Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1024_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$\{f_n\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>f</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> be a sequence of meromorphic functions defined in a domain D, and let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1024_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\{\psi _n\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> be a sequence of holomorphic functions on D, whose zeros are multiple, such that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1024_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$\psi _n\rightarrow \psi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ψ</mi> <mi>n</mi> </msub> <mo stretchy="false">→</mo> <mi>ψ</mi> </mrow> </math></EquationSource> </InlineEquation> converges locally uniformly in D, where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1024_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$\psi (\not \equiv 0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mo>≢</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is holomorphic in D. If, (1) <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1024_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$f_n\ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1024_Article_IEq6.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$f_n^{(k)}\ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>f</mi> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>; (2) all zeros of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1024_Article_IEq7.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$f_n^{(k)}-\psi _n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>f</mi> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>-</mo> <msub> <mi>ψ</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> have multiplicities at least <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1024_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$(k+2)/k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1024_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$\{f_n\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>f</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> is normal in D.

Normality concerning the sequence of multiple functions

期刊论文
发表日期：

2025年2月10日

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Dongmei Wei,
Fei Li,
Yan Xu

摘要

Let $\{f_n\}$ ${f_{n}}$ be a sequence of meromorphic functions defined in a domain D, and let $\{\psi _n\}$ ${ψ_{n}}$ be a sequence of holomorphic functions on D, whose zeros are multiple, such that $\psi _n\rightarrow \psi $ $ψ_{n} \to ψ$ converges locally uniformly in D, where $\psi (\not \equiv 0)$ $ψ (≢ 0)$ is holomorphic in D. If, (1) $f_n\ne 0$ $f_{n} \neq 0$ and $f_n^{(k)}\ne 0$ $f_{n}^{(k)} \neq 0$ ; (2) all zeros of $f_n^{(k)}-\psi _n$ $f_{n}^{(k)} - ψ_{n}$ have multiplicities at least $(k+2)/k$ $(k + 2) / k$ , then $\{f_n\}$ ${f_{n}}$ is normal in D.