On the complex oscillation theory of f ″ + A (z)f = 0 where A (z) is analytic in the unit disc

Abstract

In this paper, we investigate the complex oscillation theory of the second order linear differential equationwhere the coefficient A(z) is an analytic function in the unit disc Δ = {z : |z| < 1}.

A unit disc analogue of the Bank-Laine conjecture does not hold

“…This settles one open problem stated in a recent paper by Cao and Yi [4]. Another open problem in [4] is stated as follows:…”

“…The proof requires that we compare the growth of characteristics T (r, ϕ) and T (r, e ϕ ) for a function ϕ analytic in D. This comparison, to be presented in Section 3, is closely related to the well known paper by Clunie [8] in the case of the complex plane. The main discussion yields solutions to two open problems recently stated by Cao and Yi in [4], see Section 2. Concluding remarks on zero-free solution bases are given in Section 5.…”

On the complex oscillation of meromorphic solutions of second order linear differential equations in the unit disc

“…Theorem 1.1. (See [6].) Let A be a nonadmissible analytic function in the unit disc D. Assume that f 1 and f 2 are two linearly independent solutions of Eq.…”

“…(1) in D can vanish. Using the methods of Bank and Laine [2,3], Cao and Yi [6] obtained some parallel oscillation results of analytic solutions of Eq. (1) in the unit disc D. When A is nonadmissible, they obtained the following result.…”

“…Nevanlinna theory has appeared to be a powerful tool in the field of complex differential equations on the complex plane C and in the unit disc D, see for example [2][3][4][5][6][7][8][10][11][12][13][14][15][16]. The active research of the complex oscillation theory of linear differential equations on the complex plane C was started to investigate the second order differential equation…”