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Volume 32, Issue 1
Normality Criteria of Meromorphic Functions

Qiong Wang, Wenjun Yuan, Wei Chen & Honggen Tian

DOI: 10.13447/j.1674-5647.2016.01.07

Commun. Math. Res., 32 (2016), pp. 88-96.

Published online: 2021-03

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  • Abstract

In this paper, we consider normality criteria for a family of meromorphic functions concerning shared values. Let $\mathcal{F}$ be a family of meromorphic functions defined in a domain $D, m, n, k$ and $d$ be four positive integers satisfying $m ≥ n + 2$ and $d ≥ \frac{k + 1}{m − n − 1}$, and $a(≠0), b$ be two finite constants. Suppose that every $f ∈ \mathcal{F}$ has all its zeros and poles of multiplicity at least $k$ and $d$, respectively. If $(f^n)^{(k)}−af^m$ and $(g^n)^{(k)}−ag^m$ share the value $b$ for every pair of functions $(f, g)$ of $\mathcal{F}$, then $\mathcal{F}$ is normal in $D$. Our results improve the related theorems of Schwick (Schwick W. Normality criteria for families of meromorphic function. $J$. $Anal$. $Math$., 1989, 52: 241–289), Li and Gu (Li Y T, Gu Y X. On normal families of meromorphic functions. $J$. $Math$. $Anal$. $Appl$., 2009, 354: 421–425).

  • Keywords

meromorphic function, shared value, normal criterion.

  • AMS Subject Headings

30D30, 30D45

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-32-88, author = {Wang , QiongYuan , WenjunChen , Wei and Tian , Honggen}, title = {Normality Criteria of Meromorphic Functions}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {32}, number = {1}, pages = {88--96}, abstract = {

In this paper, we consider normality criteria for a family of meromorphic functions concerning shared values. Let $\mathcal{F}$ be a family of meromorphic functions defined in a domain $D, m, n, k$ and $d$ be four positive integers satisfying $m ≥ n + 2$ and $d ≥ \frac{k + 1}{m − n − 1}$, and $a(≠0), b$ be two finite constants. Suppose that every $f ∈ \mathcal{F}$ has all its zeros and poles of multiplicity at least $k$ and $d$, respectively. If $(f^n)^{(k)}−af^m$ and $(g^n)^{(k)}−ag^m$ share the value $b$ for every pair of functions $(f, g)$ of $\mathcal{F}$, then $\mathcal{F}$ is normal in $D$. Our results improve the related theorems of Schwick (Schwick W. Normality criteria for families of meromorphic function. $J$. $Anal$. $Math$., 1989, 52: 241–289), Li and Gu (Li Y T, Gu Y X. On normal families of meromorphic functions. $J$. $Math$. $Anal$. $Appl$., 2009, 354: 421–425).

}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2016.01.07}, url = {http://global-sci.org/intro/article_detail/cmr/18666.html} }
TY - JOUR T1 - Normality Criteria of Meromorphic Functions AU - Wang , Qiong AU - Yuan , Wenjun AU - Chen , Wei AU - Tian , Honggen JO - Communications in Mathematical Research VL - 1 SP - 88 EP - 96 PY - 2021 DA - 2021/03 SN - 32 DO - http://doi.org/10.13447/j.1674-5647.2016.01.07 UR - https://global-sci.org/intro/article_detail/cmr/18666.html KW - meromorphic function, shared value, normal criterion. AB -

In this paper, we consider normality criteria for a family of meromorphic functions concerning shared values. Let $\mathcal{F}$ be a family of meromorphic functions defined in a domain $D, m, n, k$ and $d$ be four positive integers satisfying $m ≥ n + 2$ and $d ≥ \frac{k + 1}{m − n − 1}$, and $a(≠0), b$ be two finite constants. Suppose that every $f ∈ \mathcal{F}$ has all its zeros and poles of multiplicity at least $k$ and $d$, respectively. If $(f^n)^{(k)}−af^m$ and $(g^n)^{(k)}−ag^m$ share the value $b$ for every pair of functions $(f, g)$ of $\mathcal{F}$, then $\mathcal{F}$ is normal in $D$. Our results improve the related theorems of Schwick (Schwick W. Normality criteria for families of meromorphic function. $J$. $Anal$. $Math$., 1989, 52: 241–289), Li and Gu (Li Y T, Gu Y X. On normal families of meromorphic functions. $J$. $Math$. $Anal$. $Appl$., 2009, 354: 421–425).

Wang , QiongYuan , WenjunChen , Wei and Tian , Honggen. (2021). Normality Criteria of Meromorphic Functions. Communications in Mathematical Research . 32 (1). 88-96. doi:10.13447/j.1674-5647.2016.01.07
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