On the modulus of solutions of a first order differential equation

2024/06/30 by Zhang, Yueyang
Mathematics · #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions #Primary 34M10 #Secondary 30D35

paper · pdf · doi:10.48550/arxiv.2407.00580

openalex publication_date 2024/06/30 · openalex created_date 2024/07/05 · openalex updated_date 2026/08/03

Abstract

Let P(z)=zⁿ+a_n-2z^n-2+⋯+a₀ be a nonconstant polynomial and S(z) be a nonzero rational function and denote h(z)=S(z)e^P(z). Let θ∈(0,π/2n) be a constant and ε>0 be a small constant. It is shown that if f(z) is a solution of the first order differential equation f'(z)=h(z)f(z)+1, then there is a sequence \r_k\ such that the set E=∪_l=0^∞[r_2l,r_2l+1] has infinite logarithmic measure and for all r∈ E, † \beginsplit |f(re^iθ)|≥ (1-ε)\frac√[n]sin nθnrexp(e^{(1-ε)rncos nθ}sinε). \endsplit When h(z)=e^z, we also give a lower bound for |f(re^iθ)| for other values of r. The estimate in (†) yields that the hyper-order ς(f) of f(z) is equal to n, giving a partial answer to Brück's conjecture in uniqueness theory of meromorphic functions. An extension of the method also yields a complete description on the order of growth of entire solutions of a second order algebraic differential equation of Hayman in the autonomous case.

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