scholarly journals Value Distribution for a Class of Small Functions in the Unit Disk

2011 ◽  
Vol 2011 ◽  
pp. 1-24
Author(s):  
Paul A. Gunsul
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Unit Disk ◽  
Nevanlinna Theory ◽  
Meromorphic Functions ◽  
Finite Measure ◽  
Differential Polynomials ◽  
Proximity Function ◽  
Small Functions ◽  
Linear Differential

If is a meromorphic function in the complex plane, R. Nevanlinna noted that its characteristic function could be used to categorize according to its rate of growth as . Later H. Milloux showed for a transcendental meromorphic function in the plane that for each positive integer , as , possibly outside a set of finite measure where denotes the proximity function of Nevanlinna theory. If is a meromorphic function in the unit disk , analogous results to the previous equation exist when . In this paper, we consider the class of meromorphic functions in for which , , and as . We explore characteristics of the class and some places where functions in the class behave in a significantly different manner than those for which holds. We also explore connections between the class and linear differential equations and values of differential polynomials and give an analogue to Nevanlinna's five-value theorem.

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.

scholarly journals Value distribution of meromorphic functions concerning rational functions and differences

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mingliang Fang ◽  
Degui Yang ◽  
Dan Liu
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Rational Function ◽  
Finite Order ◽  
Meromorphic Functions ◽  
Rational Functions ◽  
Value Distribution ◽  
Exponent Of Convergence ◽  
Nonzero Constant ◽  
Zero Points

AbstractLet c be a nonzero constant and n a positive integer, let f be a transcendental meromorphic function of finite order, and let R be a nonconstant rational function. Under some conditions, we study the relationships between the exponent of convergence of zero points of $f-R$ f − R , its shift $f(z+nc)$ f ( z + n c ) and the differences $\Delta _{c}^{n} f$ Δ c n f .

scholarly journals Relation between Small Functions with Differential Polynomials Generated by Solutions of Linear Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Zhigang Huang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Entire Functions ◽  
Linear Differential Equations ◽  
Differential Polynomials ◽  
Small Functions ◽  
Linear Differential ◽  
The Relationship ◽  
Growth Of Solutions

This paper is devoted to studying the growth of solutions of second-order nonhomogeneous linear differential equation with meromorphic coefficients. We also discuss the relationship between small functions and differential polynomialsL(f)=d2f″+d1f′+d0fgenerated by solutions of the above equation, whered0(z),d1(z),andd2(z)are entire functions that are not all equal to zero.

scholarly journals The Zeros of Difference Polynomials of Meromorphic Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Junfeng Xu ◽  
Xiaobin Zhang
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Differential Polynomials ◽  
Difference Polynomials

We investigate the value distributions of difference polynomialsΔf(z)-af(z)nandf(z)nf(z+c)which related to two well-known differential polynomials, wheref(z)is a meromorphic function.

A uniqueness result related to certain non-linear differential polynomials sharing the same 1-points

2011 ◽  
Vol 61 (2) ◽  
Author(s):  
Abhijit Banerjee ◽  
Pranab Bhattacharjee
Keyword(s):  
Meromorphic Function ◽  
Recent Result ◽  
Uniqueness Result ◽  
Differential Polynomials ◽  
Non Linear ◽  
Linear Differential Polynomials ◽  
Linear Differential

AbstractThe purpose of the paper is to study the uniqueness of meromorphic function when certain non-linear differential polynomials share the same 1-points. As a consequence of the main result we improve and supplement the following recent result: [LAHIRI, I.—PAL, R.: Nonlinear differential polynomials sharing 1-points, Bull. Korean Math. Soc. 43 (2006), 161–168].

On the value distribution of f2f(k)

2005 ◽  
Vol 78 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Xiaojun Huang ◽  
Yongxing Gu
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Value Distribution ◽  
Multiple Zeros ◽  
Transcendental Meromorphic Function

AbstractIn this paper, we prove that for a transcendental meromorphic function f(z) on the complex plane, the inequality T(r, f) < 6N (r, 1/(f2 f(k)−1)) + S(r, f) holds, where k is a positive integer. Moreover, we prove the following normality criterion: Let ℱ be a family of meromorphic functions on a domain D and let k be a positive integer. If for each ℱ ∈ ℱ, all zeros of ℱ are of multiplicity at least k, and f2 f(k) ≠ 1 for z ∈ D, then ℱ is normal in the domain D. At the same time we also show that the condition on multiple zeros of f in the normality criterion is necessary.

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