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

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...

Full description

Saved in:

Bibliographic Details
Main Author:	A. Rathod
Format:	Article
Language:	deu
Published:	Ivan Franko National University of Lviv 2022-06-01
Series:	Математичні Студії
Subjects:	nevanlinna theory; valiron defect; meromorphic function; the annuli
Online Access:	http://matstud.org.ua/ojs/index.php/matstud/article/view/290
Tags:	Add Tag No Tags, Be the first to tag this record!

_version_	1849416377600835584
author	A. Rathod
author_facet	A. Rathod
author_sort	A. Rathod
collection	DOAJ
description	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.
format	Article
id	doaj-art-52b878b5311042ffb65f167b48fc4e21
institution	Kabale University
issn	1027-4634 2411-0620
language	deu
publishDate	2022-06-01
publisher	Ivan Franko National University of Lviv
record_format	Article
series	Математичні Студії
spelling	doaj-art-52b878b5311042ffb65f167b48fc4e212025-08-20T03:33:11ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202022-06-0157217218510.30970/ms.57.2.172-185290The Value distribution of meromorphic functions with relative (k; n) Valiron defect on annuliA. Rathod0KLE Societys G I Bagewadi Arts Science and Commerce College NipaniIn 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.http://matstud.org.ua/ojs/index.php/matstud/article/view/290nevanlinna theory; valiron defect; meromorphic function; the annuli
spellingShingle	A. Rathod The Value distribution of meromorphic functions with relative (k; n) Valiron defect on annuli Математичні Студії nevanlinna theory; valiron defect; meromorphic function; the annuli
title	The Value distribution of meromorphic functions with relative (k; n) Valiron defect on annuli
title_full	The Value distribution of meromorphic functions with relative (k; n) Valiron defect on annuli
title_fullStr	The Value distribution of meromorphic functions with relative (k; n) Valiron defect on annuli
title_full_unstemmed	The Value distribution of meromorphic functions with relative (k; n) Valiron defect on annuli
title_short	The Value distribution of meromorphic functions with relative (k; n) Valiron defect on annuli
title_sort	value distribution of meromorphic functions with relative k n valiron defect on annuli
topic	nevanlinna theory; valiron defect; meromorphic function; the annuli
url	http://matstud.org.ua/ojs/index.php/matstud/article/view/290
work_keys_str_mv	AT arathod thevaluedistributionofmeromorphicfunctionswithrelativeknvalirondefectonannuli AT arathod valuedistributionofmeromorphicfunctionswithrelativeknvalirondefectonannuli

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

Similar Items