Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1092_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> be a family of meromorphic functions in a domain D, and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1092_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}_k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">F</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> be a family of kth derivative functions of all <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1092_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$f\in \mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <mi mathvariant="script">F</mi> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we study normality relationships between <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1092_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1092_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}_k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">F</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation>, and obtain some normality criteria. Some applications of our results are given.

Normality relationships between two function families and their applications

期刊论文
发表日期：

2025年6月17日

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Fei Li,
Jianming Chang,
Yan Xu

摘要

Let $\mathcal {F}$ $F$ be a family of meromorphic functions in a domain D, and $\mathcal {F}_k$ $F_{k}$ be a family of kth derivative functions of all $f\in \mathcal {F}$ $f \in F$ . In this paper, we study normality relationships between $\mathcal {F}$ $F$ and $\mathcal {F}_k$ $F_{k}$ , and obtain some normality criteria. Some applications of our results are given.