A remark on distribution of zeros of solutions of linear differential equations

Abstract

The main purpose of this paper is to prove a sharp estimate of the order p(w) of a transcendental solution w in the complex plane of an n th-order linear differential equation with polynomial coefficients in terms of the distribution of its Stokes rays, under the assumption that zero is not a Nevanlinna deficient value of w . If, in addition, there are only two Stokes rays and if all the solutions of the equation have order at most p(w), then we can conclude that the coefficients of the equation are all constants.

Linear differential equations with rational coefficients

“…It follows from the proof of Lemma 1 in [4] that a ray $\arg z=\theta$ is a Stokes ray of $w$ of order $\lambda(w)$ if and only if for some $c>0$, $n(r, S, w)=cr^{\lambda_{(w)}}(1+0(1))$ , where $S$ is an arbitrary small sector containing the ray $\arg z=\theta$ . And Corollary 1 of Steinmetz [8] and Theorem 2 of Zheng [12] show that the zeros of any nontrivial solution $w$ of Eq. ( 1) are attracted to a system of its finitely many Stokes rays of order $\lambda(w)$ .…”

“…The following is well known and comes from the theory of asymptotic integration, please refer to Br\"uggemann [4] and Zheng [12] as well:…”

“…1) When $w_{j}$ has exactly two Stokes rays of order $\lambda_{j}=\lambda(w_{j})$ at $\arg z=0$ and $\pi,$ $h_{w_{j}}(\theta)$ has the form $h_{w_{j}}(\theta)=\{$ $|b_{j}|\cos(\arg b_{j}+\lambda_{j}\theta)$ , $0\leqq\theta\leqq\pi$ , $|c_{j}|\cos(\arg c_{j}+\lambda_{j}\theta)$ , $\pi<\theta<2\pi$ , (12) where $b_{j}$ and $c_{j}$ are constants and $b_{j}\neq c_{j}$ .…”

“…Moreover, for special case when the system of rays (2) is formed by the real axis, the following is also a further consequence of Theorem 1, which was listed in Br\"uggemann [4], but as pointed out in Zheng [12], Br\"uggemann's proof is incomplete. THEOREM 2.…”