<p>In this paper, we study nonlinear differential equations of Tumura-Clunie type, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ f^n + P(z, f) = h, $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>f</mi> <mi>n</mi> </msup> <mo>+</mo> <mi>P</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>h</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n \geq 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> is an integer, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ P(z, f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a differential polynomial in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ f $</EquationSource> <EquationSource Format="MATHML"><math> <mi>f</mi> </math></EquationSource> </InlineEquation> of degree <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$ \gamma_P \leq n - 1 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mi>P</mi> </msub> <mo>≤</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> with small functions as coefficients, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ h $</EquationSource> <EquationSource Format="MATHML"><math> <mi>h</mi> </math></EquationSource> </InlineEquation> is a meromorphic function. Assuming that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ h $</EquationSource> <EquationSource Format="MATHML"><math> <mi>h</mi> </math></EquationSource> </InlineEquation> satisfies a linear differential equation of order <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ p\le n $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≤</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> with rational coefficients, we establish a result that classifies the meromorphic solutions <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ f $</EquationSource> <EquationSource Format="MATHML"><math> <mi>f</mi> </math></EquationSource> </InlineEquation> into two cases based on the distribution of their zeros and poles. This result is then applied to study the zeros and critical points of entire solutions to certain higher-order linear differential equations, thereby extending some known results in the literature.</p>

Tumura-Clunie differential equations with applications to linear ODEs

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发表日期：

2026年6月19日

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M. A. Zemirni,
Z. Latreuch

摘要

In this paper, we study nonlinear differential equations of Tumura-Clunie type, $ f^n + P(z, f) = h, $ $f^{n} + P (z, f) = h,$ where $n \geq 2$ $n \geq 2$ is an integer, $ P(z, f)$ $P (z, f)$ is a differential polynomial in $ f $ $f$ of degree $ \gamma_P \leq n - 1 $ $γ_{P} \leq n - 1$ with small functions as coefficients, and $ h $ $h$ is a meromorphic function. Assuming that $ h $ $h$ satisfies a linear differential equation of order $ p\le n $ $p \leq n$ with rational coefficients, we establish a result that classifies the meromorphic solutions $ f $ $f$ into two cases based on the distribution of their zeros and poles. This result is then applied to study the zeros and critical points of entire solutions to certain higher-order linear differential equations, thereby extending some known results in the literature.