Abstract In this paper, we mainly prove: Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7214_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(f\)</EquationSource>  </InlineEquation> be a nonconstant meromorphic function of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7214_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">\(\rho_{2}(f)<1\)</EquationSource>  </InlineEquation>, let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7214_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">\(k(\geq 2)\)</EquationSource>  </InlineEquation> be a positive integer, and let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7214_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">\(a,c\)</EquationSource>  </InlineEquation> be two nonzero complex numbers. Suppose that <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7214_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="116" /> </InlineMediaObject> <EquationSource Format="TEX">\(f(z),f^{(k)}(z+c)\)</EquationSource>  </InlineEquation> share <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7214_Article_IEq6.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(a\)</EquationSource>  </InlineEquation> IM and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7214_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="99" /> </InlineMediaObject> <EquationSource Format="TEX">\(f(z),f(z+c)\)</EquationSource>  </InlineEquation> share <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7214_Article_IEq8.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\infty\)</EquationSource>  </InlineEquation> CM. If <Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7214_Article_Equa.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="512" /> </MediaObject> <EquationSource Format="TEX">\(N\left(r,\frac{1}{f(z)}\right)+N\left(r,\frac{1}{f^{(k)}(z+c)}\right)=S(r,f),\quad\text{then }f(z)\equiv f^{(k)}(z+c).\)</EquationSource>  </Equation> The results obtained in this paper improve and extend the results due to Majumder [Commun. Math. Stat., 2017] and Kaish and Mobdal [Commun. Korean Math. Soc., 2024].

Uniqueness of Meromorphic Functions Concerning Derivatives and Shifts

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发表日期：

2025年4月1日

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G. Wang,
Z. Y. He,
M. L. Fang

摘要

Abstract

In this paper, we mainly prove: Let \(f\) be a nonconstant meromorphic function of \(\rho_{2}(f)<1\) , let \(k(\geq 2)\) be a positive integer, and let \(a,c\) be two nonzero complex numbers. Suppose that \(f(z),f^{(k)}(z+c)\) share \(a\) IM and \(f(z),f(z+c)\) share \(\infty\) CM. If

\(N\left(r,\frac{1}{f(z)}\right)+N\left(r,\frac{1}{f^{(k)}(z+c)}\right)=S(r,f),\quad\text{then }f(z)\equiv f^{(k)}(z+c).\)

The results obtained in this paper improve and extend the results due to Majumder [Commun. Math. Stat., 2017] and Kaish and Mobdal [Commun. Korean Math. Soc., 2024].