Nonconstant Meromorphic Functions Research Articles

Meromorphic Functions Sharing Three Values with Their Linear Differential Polynomials in an Angular Domain

Let f be a nonconstant meromorphic function of lower order µ (f) > 1/2 in ℂ, and let aj (j = 1, 2, 3) be three distinct finite complex numbers. We show that there exists an angular domain D = {z: α ≤ arg z ≤ β}, where 0 < β − α ≤ 2π, such that if f share aj (j = 1, 2, 3) CM with its k-th linear differential polynomial L[f] in D, then f = L[f]. This generalizes the corresponding results from Frank and Schwick, Zheng and Li-Liu-Yi.

A Note on a Unicity Theorem for the Gauss Maps of Complete Minimal Surfaces in Euclidean Four-space

AbstractThe classical result of Nevanlinna states that two nonconstantmeromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.

Meromorphic functions that share a polynomial with their difference operators

In this paper, we prove the following result: Let f be a nonconstant meromorphic function of finite order, p be a nonconstant polynomial, andc be a nonzero constant. If f, Delta _{c}f, and Delta_{c}^{n}f (nge 2) share ∞ and p CM, then fequiv Delta_{c}f. Our result provides a difference analogue of the result of Chang and Fang in 2004 (Complex Var. Theory Appl. 49(12):871–895, 2004).

Some uniqueness results related to the Brück conjecture

Abstract Let f be a non-constant meromorphic function and let {a=a(z)} ( {\not\equiv 0,\infty} ) be a small function of f. Under certain essential conditions, we obtained a conclusion similar to the Brück Conjecture, when f and its differential polynomial {P[f]} shares a with weight l ( {\geq 0} ). Our result improves and generalizes a recent result of Li, Yang and Liu [N. Li, L. Yang and K. Liu, A further result related to a conjecture of R. Brück, Kyungpook Math. J. 56 2016, 2, 451–464].

Some further studies on the uniqueness of meromorphic functions sharing three sets

In this paper we improve and rectify two recent results of Banerjee and Majumder (Analysis 34(2):143–162, 2014) relating to the uniqueness of meromorphic functions sharing three sets. We also show that the uniqueness of two nonconstant meromorphic functions can be achieved without any condition on deficiency when at least one of the functions under consideration assumes a certain value.

Meromorphic Functions that Share Four or Five Pairs of Values

It is shown that two non-constant meromorphic functions that share two pairs of values counting multiplicities and share three other pairs of values ignoring multiplicities are Mobius transformations of each other. This result is sharp. Examples of meromorphic functions that share four or five pairs of values are discussed.

Meromorphic Functions Sharing One Value with Their Derivatives Concerning the Difference Operator

Let f(z) be a non-constant meromorphic function of finite order, $$c\in \mathbb {C}\setminus \{0\}$$ and $$k\in \mathbb {N}$$ . Suppose f(z) and $$f^{(k)}(z+c)$$ share 1 CM (IM), f(z) and $$f(z+c)$$ share $$\infty $$ CM. If $$N(r,0;f)=S(r,f) \left( N\left( r,0;f(z)\right) +N\left( r,0;f^{(k)}(z+c)\right) =S(r,f)\right) $$ , then either $$f(z)\equiv f^{(k)}(z+c)$$ or f(z) is a solution of the following equation: $$\begin{aligned}f\; \hbox {and}\\&\quad N\left( r,0;f(z)+\frac{1}{a(z)}\right) =S(r,f)\\&\quad \left( f'(z+c)-1=a(z)\left( f(z)-1\right) \left( f(z)+\frac{1}{a(z)}\right) \right) \end{aligned}$$ where $$a(z)\left( \not \equiv -\,1,0,\infty \right) \left( a(z)\left( \not \equiv 0,\infty \right) \right) $$ is a meromorphic function satisfying $$T(r,a)=S(r,f)$$ . Also we exhibit some examples to show that the conditions of our results are the best possible.

On arithmetic toroidal groups

Andreotti, Gherardelli, and Abe constructed toroidal groups from algebraic number fields. We call these toroidal groups arithmetic toroidal groups. We study the algebraic structures of arithmetic toroidal groups and we show two examples of arithmetic toroidal groups where one is a quasi-Abelian variety and the other has no nonconstant meromorphic functions on it.

MEROMORPHIC FUNCTIONS SHARING FOUR VALUES WITH THEIR DIFFERENCE OPERATORS OR SHIFTS

We prove a uniqueness theorem of nonconstant meromorphic functions sharing three distinct values IM and a fourth value CM with their shifts, and prove a uniqueness theorem of nonconstant entire functions sharing two distinct small functions IM with their shifts, which respectively improve Corollary 3.3(a) and Corollary 2.2(a) from [12], where the meromorphic functions and the entire functions are of hyper order less than 1. An example is provided to show that the above results are the best possible. We also prove two uniqueness theorems of nonconstant meromorphic functions sharing four distinct values with their difference operators.

“2CM+1IM” Theorem for Periodic Meromorphic Functions

Generally, the concrete relations between two nonconstant meromorphic functions that share two values CM and one value IM are hard to determine. However, for the class \({\mathcal{F}}\) of all nonconstant meromorphic functions with the same period \({c\neq0}\), we prove a result in this paper that: let \({f(z), g(z) \in \mathcal{F}}\) such that the hyper-order \({\rho_2(f) < 1}\), if \({f(z), g(z)}\) share \({0, \infty}\) CM and 1 IM, then either \({f(z)\equiv g(z)}\) or \({f(z)=e^{az+b}g(z)}\) and \({\mu(f)=\mu(g)=1}\), where \({a=\frac{2k\pi i}{c}}\) and k is some integer. As an application of this result, we obtain an uniqueness theorem for elliptic meromorphic functions. Moreover, examples are given to illustrate that all the conditions are necessary and sharp.

SOME RESULTS ON THE QUESTIONS OF KIT-WING YU

Abstract. The paper deals with the problem of meromorphic functionssharing a small function with its differential polynomials and improvesthe results of Liu and Gu [9], Lahiri and Sarkar [8], Zhang [13] and Zhangand Yang [14] and also answer some open questions posed by Kit-Wing Yu[16]. In this paper we provide some examples to show that the conditionsin our results are the best possible. 1. Introduction, definition and resultsIn this paper by meromorphic functions we will always mean meromorphicfunctions in the complex plane.Let f and g be two non-constant meromorphic functions and let a be a finitecomplex number. We say that f and g share a CM, provided that f − a andg − a have the same zeros with the same multiplicities. Similarly, we say thatf and g share a IM, provided that f −a and g−a have the same zeros ignoringmultiplicities. In addition we say that f and g share ∞ CM, if 1/f and 1/gshare 0 CM, and we say that f and g share ∞ IM, if 1/f and 1/g share 0 IM.We adopt the standard notations in Nevanlinna’s value distribution theoryof meromorphic functions such as the characteristic function T(r,f), the count-ing function of the poles N(r,∞;f) and the proximity function m(r,∞;f) (see[10]). For a non-constant meromorphic function f we denote by S(r,f) anyquantity satisfying S(r,f) = o(T(r,f)) as r → ∞, outside of a possible ex-ceptional set of finite linear measure. Let k ∈ N and a ∈ C∪ {∞}. We useN

English

The problem of uniqueness of an entire or a meromorphic function when it shares a value or a small function with its derivative became popular among the researchers after the work of Rubel and Yang (1977). Several authors extended the problem to higher order derivatives. Since a linear differential polynomial is a natural extension of a derivative, in the paper we study the uniqueness of a meromorphic function that shares one small function CM with a linear differential polynomial, and prove the following result: Let $f$ be a nonconstant meromorphic function and $L$ a nonconstant linear differential polynomial generated by $f$. Suppose that $a = a(z)$ ($\not \equiv 0, \infty $) is a small function of $f$. If $f-a$ and $L-a$ share $0$ CM and \[ (k+1)\overline N(r, \infty ; f)+ \overline N(r, 0; f')+ N_{k}(r, 0; f')< \lambda T(r, f')+ S(r, f') \] for some real constant $\lambda \in (0, 1)$, then $ f-a=(1+ {c}/{a})(L-a)$, where $c$ is a constant and $1+{c}/{a} \not \equiv 0$.

Uniqueness of $p(f)$ and $P[f

Let $f$ be a nonconstant meromorphic function, $a (\not\equiv 0, \infty)$ be a meromorphic function satisfying $T(r,a) = o(T(r,f))$ as $r \rightarrow \infty$, and $p(z)$ be a polynomial of degree $n \geq 1$ with $p(0) = 0$. Let $P[f]$ be a nonconstant differential polynomial of $f$. Under certain essential conditions, we prove that $p(f) \equiv P[f]$, when $p(f)$ and $P[f]$ share $a$ with weight $l \geq 0$. Our result generalizes the results due to Zhang and L$\ddot{\text{u}}$, Banerjee and Majumdar, and Bhoosnurmath and Kabbur and answers a question asked by Zhang and L$\ddot{\text{u}}$.

UNIQUENESS OF MEROMORPHIC FUNCTIONS SHARING A CERTAIN NONZERO POLYNOMIAL WITH WEIGHT

In this paper we investigate the uniqueness problem of meromorphic functions when certain non-linear dierential polynomi- als share a nonzero polynomial. We prove a uniqueness result which signicantly rectify, improves and generalizes the recent results of P. Sahoo and S. Seikh (12) as well as X. M. Li and L. Cao (10). 1. Introduction, definitions and results In this paper by meromorphic functions we shall always mean meromor- phic functions in the complex plane. Let f and g be two non-constant meromorphic functions and let a be a

A power of a meromorphic function sharing a set with its derivative

If f is a non-constant meromorphic function such that a power of f shares certain set counting multiplicities in the weak sense with the derivative of the power, then f becomes an exponential function.

THE UNIQUENESS OF MEROMORPHIC FUNCTIONS WHOSE DIFFERENTIAL POLYNOMIALS SHARE SOME VALUES

In this article, we deal with the uniqueness problems of meromorphic functions concerning differential polynomials and prove the following theorem. Let f and g be two nonconstant meromorphic functions, n ≥ 12 a positive integer. If fn(f3 - 1)f′ and gn(g3 - 1)g′ share (1, 2), f and g share ∞ IM, then f ≡ g. The results in this paper improve and generalize the results given by Meng (C. Meng, Uniqueness theorems for differential polynomials concerning fixed-point, Kyungpook Math. J. 48(2008), 25-35), I. Lahiri and R. Pal (I. Lahiri and R. Pal, Nonlinear differential polynomials sharing 1-points, Bull. Korean Math. Soc. 43(2006), 161-168), Meng (C. Meng, On unicity of meromorphic functions when two differential polynomials share one value, Hiroshima Math.J. 39(2009), 163-179).

RESULTS ON MEROMORPHIC FUNCTIONS SHARING THREE VALUES WITH THEIR DIFFERENCE OPERATORS

Under the restriction of finite order, we prove two uniqueness theorems of nonconstant meromorphic functions sharing three values with their difference operators, which are counterparts of Theorem 2.1 in [6] for a finite-order meromorphic function and its shift operator.

Uniqueness of meromorphic functions sharing two finite sets in $${\mathbb {C}}$$ C with finite weight II

In the paper we find two finite sets in \({\mathbb {C}}\) such that two non-constant meromorphic functions sharing the sets with finite weight become identical. Our result will improve a recent result in Banerjee and Halder (Konuralp J Math 22:42–52, 2014).

A NOTE ON NEVANLINNA'S FIVE VALUE THEOREM

Abstract. In the paper we prove a uniqueness theorem which improvesand generalizes a number of uniqueness theorems for meromorphic func-tions related to Nevanlinna’s five value theorem. 1. Introduction, definitions and resultsIn the paper, by meromorphic functions we always mean meromorphic func-tions in the open complex plane C. Let f be a non-constant meromorphicfunction. A meromorphic function a = a(z) is said to be a small function of fif either a ≡ ∞ or T(r,a) = S(r,f). We denote by S(f) the collection of allsmall functions of f. Clearly C∪ {∞} ⊂ S(f) and S(f) is a field over the setof complex numbers.For a positive integer p and a ∈ S(f) we denote by E p) (a;f) the set of thosedistinct zeros of f −a whose multiplicities do not exceed p, where we mean bya zero of f −∞ a pole of f. Also by E ∞) (a;f) we denote the set of all distinctzeros of f −a.For A ⊂ Cwe denote by N A (r,a;f) the reduced counting function of thosezeros of f −a which belong to the set A, where a ∈ S(f).For a positive integer p and a ∈ S(f) we denote by N

A new Picard type theorem concerning elliptic functions

Let k ≥ 2 be an integer, let h be a nonconstant elliptic function, and let f be a nonconstant meromorphic function in C, all of whose zeros have multiplicity at least k + 1, except possibly finitely many. If T(r,h) = o{T(r,f)} as r → ∞, then f (k) = h has infinitely many solutions (including the possibility of infinitely many common poles of f and h).