Uniqueness of Meromorphic Functions with Respect to Their Shifts Concerning Derivatives

Pub Date : 2024-04-25 DOI:10.3103/s1068362324700031

X. H. Huang

{"title":"Uniqueness of Meromorphic Functions with Respect to Their Shifts Concerning Derivatives","authors":"X. H. Huang","doi":"10.3103/s1068362324700031","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3>An example in the article shows that the first derivative of \$f(z)=\\frac{2}{1-e^{-2z}}\$ sharing \$0\$ CM and \$1,\\infty\$ IM with its shift \$\\pi i\$ cannot obtain they are equal. In this paper, we study the uniqueness of meromorphic function sharing small functions with their shifts concerning its \$k\$th derivatives. We use a different method from Qi and Yang [1] to improves entire function to meromorphic function, the first derivative to the \$k\$th derivatives, and also finite values to small functions. As for \$k=0\$, we obtain: Let \$f(z)\$ be a transcendental meromorphic function of \$\\rho_{2}(f)<1\$, let \$c\$ be a nonzero finite value, and let \$a(z)\\not\\equiv\\infty,b(z)\\not\\equiv\\infty\\in\\hat{S}(f)\$ be two distinct small functions of \$f(z)\$ such that \$a(z)\$ is a periodic function with period \$c\$ and \$b(z)\$ is any small function of \$f(z)\$. If \$f(z)\$ and \$f(z+c)\$ share \$a(z),\\infty\$ CM, and share \$b(z)\$ IM, then either \$f(z)\\equiv f(z+c)\$ or$$e^{p(z)}\\equiv\\frac{f(z+c)-a(z+c)}{f(z)-a(z)}\\equiv\\frac{b(z+c)-a(z+c)}{b(z)-a(z)},$$where \$p(z)\$ is a nonconstant entire function of \$\\rho(p)<1\$ such that \$e^{p(z+c)}\\equiv e^{p(z)}\$.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3103/s1068362324700031","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

Abstract

An example in the article shows that the first derivative of $f(z)=\frac{2}{1-e^{-2z}}$ sharing $0$ CM and $1,\infty$ IM with its shift $\pi i$ cannot obtain they are equal. In this paper, we study the uniqueness of meromorphic function sharing small functions with their shifts concerning its $k$th derivatives. We use a different method from Qi and Yang [1] to improves entire function to meromorphic function, the first derivative to the $k$th derivatives, and also finite values to small functions. As for $k=0$, we obtain: Let $f(z)$ be a transcendental meromorphic function of $\rho_{2}(f)<1$, let $c$ be a nonzero finite value, and let $a(z)\not\equiv\infty,b(z)\not\equiv\infty\in\hat{S}(f)$ be two distinct small functions of $f(z)$ such that $a(z)$ is a periodic function with period $c$ and $b(z)$ is any small function of $f(z)$. If $f(z)$ and $f(z+c)$ share $a(z),\infty$ CM, and share $b(z)$ IM, then either $f(z)\equiv f(z+c)$ or

$$e^{p(z)}\equiv\frac{f(z+c)-a(z+c)}{f(z)-a(z)}\equiv\frac{b(z+c)-a(z+c)}{b(z)-a(z)},$$

where $p(z)$ is a nonconstant entire function of $\rho(p)<1$ such that $e^{p(z+c)}\equiv e^{p(z)}$.

查看原文

微变函数关于其偏移的唯一性

Abstract 文章中的一个例子表明，共享 $0$ CM 和 $1,\infty$ IM 的 $f(z)=\frac{2}{1-e^{-2z}}\ 的第一导数与它的移\(\pi i$ 不能得到它们相等。在本文中，我们研究了分担小函数与它们的移（$k$th derivatives）的微函数的唯一性。我们采用了与齐和杨[1]不同的方法，将整个函数改进为分形函数，将第一导数改进为$k$三次导数，同时将有限值改进为小函数。对于 $k=0$, 我们得到：设 $f(z)$ 是 $\rho_{2}(f)<；1), let \(c$ be a nonzero finite value, and let $a(z)\not\equiv\infty、b(z)\not\equiv\infty\inhat{S}(f)$ 是两个不同的小函数，使得（a(z)\）是一个周期为（c）的周期函数，而（b(z)\）是（f(z)\）的任何小函数。如果 $f(z)$ 和 $f(z+c)$ 共享 $a(z),\infty$CM, and share $b(z)$IM, then either $f(z)\equiv f(z+c)$ or$$e^{p(z)}equiv\frac{f(z+c)-a(z+c)}{f(z)-a(z)}equiv\frac{b(z+c)-a(z+c)}{b(z)-a(z)},$$where $p(z)$ is a nonconstant entire function of $\rho(p)<；1$ such that $e^{p(z+c)}\equiv e^{p(z)}$.

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文