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 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 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 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 Bruck conjecture and certain solution of some differential equation.

2020 ◽  
Vol 51 (3) ◽  
pp. 245-259
Author(s):  
Molla Basir Ahamed
Keyword(s):  
Differential Equation ◽  
Meromorphic Function ◽  
Differential Polynomial ◽  
Small Function ◽  
Brück Conjecture

We investigate on the famous Br$ \ddot{u} $ck conjecture further, and improved some of the existing results by extending them up to a differential monomial $ M[f] $ sharing small function with certain power of $ f^{d_M} $ of a meromorphic function. We have found the class of the meromorphic function satisfying the relation $ f^{d_M}\equiv M[f] $. For the generalizing our main results further up to a differential polynomial $ P[f] $, some relevant  questions finally have been posed for further study in this direction.

Meromorphic function sharing a small function with a homogeneous differential polynomial

2019 ◽  
Vol 10 (3) ◽  
pp. 183-192
Author(s):  
Indrajit Lahiri ◽  
Bipul Pal
Keyword(s):  
Meromorphic Function ◽  
Differential Polynomial ◽  
Small Function ◽  
Type Result ◽  
Function Sharing

Abstract In this paper, we consider a Brück-type result for a homogeneous differential polynomial generated by a meromorphic function.

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