• Title/Summary/Keyword: Meromorphic

Search Result 240, Processing Time 0.025 seconds

Argument Estimates Of Certain Meromorphic Functions

  • Cho, Nak-Eun
    • Communications of the Korean Mathematical Society
    • /
    • v.15 no.2
    • /
    • pp.263-274
    • /
    • 2000
  • The object of the present paper is to obtain some argu-ment properties of certain mermorphic functions in the punctured open unit disk. Furthermore, we investigate their integral preserving properties in a sector.

  • PDF

THE UNIQUENESS OF MEROMORPHIC FUNCTIONS WHOSE DIFFERENTIAL POLYNOMIALS SHARE SOME VALUES

  • MENG, CHAO;LI, XU
    • Journal of applied mathematics & informatics
    • /
    • v.33 no.5_6
    • /
    • pp.475-484
    • /
    • 2015
  • 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).

MEROMORPHIC FUNCTIONS SHARING FOUR VALUES WITH THEIR DIFFERENCE OPERATORS OR SHIFTS

  • Li, Xiao-Min;Yi, Hong-Xun
    • Bulletin of the Korean Mathematical Society
    • /
    • v.53 no.4
    • /
    • pp.1213-1235
    • /
    • 2016
  • 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.

MEROMORPHIC SOLUTIONS OF SOME q-DIFFERENCE EQUATIONS

  • Chen, Baoqin;Chen, Zongxuan
    • Bulletin of the Korean Mathematical Society
    • /
    • v.48 no.6
    • /
    • pp.1303-1314
    • /
    • 2011
  • We consider meromorphic solutions of q-difference equations of the form $$\sum_{j=o}^{n}a_j(z)f(q^jz)=a_{n+1}(z),$$ where $a_0(z)$, ${\ldots}$, $a_{n+1}(z)$ are meromorphic functions, $a_0(z)a_n(z)$ ≢ 0 and $q{\in}\mathbb{C}$ such that 0 < |q| ${\leq}$ 1. We give a new estimate on the upper bound for the length of the gap in the power series of entire solutions for the case 0 < |q| < 1 and n = 2. Some growth estimates for meromorphic solutions are also given in the cases 0 < |q| < 1. Moreover, we investigate zeros and poles of meromorphic solutions for the case |q| = 1.

CRITERIA OF NORMALITY CONCERNING THE SEQUENCE OF OMITTED FUNCTIONS

  • Chen, Qiaoyu;Qi, Jianming
    • Bulletin of the Korean Mathematical Society
    • /
    • v.53 no.5
    • /
    • pp.1373-1384
    • /
    • 2016
  • In this paper, we research the normality of sequences of meromorphic functions concerning the sequence of omitted functions. The main result is listed below. Let {$f_n(z)$} be a sequence of functions meromorphic in D, the multiplicities of whose poles and zeros are no less than k + 2, $k{\in}\mathbb{N}$. Let {$b_n(z)$} be a sequence of functions meromorphic in D, the multiplicities of whose poles are no less than k + 1, such that $b_n(z)\overset{\chi}{\Rightarrow}b(z)$, where $b(z({\neq}0)$ is meromorphic in D. If $f^{(k)}_n(z){\neq}b_n(z)$, then {$f_n(z)$} is normal in D. And we give some examples to indicate that there are essential differences between the normal family concerning the sequence of omitted functions and the normal family concerning the omitted function. Moreover, the conditions in our paper are best possible.

MEROMORPHIC SOLUTIONS OF SOME NON-LINEAR DIFFERENCE EQUATIONS WITH THREE EXPONENTIAL TERMS

  • Min-Feng Chen;Zong-Sheng Gao;Xiao-Min Huang
    • Bulletin of the Korean Mathematical Society
    • /
    • v.61 no.3
    • /
    • pp.745-762
    • /
    • 2024
  • In this paper, we study the existence of finite order meromorphic solutions of the following non-linear difference equation fn(z) + Pd(z, f) = p1eα1z + p2eα2z + p3eα3z, where n ≥ 2 is an integer, Pd(z, f) is a difference polynomial in f of degree d ≤ n - 2 with small functions of f as its coefficients, pj (j = 1, 2, 3) are small meromorphic functions of f and αj (j = 1, 2, 3) are three distinct non-zero constants. We give the expressions of finite order meromorphic solutions of the above equation under some restrictions on αj (j = 1, 2, 3). Some examples are given to illustrate the accuracy of the conditions.

Class of Meromorphic Functions Partially Shared Values with Their Differences or Shifts

  • Ahamed, Molla Basir
    • Kyungpook Mathematical Journal
    • /
    • v.61 no.4
    • /
    • pp.745-763
    • /
    • 2021
  • For a value s ∈ ℂ∪ {∞}, two meromorphic functions f and g are said to share the value s, CM, (or IM), provided that f(z)-s and g(z)-s have the same set of zeros, counting multiplicities, (respectively, ignoring multiplicities). We say that a meromorphic function f shares s ∈ Ŝ partially with a meromorphic function g if E(s, f) ⊆ E(s, g). It is easy to see that "partially shared values CM" are more general than "shared values CM". With the help of partially shared values, in this paper, we prove some uniqueness results between a non-constant meromorphic function and its generalized differences or shifts. We exhibit some examples to show that the result of Charak et al. [8] is not true for k = 2 or k = 3. We find some gaps in proof of the result of Lin et al. [24]. We not only correct these resuts, but also generalize them in a more convenient way. We give a number of examples to validate certain claims of the main results of this paper and also to show that some of conditions are sharp. Finally, we pose some open questions for further investigation.