The shared set and uniqueness of meromorphic functions in an angular domain

Rathod, Ashok

doi:10.32513/tmj/19322008145

Tbilisi Math. J.

2021

DOI: 10.32513/tmj/19322008145

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The shared set and uniqueness of meromorphic functions in an angular domain

Ashok Rathod¹

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2022

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

Rathod

2022

Mat. Stud.

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The shared set and uniqueness of meromorphic functions in an angular domain

Cited by 1 publication

References 16 publications

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

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

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