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 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 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.

scholarly journals On the value distribution of a differential monomial and some normality criteria

2021 ◽  
Vol 56 (1) ◽  
pp. 55-60
Author(s):  
W. Lü ◽  
B. CHAKRABORTY
Keyword(s):  
Meromorphic Function ◽  
Analytic Functions ◽  
Differential Polynomial ◽  
Value Distribution ◽  
Zero Distribution ◽  
Small Functions ◽  
Transcendental Meromorphic Function ◽  
Distribution Result ◽  
Positive Integers

The aim of this paper is to study the zero distribution of the differential polynomial $\displaystyle af^{q_{0}}(f')^{q_{1}}...(f^{(k)})^{q_{k}}-\varphi,$where $f$ is a transcendental meromorphic function and $a=a(z)(\not\equiv 0,\infty)$ and $\varphi(\not\equiv 0,\infty)$ are small functions of $f$. Moreover, using this value distribution result, we prove the following normality criterion for family of analytic functions:\\ {\it Let $\mathscr{F}$ be a family of analytic functions on a domain $D$ and let $k \geq1$, $q_{0}\geq 2$, $q_{i} \geq 0$ $(i=1,2,\ldots,k-1)$, $q_{k}\geq 1$ be positive integers. If for each $f\in \mathscr{F}$: i.\ $f$ has only zeros of multiplicity at least $k$,\ ii.\ $\displaystyle f^{q_{0}}(f')^{q_{1}}\ldots(f^{(k)})^{q_{k}}\not=1$,then $\mathscr{F}$ is normal on domain $D$.

scholarly journals On the Zeros of the Differential Polynomial φ(z)f2(z)f′(z)2 − 1

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 87 ◽  
Author(s):  
Junfeng Xu ◽  
Shuichao Ye
Keyword(s):  
Meromorphic Function ◽  
Counting Function ◽  
Differential Polynomial ◽  
Value Distribution ◽  
Small Function ◽  
Transcendental Meromorphic Function

In this study, the value distribution of the differential polynomial φ f 2 f ′ 2 − 1 is considered, where f is a transcendental meromorphic function, φ ( ≢ 0 ) is a small function of f by the reduced counting function. This result improves the existed theorems which obtained by Jiang (Bull Korean Math Soc 53: 365-371, 2016) and also give a quantitative inequality of φ f f ′ − 1 .

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.

Picard values and normal families of meromorphic functions

2009 ◽  
Vol 139 (5) ◽  
pp. 1091-1099 ◽  
Author(s):  
Yan Xu ◽  
Fengqin Wu ◽  
Liangwen Liao
Keyword(s):  
Meromorphic Function ◽  
Complex Plane ◽  
Complex Number ◽  
Meromorphic Functions ◽  
Normal Families ◽  
Finite Complex ◽  
Transcendental Meromorphic Function ◽  
Positive Integers ◽  
Normal Criterion

Let f be a transcendental meromorphic function on the complex plane ℂ, let a be a non-zero finite complex number and let n and k be two positive integers. In this paper, we prove that if n≥k+1, then $\smash{f+a(f^{(k)})^n}$ assumes each value b∈ℂ infinitely often. Also, the related normal criterion for families of meromorphic functions is given. Our results generalize the related results of Fang and Zalcman.

scholarly journals Some uniqueness results related to the Brück conjecture

Analysis ◽  
2018 ◽  
Vol 38 (2) ◽  
pp. 91-100
Author(s):  
Bikash Chakraborty
Keyword(s):  
Meromorphic Function ◽  
Recent Result ◽  
Differential Polynomial ◽  
Small Function ◽  
Uniqueness Results ◽  
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].

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 .

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