Estimates on the growth of meromorphic solutions of linear differential equations

Abstract

We give a pointwise estimate of meromorphic solutions of linear differential equations with coefficients meromorphic in a finite disk or in the open plane. Our results improve some earlier estimates of Bank and Laine. In particular we show that the growth of meromorphic solutions with δ(∞)>0 can be estimated in terms of initial conditions of the solution at or near the origin and the characteristic functions of the coefficients. Examples show that the estimates are sharp in a certain sense. Our results give an affirmative answer to a question of Milne Anderson.

Our method consists of two steps. In Theorem 2.1 we construct a path Γ(θ₀, ρ, t) consisting of the ray $z=\tau e^{i\theta_0},\quad \rho\le\tau\le t,$ followed by the circle $z=t e^{i\theta},\quad \theta_0\le \theta\le \theta_0+2\pi,$ on which the coefficients are all bounded in terms of the sum of their characteristic functions on a larger circle. In Theorem 2.2 we show how such an estimate for the coefficients leads to a corresponding bound for the solution on ∣z∣ = t. Putting these two steps together we obtain our main result, Theorem 2.3.