scholarly journals On value distribution and uniqueness of meromorphic function with finite logarithmic order concerning its derivative and q-shift difference

2014 ◽  
Vol 2014 (1) ◽  
Author(s):  
Xiu-Min Zheng ◽  
Hong-Yan Xu
Keyword(s):  
Meromorphic Function ◽  
Value Distribution ◽  
Logarithmic Order

scholarly journals On Properties of Differences Polynomials about Meromorphic Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Jianming Qi ◽  
Jie Ding ◽  
Wenjun Yuan
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Special Class ◽  
Meromorphic Functions ◽  
Value Distribution ◽  
Class Difference ◽  
Difference Polynomial

We study the value distribution of a special class difference polynomial about finite order meromorphic function. Our methods of the proof are also different from ones in the previous results by Chen (2011), Liu and Laine (2010), and Liu and Yang (2009).

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 Value distribution of certain monomials of algebroid functions

1997 ◽  
Vol 62 (3) ◽  
pp. 398-404
Author(s):  
Kari Katajamäki
Keyword(s):  
Meromorphic Function ◽  
Value Distribution ◽  
Algebroid Function ◽  
Transcendental Meromorphic Function ◽  
Algebroid Functions

AbstractHayman has shown that if f is a transcendental meromorphic function and n ≽ 3, then fn f′ assumes all finite values except possibly zero infinitely often. We extend his result in three directions by considering an algebroid function ω, its monomial ωn0 ω′n1, and by estimating the growth of the number of α-points of the monomial.

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.

scholarly journals A note on the value distribution of $\phi f^2 f^{(k)}-1$

2021 ◽  
Vol 55 (1) ◽  
pp. 64-75
Author(s):  
P. Sahoo ◽  
G. Biswas
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Characteristic Function ◽  
Counting Function ◽  
Differential Polynomial ◽  
Value Distribution ◽  
Small Function ◽  
Nevanlinna Characteristic ◽  
Transcendental Meromorphic Function

In this paper, we study the value distribution of the differential polynomial $\varphi f^2f^{(k)}-1$, where $f(z)$ is a transcendental meromorphic function, $\varphi (z)\;(\not\equiv 0)$ is a small function of $f(z)$ and $k\;(\geq 2)$ is a positive integer. We obtain an inequality concerning the Nevanlinna Characteristic function $T(r,f)$ estimated by reduced counting function only. Our result extends the result due to J.F. Xu and H.X. Yi [J. Math. Inequal., 10 (2016), 971-976].

scholarly journals On the value distribution of the differential polynomial Afn f(k) + Bfn+1 -1

Filomat ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 579-589
Author(s):  
Pulak Sahoo ◽  
Anjan Sarkar
Keyword(s):  
Meromorphic Function ◽  
Characteristic Function ◽  
Recent Result ◽  
Counting Function ◽  
Differential Polynomial ◽  
Value Distribution ◽  
Nevanlinna Characteristic ◽  
Transcendental Meromorphic Function ◽  
Positive Integers

In the paper, we study the value distribution of the differential polynomial Afn f(k) + Bf n+1 -1, where f is a transcendental meromorphic function and n(? 2),k(?2) are positive integers. We prove an inequality for the Nevanlinna characteristic function T(r,f) in terms of reduced counting function only. The result of the paper not only improves the result due to Q.D. Zhang [J. Chengdu Ins. Meteor., 20(1992), 12-20], also partially improves a recent result of H. Karmakar and P. Sahoo [Results Math., (2018),73:98].

scholarly journals Value Distribution and Linear Operators

2013 ◽  
Vol 57 (2) ◽  
pp. 493-504 ◽  
Author(s):  
R. Halburd ◽  
R. Korhonen
Keyword(s):  
Meromorphic Function ◽  
Linear Operator ◽  
Meromorphic Functions ◽  
Differential Operators ◽  
Value Distribution ◽  
Linear Operators ◽  
Difference Operators ◽  
Second Main Theorem ◽  
Picard’S Theorem ◽  
Picard's Theorem

AbstractNevanlinna's second main theorem is a far-reaching generalization of Picard's theorem concerning the value distribution of an arbitrary meromorphic function f. The theorem takes the form of an inequality containing a ramification term in which the zeros and poles of the derivative f′ appear. We show that a similar result holds for special subfields of meromorphic functions where the derivative is replaced by a more general linear operator, such as higher-order differential operators and differential-difference operators. We subsequently derive generalizations of Picard's theorem and the defect relations.

scholarly journals Value distribution of some differential monomials

Filomat ◽  
2020 ◽  
Vol 34 (13) ◽  
pp. 4287-4295 ◽  
Author(s):  
Bikash Chakraborty ◽  
Sudip Saha ◽  
Amit Pal ◽  
Jayanta Kamila
Keyword(s):  
Meromorphic Function ◽  
Characteristic Function ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Quantitative Estimation ◽  
Differential Polynomial ◽  
Value Distribution ◽  
Transcendental Meromorphic Function

Let f be a transcendental meromorphic function defined in the complex plane C and k ? N. We consider the value distribution of the differential polynomial fq0(f(k))qk, where q0(?2), qk(?1) are integers. We obtain a quantitative estimation of the characteristic function T(r,f) in terms of N?(r, 1/fq0(f(k))qk-1). Our result generalizes the results obtained by Xu et al. (Math. Inequal. Appl., Vol. 14, PP. 93-100, 2011); Karmakar and Sahoo (Results Math., Vol. 73, 2018) for a particular class of transcendental meromorphic functions.

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