Characteristic function and deficiency of algebroid functions on annuli

Rathod, Ashok

doi:10.13108/2019-11-1-121

Cited by 5 publications

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“…In 2009, Cao and Yi [8] studied the uniqueness of meromorphic functions sharing some values on annuli. In 2015, Yang Tan [2], Yang Tan and Yue Wang [1] proved some interesting results on the multiple values and uniqueness of algebroid functions on annuli and also others have proved several results for algebroid functions on annuli, see [9], [11], [16], [18], [19], [20], [22], [23], [24], [25], [26], [27], [29], [30]. Thus, it is interesting to consider the uniqueness problem of algebroid functions in multiply connected domains.…”

Section: Introductionmentioning

confidence: 99%

Uniqueness theorems for meromorphic functions on annuli

Rathod

2020

Ufimsk. Mat. Zh.

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In this paper, we discuss the uniqueness problems of meromorphic functions on annuli. We prove a general theorem on the uniqueness of meromorphic functions on annuli. An analogue of a famous Nevanlinna's five-value theorem is proposed. The main result in this paper is an analog of a result on the plane C obtained by H.S. Gopalkrishna and Subhas S. Bhoosnurmath for an annuli. That is, let 1 ( ) and 2 ( ) be two transcendental meromorphic functions on the annulus A = {︁:, where 1 < 0 +∞. Let , = 1, 2, . . . , ), be distinct complex numbers in C, and , = 1, 2, . . . , be positive integers or ∞ satisfying 1 2 then 1 ( ) ≡ 2 ( ).

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Section: Introductionmentioning

confidence: 99%

Uniqueness theorems for meromorphic functions on annuli

Rathod

2020

Ufimsk. Mat. Zh.

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“…The uniqueness theory of algebroid functions is an interesting problem in the value distribution theory. The uniqueness problem of algebroid functions was firstly considered by Valiron, afterwards some scholars have got several uniqueness theorems of algebroid functions in the complex plane C, (see [1]- [3], [8]- [11], [14], [16], [18]- [35]). In 2005, Khrystiyanyn and Kondratyuk [6]- [7] gave an extension of the Nevanlinna value distribution theory for meromorphic functions in annuli.…”

Section: Introductionmentioning

confidence: 99%

Value Distribution of a Algebroid Function and Its Linear Combinations of Derivatives on Annuli Ashok Rathod

2020

Electronic Journal of Mathematical Analysis and Applications

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“…In 2009, Cao and Yi [1] investigated the uniqueness of meromorphic functions sharing some values on annuli. On the characteristic function of derivative of f(z) with maximum deficiency sum has been studied by S. K. Singh, Kulkarni and A. Weitsman [18,19] and others have done lots of work in this area( [2][3][4], [7][8][9][10][11][12][13][14][15][16][17] and [20][21][22][23][24][25][26][27][28][29][30][31][32][33]). After this work, it is natural to ask whether we study and compare relative (k, n) Valiron defect with the relative Nevanlinna defect for meromorphic function on annuli where k and n are both non negative integers.…”

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The Value distribution of meromorphic functions with relative (k; n) Valiron defect on annuli

Rathod

2022

Mat. Stud.

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In the paper, we study and compare relative $(k,n)$ Valiron defect with the relative Nevanlinna defect for meromorphic function where $k$ and $n$ are both non negative integers on annuli. The results we proved are as follows \\1. Let $f(z)$ be a transcendental or admissible meromorphic function of finite order in $\mathbb{A}(R_0),\,$ where $1<R_0\leq +\infty$ and $\sum\nolimits_{a\not=\infty}^{}\delta_{0}(a,f)+\delta_{0}(\infty,f)=2.$Then\centerline{$\displaystyle\lim\limits_{R\rightarrow\infty}^{}\frac{T_{0}(R,f^{(k)})}{T_{0}(R,f)}=(1+k)-k\delta_{0}(\infty,f).$}\noi 2. Let $f(z)$ be a transcendental or admissible meromorphic function of finite order in $\mathbb{A}(R_0),\,$ where $1<R_0\leq +\infty$ such that $m_{0}(r,f)=S(r,f)$. If $a$, $b$ and $c$ are three distinct complex numbers, then for any two positive integer $k$ and $n$ \smallskip\centerline{$\displaystyle 3 _{R}\delta_{0(n)}^{(0)}(a,f)+2 _{R}\delta_{0(n)}^{(0)}(b,f)+3 _{R}\delta_{0(n)}^{(0)}(c,f)+5 _{R}\Delta_{0(n)}^{(k)}(\infty ,f)\leq 5 _{R}\Delta_{0(n)}^{(0)}(\infty,f)+5 _{R}\Delta_{0(n)}^{(k)}(0,f).$} \noi 3. Let $f(z)$ be a transcendental or admissible meromorphic function of finite order in $\mathbb{A}(R_0),\,$ where $1<R_0\leq +\infty$ such that $m_{0}(r,f)=S(r,f)$. If $a$, $b$ and $c$ are three distinct complex numbers, then for any two positive integer $k$ and $n$\smallskip\centerline{$\displaystyle_{R}\delta_{0(n)}^{(0)}(0,f)+_{R}\Delta_{0(n)}^{(k)}(\infty,f)+_{R}\delta_{0(n)}^{(0)}(c,f)\leq _{R}\Delta_{0(n)}^{(0)}(\infty,f)+2_{R}\Delta_{0(n)}^{(k)}(0,f).$} \noi 4. Let $f(z)$ be a transcendental or admissible meromorphic function of finite order in $\mathbb{A}(R_0),\,$ where $1<R_0\leq +\infty$ such that $m_{0}(r,f)=S(r,f)$. If $a$ and $d$ are two distinct complex numbers, then for any two positive integer $k$ and $p$ with $0\leq k\leq p$\smallskip\centerline{$\displaystyle_{R}\delta_{0(n)}^{(0)}(d,f)+_{R}\Delta_{0(n)}^{(p)}(\infty,f)+_{R}\delta_{0(n)}^{(k)}(a,f)\leq _{R}\Delta_{0(n)}^{(k)}(\infty,f)+_{R}\Delta_{0(n)}^{(p)}(0,f)+_{R}\Delta_{0(n)}^{(k)}(0,f),$} \noi where $n$ is any positive integer.\\5.Let $f(z)$ be a transcendental or admissible meromorphic function of finite order in $\mathbb{A}(R_0),\,$ where $1<R_0\leq +\infty$ . Then for any two positive integers $k$ and $n$,\smallskip\centerline{$\displaystyle_{R}\Delta_{0(n)}^{(0)}(\infty,f)+_{R}\Delta_{0(n)}^{(k)}(0,f) \geq _{R}\delta_{0(n)}^{(0)}(0,f)+_{R}\delta_{0(n)}^{(0)}(a,f)+_{R}\Delta_{0(n)}^{(k)}(\infty,f),$}\noi where $a$ is any non zero complex number.

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Characteristic function and deficiency of algebroid functions on annuli

Cited by 5 publications

References 6 publications

Uniqueness theorems for meromorphic functions on annuli

Uniqueness theorems for meromorphic functions on annuli

Value Distribution of a Algebroid Function and Its Linear Combinations of Derivatives on Annuli Ashok Rathod

The Value distribution of meromorphic functions with relative (k; n) Valiron defect on annuli

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