## 亚纯函数关于单叶离散值的正规定理

1 广东培正学院数据科学与计算机学院 广州 510830

2 广东金融学院金融数学与统计学院 广州 510521

3 华南师范大学数学科学学院 广州 510631

## Normal Family Theorems for Meromorphic Functions with Discrete Values of One Leaf

Guo Xiaojing,1, Chai Fujie,2, Sun Daochun,3

1 School of Data Science and Computer Science, Guangdong Peizheng College, Guangzhou 510830

2 School of Mathematics and Statistics, Guangdong University of Finance, Guangzhou 510521

3 School of Mathematics Science, South China Normal University, Guangzhou 510631

 基金资助: 国家自然科学基金.  11501127广州民航学院基金.  18X0428

 Fund supported: the NSFC.  11501127the Guangzhou Civil Aviation College.  18X0428

Abstract

In this paper, the normal theorems of meromorphic functions involving discrete values are studied by using the theory of Ahlfors covering surfaces. Firstly, the discrete values with one leaf of meromorphic functions are defined, then the inequalities about islands are investigated and two precise inequalities about islands are obtained. Finally, the inequalities are used to study the discrete values and the normal family of meromorphic functions, then a normal theorem involving a monophyletic island and a normal theorem involving discrete values of one leaf are obtained. All these theorems promote the famous Ahlfors' five islands theorem and five single valued theorem of Nevanlinna.

Keywords： Meromorphic functions ; Normal family ; Covering surface ; Island ; Discrete values of one Leaf

Guo Xiaojing, Chai Fujie, Sun Daochun. Normal Family Theorems for Meromorphic Functions with Discrete Values of One Leaf. Acta Mathematica Scientia[J], 2021, 41(6): 1598-1605 doi:

## 1 引言与主要结果

$V$是直径为1的Riemann球面, 本文中我们认为扩充复平面$\hat{{\Bbb C}} $$V 是等同的, 并且在使用的时候不加区分. 设 F$$ \hat{{\Bbb C}}$的有限连通覆盖曲面, 其边界$\partial F$由有限条解析Jordan曲线组成, $\partial F$的长度记为$L$. 设区域$D\subset \hat{{\Bbb C}}$是一个Jordan区域, 我们用$|D|$表示区域$D$的球面面积, 用$|a, b|$表示$a, b(\in D)$二点间的球面距离. 记$F$盖在$D$上的部分为$F(D)$. 我们分别称

$F $$V$$ D$的平均覆盖次数. 设$F(D)$由有限个连通曲面$\{F_k(D)\}$组成. 若$F_k(D)\in\{F_k(D)\}$没有对$D$的相对边界, 我们就称之为岛, 记为$F_k^i(D)$; 否则就称之为半岛, 记为$F_k^p(D)$. 关于岛, Ahlfors曾得到如下著名的五岛定理:

Bloch和Valiron分别应用Nevanlinna理论得到了如下与Ahlfors五岛定理相应的Nevanlinna五单值定理:

$\# E = 0, 1$, 则令$d_f(a): = 1$, 再令

$$$\sum\limits^{q}_{v = 1}n_v\geq \sum\limits^{N}_{t = 1}\sum\limits_\mu \rho^+(F^*_{t\mu})+N(F^\prime)-\sum\limits^{N}_{t = 1}N(F^*_t).$$$

(ⅰ) 若每个曲面$F^\prime_t $$\{D_v\} 上均没有岛, 即 \sum\limits^{q}_{v = 1}n_v = 0 , 这时 F^\prime_t = F^*_t = F^*_{t\mu} , 于是 (ⅱ) 若至少有一个曲面 F^\prime_t$$ \{D_v\}$上有岛, 则对任何一个有岛的曲面$F^\prime_t$, 由于$F^\prime_t$是单连通的, 其边界$\partial F^\prime_t$是连通的, 所以$F^*_t = \{F^*_{t\mu }\}$中仅有一曲面$A\in \{F^*_{t\mu}\}$的边界$\partial A$包含$\partial F^\prime_t$, 而$A$上因挖走了岛必然有“洞”, 故$A$不是单连通的. 其余的曲面$B\in \{F^*_{t\mu}\}-A$的边界$\partial B $$\partial F^\prime_t 无公共点, 因此 B$$ F_0$的相对边界长$L(F_0) = 0$. 由引理2.2可知$\rho^+(B)\geq (q-2)S(F_0)>0$. 所以这些曲面都不是单连通的. 因此对任何一个有岛的曲面$F^\prime_t$, 在$\{F^*_{t\mu }\}$中没有单连通曲面. 这样在挖掉所有的岛之后, $F^\prime$中至少减少了一个单连通曲面, 于是$N(F^\prime)\geq \sum\limits^{N}_{t = 1}N(F^*_t)+1$. 再结合(2.1) 式便得到

$\begin{eqnarray} \sum\limits^{N}_{t = 1}\sum\limits_\mu \rho^+(F^*_{t\mu})&>&(q-2)\sum\limits^{N}_{t = 1}\sum\limits_\mu S_{t\mu }(F_0)-\frac{2^5\pi^2}{\delta^3} \sum\limits^{N}_{t = 1}\sum\limits_\mu L_{t\mu }(F_0){}\\ & = &(q-2)S(F_0)-\frac{2^5\pi^2}{\delta^3}L(F_0), \end{eqnarray}$

$L_{t\mu}(F_0) $$F^*_{t\mu }$$ F_0$的相对边界长, $L(F_0) = \sum\limits^{N}_{t = 1}\sum\limits_\mu L_{t\mu }(F_0) $$F$$ F_0$的相对边界长. 结合(2.2) 式及引理2.2, 并注意$q(\frac{\delta}{2})^2<|F_0|$便得到

\begin{align} 0>(q-2)S-\frac{2^6\pi^2}{\delta^3}L. \end{align}

\begin{align} \sum\limits^{q}_{v = 1}n_v-1>(q-2)S-\frac{2^6\pi^2}{\delta^3}L. \end{align}

设$n^{\prime\prime}_v$表示$F(D_v)$中所含多叶单连通岛的个数, 于是有

\begin{align} qS+qhL+\sum\limits^q_{v = 1}n^{\prime}_v\geq 2\sum\limits^q_{v = 1}n_v. \end{align}

$\sum\limits^q_{v = 1}n_v\geq 1$

\begin{align} \sum\limits^q_{v = 1}n^{\prime}_v\geq (q-4)S-h_1L . \end{align}

$\sum\limits^q_{v = 1}n_v = 0$时, 有$\sum\limits^q_{v = 1}n^\prime_v = 0$, 于是

\begin{align} 0\geq (q-4)S-h_2L . \end{align}

(ⅰ) 若对任意的$t\in (r, R)$, 恒有

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