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

• A. Rathod KLE Societys G I Bagewadi Arts Science and Commerce College Nipani
Keywords: Nevanlinna theory; Valiron defect; meromorphic function; the annuli

### Abstract

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.

### Author Biography

A. Rathod, KLE Societys G I Bagewadi Arts Science and Commerce College Nipani

KLE Societys G I Bagewadi Arts Science and Commerce College Nipani

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Published
2022-06-27
How to Cite
Rathod, A. (2022). The Value distribution of meromorphic functions with relative (k; n) Valiron defect on annuli. Matematychni Studii, 57(2), 172-185. https://doi.org/10.30970/ms.57.2.172-185
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