Tumura-Clunie Differential Equations with Applications to Linear ODE's
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909617391403008 |
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| author | Zemirni, Mohamed Amine Latreuch, Zinelaabidine |
| author_facet | Zemirni, Mohamed Amine Latreuch, Zinelaabidine |
| contents | In this paper, we study nonlinear differential equations of Tumura-Clunie type, $ f^n + P(z, f) = h, $ where \( n \geq 2 \) is an integer, \( P(z, f) \) is a differential polynomial in \( f \) of degree \( γ_P \leq n - 1 \) with small functions as coefficients, and \( h \) is a meromorphic function. Assuming that $ h $ satisfies a linear differential equation of order $ p\le n $ with rational coefficients, we establish a result that classifies the meromorphic solutions \( f \) 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09143 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tumura-Clunie Differential Equations with Applications to Linear ODE's Zemirni, Mohamed Amine Latreuch, Zinelaabidine Complex Variables 34M05 30D35 34M05, 30D35 In this paper, we study nonlinear differential equations of Tumura-Clunie type, $ f^n + P(z, f) = h, $ where \( n \geq 2 \) is an integer, \( P(z, f) \) is a differential polynomial in \( f \) of degree \( γ_P \leq n - 1 \) with small functions as coefficients, and \( h \) is a meromorphic function. Assuming that $ h $ satisfies a linear differential equation of order $ p\le n $ with rational coefficients, we establish a result that classifies the meromorphic solutions \( f \) 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. |
| title | Tumura-Clunie Differential Equations with Applications to Linear ODE's |
| topic | Complex Variables 34M05 30D35 34M05, 30D35 |
| url | https://arxiv.org/abs/2505.09143 |