharing of entire functions, Comput. Math. Appl. 44 (2002) 823–831. [7]], Fang and Hong [M.L. Fang, W. Hong, A unicity theorem for entire functions concerning differential polynomials, Indian J. Pure Appl. Math. 32 (9) (2001) 1343–1348. [8]] and Lin and Yi [W.-C. Lin, H.-X. Yi, Uniqueness theorems for meromorphic function, Indian J. Pure Appl. Math. 32 (9) (2004) 121–132. [9]].

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.

On the characteristics of meromorphic functions with three weighted sharing values

In this paper, we deal with the relation between the characteristic function of two nonconstant meromorphic functions with three weighted sharing values, which improves a result given by H.X. Yi and Y.H. Li. From this we establish a theorem which improves a result given by P. Li and C.C. Yang.

On a result of G. Brosch

We prove a uniqueness theorem for non-constant meromorphic functions f, g which share three values 0, 1, ∞ and f − a , g − b share the value 0 for a , b ∉ { 0 , 1 , ∞ } . Our theorem improves a result of G. Brosch.

Unique range sets for meromorphic functions on §

A finite set S in the extended complex plane is called a unique range set counting multiplicities (URSCM) (unique range set ignoring multiplicities, URSIM) for meromorphic functions on if any two nonconstant meromorphic functions f and g in satisfying counting multiplicities (CM) (ignoring multiplicities IM), then f = g. The question ‘What is the smallest cardinality for such a set?’ is still a challenging and open one. In this article, for meromorphic functions on we exhibit a URSCM and a URSIM of 13 and 19 elements, respectively. We also characterize a unique range set for entire functions of order <1 on . Some conjectures are proposed for further studies. §Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.

Compact Quotients of Large Domains in a Complex Projective 3-space

In a complex projective 3-space, we consider a domain with a projective line. If there is a compact non-singular quotient of the domain and the quotient manifold admits a non-constant meromorphic function, then the domain is dense in the projective 3-space and its complement is properly contained in a finite union of complex hypersurfaces and a set with Hausdorff dimension not more than two. Further, if the complement admits a certain fiber space structure, then it is either a disjoint union of two projective lines, a projective line, or an empty set.

Meromorphic functions that weighted sharing three values and one pair

G. Brosch improved the theorem of Nevanlinna for four values theorem and proved that let f and g be two nonconstant meromorphic functions sharing 0, 1, ∞ CM, and let a and b be two finite complex numbers such that $a,b\not\in \{0,1\}$. If $f=a\Leftrightarrow g=b$, then f is a fractional linear transformation of g. In this paper we extend this theorem by using the idea of weighted sharing.