Abstract <p>In this paper, we consider the relationship between the difference operator and the first derivative from the perspective of shared values. And as a result of that, we get: Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7203_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">\(f(z)\)</EquationSource>  </InlineEquation> be a transcendental meromorphic function of hyperorder strictly less than <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7203_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="8" /> </InlineMediaObject> <EquationSource Format="TEX">\(\frac{3}{4}\)</EquationSource>  </InlineEquation>, and let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7203_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(a,b\)</EquationSource>  </InlineEquation> be two distinct constants. If <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7203_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">\(f^{\prime}(z)\)</EquationSource>  </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7203_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Delta f\)</EquationSource>  </InlineEquation> share <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7203_Article_IEq6.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(a\)</EquationSource>  </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7203_Article_IEq7.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\infty\)</EquationSource>  </InlineEquation> CM and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7203_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(b\)</EquationSource>  </InlineEquation> IM, then <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7203_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="84" /> </InlineMediaObject> <EquationSource Format="TEX">\(f^{\prime}(z)=\Delta f\)</EquationSource>  </InlineEquation>. The research also includes some improvements of earlier results of such studies in [<CitationRef CitationID="CR1">1</CitationRef>, <CitationRef CitationID="CR2">2</CitationRef>].</p>

The Relationship Between the Difference Operator and the Derivative Is Viewed from the Perspective of Shared Values

期刊论文
发表日期：

2025年2月1日

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X. Cao,
X. Qi

摘要

Abstract

In this paper, we consider the relationship between the difference operator and the first derivative from the perspective of shared values. And as a result of that, we get: Let \(f(z)\) be a transcendental meromorphic function of hyperorder strictly less than \(\frac{3}{4}\) , and let \(a,b\) be two distinct constants. If \(f^{\prime}(z)\) and \(\Delta f\) share \(a\) , \(\infty\) CM and \(b\) IM, then \(f^{\prime}(z)=\Delta f\) . The research also includes some improvements of earlier results of such studies in [1, 2].