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<p>In this paper, we study the uniqueness of meromorphic functions sharing small functions with their kth order derivatives. We prove: let <i>f</i> be a non-constant meromorphic function, let <i>k</i> be a positive integer, and let <i>a</i>,&#xa0;<i>b</i>,&#xa0;<i>c</i> be three distinct small functions such that none of them is identical to <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1717_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation>. If <i>f</i> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1717_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(f^{(k)}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> share <i>a</i>, <i>b</i> and <i>c</i> IM, then <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1717_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">\(f\equiv f^{(k)}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>≡</mo> <msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>. This improves a result due to Frank and Schwick (Results Math 22:679–684, 1992).</p>

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Meromorphic functions that share three small functions with their kth order derivatives

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  • 发表日期：

    2025年3月9日

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  • Xiaohuang Huang

摘要

In this paper, we study the uniqueness of meromorphic functions sharing small functions with their kth order derivatives. We prove: let f be a non-constant meromorphic function, let k be a positive integer, and let a, b, c be three distinct small functions such that none of them is identical to \(\infty \) ∞ . If f and \(f^{(k)}\) f ( k ) share a, b and c IM, then \(f\equiv f^{(k)}\) f ≡ f ( k ) . This improves a result due to Frank and Schwick (Results Math 22:679–684, 1992).