Characteristic function of a meromorphic function and its derivative

Singh, Surjeet; Kulkarni, Veena

doi:10.4064/ap-28-2-123-133

Cited by 22 publications

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“…In 1973, S. K. Singh and V. N. Kulkarni [1] proved the following result: Theorem 1.1. Suppose that f is a transcendental meromorphic function of finite order satisfying X…”

Section: Introduction and Resultsmentioning

confidence: 96%

Meromorphic function of infinite order with maximum deficiency sum

Xiong¹

2004

Kodai Math. J.

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In this paper we prove the following theorem: Let f ðzÞ be a meromorphic function of infinite order. If P a0y dða; f Þ þ dðy; f Þ ¼ 2, then for each positive integer k, we have Kð f ðkÞ Þ ¼ 2kð1 À dðy; f ÞÞ=ð1 þ k À kdðy; f ÞÞ, where Kð f ðkÞ Þ ¼ lim r!y ðNðr; 1=f ðkÞ Þ þ Nðr; f ðkÞ ÞÞ=Tðr; f ðkÞ Þ exists. This result improves the results by S. K. Singh and V. N. Kulkarni [1] and Mingliang Fang [2].

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“…In 1973, S. K. Singh and V. N. Kulkarni [1] proved the following result: Theorem 1.1. Suppose that f is a transcendental meromorphic function of finite order satisfying X…”

Section: Introduction and Resultsmentioning

confidence: 96%

Meromorphic function of infinite order with maximum deficiency sum

Xiong¹

2004

Kodai Math. J.

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“…In 2005, A.Ya. Khrystiyanyn and A.A. Kondratyuk have proposed the Nevanlinna theory for meromorphic functions on annuli (see [4], [5]) and after this work others have done lot of work in this area (see [8], [12], [13]- [36]). In 2009, Cao and Yi [1] studied the uniqueness of meromorphic functions sharing some values on annuli.…”

Section: Introductionmentioning

confidence: 99%

Characteristic function and deficiency of algebroid functions on annuli

Rathod¹

2019

Ufimskii Matematicheskii Zhurnal

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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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confidence: 99%

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 of a meromorphic function and its derivative

Cited by 22 publications

References 0 publications

Meromorphic function of infinite order with maximum deficiency sum

Meromorphic function of infinite order with maximum deficiency sum

Characteristic function and deficiency of algebroid functions on annuli

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

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