2025/05/14 by Zemirni, Mohamed Amine, Latreuch, Zinelaabidine
#30D35 #34M05 30D35 34M05 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2505.09143
In this paper, we study nonlinear differential equations of Tumura-Clunie type, fn + P(z, f) = h, where \( n ≥ 2 \) is an integer, \( P(z, f) \) is a differential polynomial in \( f \) of degree \( γP ≤ n - 1 \) with small functions as coefficients, and \( h \) is a meromorphic function. Assuming that h satisfies a linear differential equation of order p≤ 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.