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Abstract <p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(k\geq 2\)</EquationSource> <!--ContMath2670004Yang-m1--> </InlineEquation> be a positive integer. Let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal{F}\)</EquationSource> <!--ContMath2670004Yang-m2--> </InlineEquation> be a family of meromorphic functions in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(D\)</EquationSource> <!--ContMath2670004Yang-m3--> </InlineEquation>, whose zeros have multiplicity at least <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(k+1\)</EquationSource> <!--ContMath2670004Yang-m4--> </InlineEquation>. Let <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(h(\not\equiv 0)\)</EquationSource> <!--ContMath2670004Yang-m5--> </InlineEquation> be a holomorphic function in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(D\)</EquationSource> <!--ContMath2670004Yang-m6--> </InlineEquation>, whose zeros are multiple. If each <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(f\in\mathcal{F}\)</EquationSource> <!--ContMath2670004Yang-m7--> </InlineEquation> and each <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(z\in D\)</EquationSource> <!--ContMath2670004Yang-m8--> </InlineEquation>, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(f^{(k)}(z)\neq h(z)\)</EquationSource> <!--ContMath2670004Yang-m9--> </InlineEquation>, then <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\mathcal{F}\)</EquationSource> <!--ContMath2670004Yang-m10--> </InlineEquation> is normal at points for which <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(h(z)=0\)</EquationSource> <!--ContMath2670004Yang-m11--> </InlineEquation>.</p>

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Normal Criterion of Meromorphic Functions Whose Derivatives Omit a Holomorphic Function

  • 期刊论文

  • 发表日期：

    2026年6月20日

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  • P. Yang,
  • L. Wu,
  • J. Yang

摘要

Abstract

Let \(k\geq 2\) be a positive integer. Let \(\mathcal{F}\) be a family of meromorphic functions in \(D\) , whose zeros have multiplicity at least \(k+1\) . Let \(h(\not\equiv 0)\) be a holomorphic function in \(D\) , whose zeros are multiple. If each \(f\in\mathcal{F}\) and each \(z\in D\) , \(f^{(k)}(z)\neq h(z)\) , then \(\mathcal{F}\) is normal at points for which \(h(z)=0\) .