Tumura-Clunie Differential Equations with Applications to Linear ODE's
Mohamed Amine Zemirni, Zinelaabidine Latreuch
TL;DR
This work advances the study of nonlinear Tumura–Clunie type equations $f^n + P(z,f) = h(z)$ by assuming $h$ solves a linear differential equation of order $p \le n$ with rational coefficients and that the coefficients in $P(z,f)$ are small relative to $f$. It establishes a two-pronged classification: either $f$ has an exponential form $f = q e^{\alpha}$ with a precise relation between $T(r,h)$ and $T(r,f)$, or $f$ obeys a Nevanlinna-type growth bound in terms of counting functions of $f$ and $f f'$, with the optimal bound achieved at $j = n-\gamma_P$. These results are then applied to linear differential equations for which $h$ is a solution of a higher-order ODE, yielding detailed information on the zeros and critical points of entire solutions and providing corollaries such as a $1/16$-theorem and zero-free solution structures. The paper further translates these Tumura–Clunie findings into the setting of higher-order linear differential equations with polynomial coefficients, revealing when nontrivial solutions must take exponential-type forms or exhibit controlled growth, and it clarifies how the zeros of $f$ and $f f'$ influence the growth of the solutions. Overall, the results extend previous HLWZ-type analyses to the regime $p\le n$ with broader classes of $h$, sharpening the understanding of the zeros, poles, and growth behavior of solutions in this nonlinear–linear differential framework.
Abstract
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.