scholarly journals A note on power of meromorphic function and its shift operator of certain hyper-order sharing one small function and a value

2021 ◽  
Vol 55 (1) ◽  
pp. 57-63
Author(s):  
A. Banerjee ◽  
A. Roy
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Shift Operator ◽  
Small Function ◽  
A Value ◽  
Hyper Order

In this article, we obtain two results on $n$ the power of a meromorphic function and its shift operator sharing a small function together with a value which improve and complement some earlier results. In particular, more or less we have improved and extended two results of Qi-Yang [Meromorphic functions that share values with their shifts or their $n$-th order differences, Analysis Math., 46(4)2020, 843-865] by dispelling the superfluous conclusions in them.

scholarly journals Some Note on Exceptional Values of Meromorphic Functions

1963 ◽  
Vol 22 ◽  
pp. 189-201 ◽  
Author(s):  
Kikuji Matsumoto
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Essential Singularity ◽  
Compact Set ◽  
A Value ◽  
Cluster Set ◽  
Totally Disconnected ◽  
Null Set

LetEbe a totally-disconnected compact set in thez-plane and letΩbe its complement with respect to the extendedz-plane. ThenΩis a domain and we can consider a single-valued meromorphic functionw = f(z)onΩwhich has a transcendental singularity at each point ofE. Suppose thatEis a null-set of the classWin the sense of Kametani [4] (the classNBin the sense of Ahlfors and Beurling [1]). Then the cluster set off(z)at each transcendental singularity is the wholew-plane, and hencef(z)has an essential singularity at each point ofE. We shall say that a valuewis exceptional forf(z)at an essential singularity ζ ∈Eif there exists a neighborhood of ζ where the functionf(z)does not take this valuew.

scholarly journals On Exceptional Values of Meromorphic Functions with the Set of Singularities of Capacity Zero

1961 ◽  
Vol 18 ◽  
pp. 171-191 ◽  
Author(s):  
Kikuji Matsumoto
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Essential Singularity ◽  
Compact Set ◽  
Capacity Zero ◽  
A Value

LetEbe a compact set in thez-plane and letΩbe its complement with respect to the extendedz-plane. Suppose thatEis of capacity zero. ThenΩis a domain and we shall consider a single-valued meromorphic functionw=f(z) onΩwhich has an essential singularity at each point ofE. We shall say that a valuewis exceptional forf(z)at a point ζ ∈Eif there exists a neighborhood of C where the functionf(z)does not take this valuew.

ZEROS OF ULTRAMETRIC MEROMORPHIC FUNCTIONS f′ fn(f − a)k − α

2008 ◽  
Vol 01 (03) ◽  
pp. 415-429 ◽  
Author(s):  
Jacqueline Ojeda
Keyword(s):  
Meromorphic Function ◽  
Characteristic Zero ◽  
Meromorphic Functions ◽  
Sufficient Conditions ◽  
Small Function ◽  
Closed Field ◽  
Open Disk ◽  
Algebraically Closed Field ◽  
Absolute Value ◽  
Special Value

Let 𝕂 be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. Similarly to the Hayman problem, here we study meromorphic functions in 𝕂 or in an open disk that are of the form f′ fn(f − a)k − α with α a small function, in order to find sufficient conditions on n, k assuring that they have infinitely many zeros. We first define and characterize a special value for a meromorphic function and check that, if it exists, it is unique. So, such values generalize Picard exceptional values.

Uniqueness of meromorphic functions sharing a value or small function

2016 ◽  
Vol 66 (4) ◽  
Author(s):  
Nguyen Van Thin ◽  
Ha Tran Phuong
Keyword(s):  
Nevanlinna Theory ◽  
Complex Analysis ◽  
Meromorphic Functions ◽  
Small Function ◽  
Uniqueness Problem ◽  
Differential Polynomials ◽  
A Value

AbstractThe paper concerns interesting problems related to the field of Complex Analysis, in particular Nevanlinna theory of meromorphic functions. The author have studied certain uniqueness problem on differential polynomials of meromorphic functions sharing a value or small function. The results of this paper are generalizations of some problems studied in [BOUSSAF, K.—ESCASSUT, A.—OJEDA, J.:

scholarly journals Results on the deficiencies of some differential-difference polynomials of meromorphic functions

2016 ◽  
Vol 14 (1) ◽  
pp. 100-108 ◽  
Author(s):  
Xiu-Min Zheng ◽  
Hong-Yan Xu
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Meromorphic Functions ◽  
Value Distribution ◽  
Small Function ◽  
Complex Constant ◽  
Difference Polynomials

Abstract In this paper, we study the relation between the deficiencies concerning a meromorphic function f(z), its derivative f′(z) and differential-difference monomials f(z)mf(z+c)f′(z), f(z+c)nf′(z), f(z)mf(z+c). The main results of this paper are listed as follows: Let f(z) be a meromorphic function of finite order satisfying $$\mathop {\lim \,{\rm sup}}\limits_{r \to + \infty } {{T(r,\,f)} \over {T(r,\,f')}}{\rm{ < }} + \infty ,$$ and c be a non-zero complex constant, then δ(∞, f(z)m f(z+c)f′(z))≥δ(∞, f′) and δ(∞,f(z+c)nf′(z))≥ δ(∞, f′). We also investigate the value distribution of some differential-difference polynomials taking small function a(z) with respect to f(z).

Uniqueness of meromorphic functions of differential polynomials sharing a small function with finite weight

2019 ◽  
Vol 25 (2) ◽  
pp. 141-153
Author(s):  
Harina P. Waghamore ◽  
Vijaylaxmi Bhoosnurmath
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Differential Polynomial ◽  
Small Function ◽  
Differential Polynomials ◽  
Weighted Sharing ◽  
Linear Differential Polynomials ◽  
Linear Differential

Abstract Let f be a non-constant meromorphic function and {a=a(z)} ( {\not\equiv 0,\infty} ) a small function of f. Here, we obtain results similar to the results due to Indrajit Lahiri and Bipul Pal [Uniqueness of meromorphic functions with their homogeneous and linear differential polynomials sharing a small function, Bull. Korean Math. Soc. 54 2017, 3, 825–838] for a more general differential polynomial by introducing the concept of weighted sharing.

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