﻿ 整函数涉及一个IM分担值的唯一性定理 Uniqueness Theorem of Entire Functions with Share on Value IM

Pure Mathematics
Vol. 11  No. 01 ( 2021 ), Article ID: 39931 , 5 pages
10.12677/PM.2021.111011

Uniqueness Theorem of Entire Functions with Share on Value IM

Haoxiang Zhi, Li Jia

School of Mathematics, Yunnan Normal University, Kunming Yunnan

Received: Dec. 13th, 2020; accepted: Jan. 14th, 2021; published: Jan. 22nd, 2021

ABSTRACT

In this paper, we shall prove that if two non-constant entire function $f\left(z\right)$ and $g\left(z\right)$ share the value 1 IM, and $\underset{r\to \infty }{\stackrel{¯}{lim}}\frac{\stackrel{¯}{N}\left(r,f\right)+{\stackrel{¯}{N}}_{\left(2}\left(r,\frac{1}{f}\right)}{T\left(r,f\right)}\le \frac{1}{6}$,$\underset{r\to \infty }{\stackrel{¯}{lim}}\frac{\stackrel{¯}{N}\left(r,\frac{1}{g}\right)+{\stackrel{¯}{N}}_{\left(2}\left(r,\frac{1}{g}\right)}{T\left(r,g\right)}\le \frac{1}{6}$, then $f\left(z\right)\equiv g\left(z\right)$ or $f\left(z\right)\cdot g\left(z\right)\equiv 1$.

Keywords:Entire Function, Shared Value, Deficient Value, Uniqueness

1. 引言及主要结果

1976年，M. Ozuwa证明了如下定理：

1987年，仪洪勋证明了：

$\underset{r\to +\infty }{\stackrel{¯}{\mathrm{lim}}}\frac{\stackrel{¯}{N}\left(r,f\right)+{\stackrel{¯}{N}}_{\left(2}\left(r,\frac{1}{f}\right)}{T\left(r,f\right)}\le \frac{1}{6}$, (1.1)

$\underset{r\to +\infty }{\stackrel{¯}{\mathrm{lim}}}\frac{\stackrel{¯}{N}\left(r,g\right)+{\stackrel{¯}{N}}_{\left(2}\left(r,\frac{1}{g}\right)}{T\left(r,g\right)}\le \frac{1}{6}$, (1.2)

$f\left(z\right)\equiv g\left(z\right)$$f\left(z\right)\cdot g\left(z\right)\equiv 1$

2. 几个引理

$N\left(r,\frac{1}{{f}^{\left(k\right)}}\right).

${\stackrel{¯}{N}}_{L}\left(r,\frac{\text{1}}{f-1}\right)\le \stackrel{¯}{N}\left(r,f\right)+S\left(r,f\right)$,${\stackrel{¯}{N}}_{L}\left(r,\frac{\text{1}}{g-1}\right)\le \stackrel{¯}{N}\left(r,g\right)+S\left(r,g\right)$.

$\underset{r\to \infty }{\stackrel{¯}{\mathrm{lim}}}\frac{\stackrel{¯}{N}\left(r,f\right)+{\stackrel{¯}{N}}_{\left(2}\left(r,\frac{1}{f}\right)}{T\left(r,f\right)}\le \frac{1}{\text{2}}$,$\underset{r\to \infty }{\stackrel{¯}{\mathrm{lim}}}\frac{\stackrel{¯}{N}\left(r,g\right)+{\stackrel{¯}{N}}_{\left(2}\left(r,\frac{1}{g}\right)}{T\left(r,g\right)}\le \frac{1}{2}$,

$f\left(z\right)\equiv g\left(z\right)$$f\left(z\right)\cdot g\left(z\right)\equiv 1$

3. 定理1的证明

$H=\left(\frac{{f}^{″}}{{f}^{\prime }}-\frac{{g}^{″}}{{g}^{\prime }}\right)-2\left(\frac{{f}^{\prime }}{f-1}-\frac{{g}^{\prime }}{g-1}\right)$.

${N}_{E}^{1\right)}\left(r,\frac{1}{f-1}\right)\le N\left(r,\frac{1}{H}\right)\le T\left(r,H\right)+O\left(1\right)\le N\left(r,H\right)+S\left(r,f\right)+S\left(r,g\right)$. (1.3)

$N\left(r,H\right)\le {\stackrel{¯}{N}}_{\left(2}\left(r,\frac{1}{f}\right)+{\stackrel{¯}{N}}_{\left(2}\left(r,\frac{1}{g}\right)+{\stackrel{¯}{N}}_{L}\left(r,\frac{1}{f-1}\right)+{\stackrel{¯}{N}}_{L}\left(r,\frac{1}{g-1}\right)+{N}_{0}\left(r,\frac{1}{{f}^{\prime }}\right)+{N}_{0}\left(r,\frac{1}{{g}^{\prime }}\right)$. (1.4)

$\stackrel{¯}{N}\left(r,\frac{1}{f-1}\right)=\stackrel{¯}{N}\left(r,\frac{1}{g-1}\right)\le {N}_{E}^{1\right)}\left(r,\frac{1}{f-1}\right)+{N}_{E}^{\left(2}\left(r,\frac{1}{f-1}\right)+{\stackrel{¯}{N}}_{L}\left(r,\frac{1}{f-1}\right)+{\stackrel{¯}{N}}_{L}\left(r,\frac{1}{g-1}\right)$. (1.5)

$\begin{array}{l}\stackrel{¯}{N}\left(r,\frac{1}{f-1}\right)+\stackrel{¯}{N}\left(r,\frac{1}{g-1}\right)\\ \le \text{2}{N}_{E}^{1\right)}\left(r,\frac{1}{f-1}\right)+\text{2}{N}_{E}^{\left(2}\left(r,\frac{1}{f-1}\right)+\text{2}{\stackrel{¯}{N}}_{L}\left(r,\frac{1}{f-1}\right)+\text{2}{\stackrel{¯}{N}}_{L}\left(r,\frac{1}{g-1}\right)\\ \le {\stackrel{¯}{N}}_{\left(2}\left(r,\frac{1}{f}\right)+{\stackrel{¯}{N}}_{\left(2}\left(r,\frac{1}{g}\right)+{\stackrel{¯}{N}}_{L}\left(r,\frac{1}{f-1}\right)+{\stackrel{¯}{N}}_{L}\left(r,\frac{1}{g-1}\right)+{N}_{0}\left(r,\frac{1}{{f}^{\prime }}\right)+{N}_{0}\left(r,\frac{1}{{g}^{\prime }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{N}_{E}^{1\right)}\left(r,\frac{1}{f-1}\right)+\text{2}{N}_{E}^{\left(2}\left(r,\frac{1}{f-1}\right)+\text{2}{\stackrel{¯}{N}}_{L}\left(r,\frac{1}{f-1}\right)+\text{2}{\stackrel{¯}{N}}_{L}\left(r,\frac{1}{g-1}\right)+S\left(r,f\right)+S\left(r,g\right)\end{array}$. (1.6)

$\begin{array}{l}{N}_{E}^{\text{1}\right)}\left(r,\frac{1}{f-1}\right)+\text{2}{N}_{E}^{\left(2}\left(r,\frac{1}{f-1}\right)+{\stackrel{¯}{N}}_{L}\left(r,\frac{1}{f-1}\right)+{\stackrel{¯}{N}}_{L}\left(r,\frac{1}{g-1}\right)\\ \le \frac{\text{1}}{\text{2}}\left(N\left(r,\frac{1}{f-1}\right)+N\left(r,\frac{1}{g-1}\right)\right)\le \frac{\text{1}}{\text{2}}\left(T\left(r,f\right)+T\left(r,g\right)\right)+O\left(1\right)\end{array}$.(1.7)

$\begin{array}{l}\stackrel{¯}{N}\left(r,\frac{1}{f-1}\right)+\stackrel{¯}{N}\left(r,\frac{1}{g-1}\right)\\ \le {\stackrel{¯}{N}}_{\left(2}\left(r,\frac{1}{f}\right)+{\stackrel{¯}{N}}_{\left(2}\left(r,\frac{1}{g}\right)+2{\stackrel{¯}{N}}_{L}\left(r,\frac{1}{f-1}\right)+2{\stackrel{¯}{N}}_{L}\left(r,\frac{1}{g-1}\right)+{N}_{0}\left(r,\frac{1}{{f}^{\prime }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{N}_{0}\left(r,\frac{1}{{g}^{\prime }}\right)+\frac{\text{1}}{\text{2}}\left(T\left(r,f\right)+T\left(r,g\right)\right)+S\left(r,f\right)+S\left(r,g\right)\end{array}$.(1.8)

$\begin{array}{l}T\left(r,f\right)+T\left(r,g\right)\\ \le \stackrel{¯}{N}\left(r,\frac{1}{f-1}\right)+\stackrel{¯}{N}\left(r,\frac{1}{g-1}\right)+\stackrel{¯}{N}\left(r,\frac{1}{f}\right)+\stackrel{¯}{N}\left(r,\frac{1}{g}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{N}_{0}\left(r,\frac{1}{{f}^{\prime }}\right)-{N}_{0}\left(r,\frac{1}{{g}^{\prime }}\right)+S\left(r,f\right)+S\left(r,g\right)\end{array}$. (1.9)

$\begin{array}{l}\frac{\text{1}}{\text{2}}T\left(r,f\right)+\frac{1}{2}T\left(r,g\right)\\ \le 3\stackrel{¯}{N}\left(r,\frac{1}{f}\right)+3\stackrel{¯}{N}\left(r,\frac{1}{g}\right)+{\stackrel{¯}{N}}_{\left(2}\left(r,\frac{1}{f}\right)+{\stackrel{¯}{N}}_{\left(2}\left(r,\frac{1}{g}\right)+S\left(r,f\right)+S\left(r,g\right)\\ \le 3\left[\stackrel{¯}{N}\left(r,\frac{1}{f}\right)+{\stackrel{¯}{N}}_{\left(2}\left(r,\frac{1}{f}\right)\right]+3\left[\stackrel{¯}{N}\left(r,\frac{1}{g}\right)+{\stackrel{¯}{N}}_{\left(2}\left(r,\frac{1}{g}\right)\right]+S\left(r,f\right)+S\left(r,g\right)\end{array}$.(1.10)

$\stackrel{¯}{N}\left(r,\frac{1}{f}\right)+{\stackrel{¯}{N}}_{\left(2}\left(r,\frac{1}{f}\right)<\left(\frac{1}{6}-\epsilon \right)T\left(r,f\right)$,$\stackrel{¯}{N}\left(r,\frac{1}{g}\right)+{\stackrel{¯}{N}}_{\left(2}\left(r,\frac{1}{g}\right)<\left(\frac{1}{6}-\epsilon \right)T\left(r,g\right)$. (1.11)

$\text{6}\epsilon \left(T\left(r,f\right)+T\left(r,g\right)\right)\le S\left(r,f\right)+S\left(r,g\right)$.

Uniqueness Theorem of Entire Functions with Share on Value IM[J]. 理论数学, 2021, 11(01): 74-78. https://doi.org/10.12677/PM.2021.111011

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