Second order differential equations with transcendental coefficients

Abstract

ABSTRACT. Let wx and W2 be two linearly independent solutions to w" + Aw = 0, where A is a transcendental entire function of order p{A) < 1. We show that the exponent of convergence \{E) of the zeros of E = wxw¿ is either infinite or satisfies p(A)_1 + \{E)~X < 2. For p{A) = i, this answers a question of Bank.

Finite order solutions of second order linear differential equations

“…Even though at least one of every two linearly independent solutions of equation (1.1) See, for example, [3,4,5,6,7,8,17,21,22]. This paper is organized as follows.…”

On the growth of solutions of $f”+gf’+hf=0$

“…Recently Rossi [15] considered the differential equation (5.1) w" + Aw = 0, where A is entire of order p(A) < 1. If wx and w2 are linearly independent solutions of (5.1) and E = wxw2, then in [15] it is shown using harmonic measure estimates that the exponent of convergence X(E) of the zero set of E is infinite if p(A) < \. We indicate a second proof of this fact using the techniques of this paper.…”

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

“…The BL-conjecture was verified in [3] in the case ρ(A) < 1/2 by means of WimanValiron theory and the cos πρ-theorem. The case ρ(A) = 1/2 was proved independently by Rossi [24] and Shen [27]. The method in [24] is based on the Beurling-Tsuji estimate for harmonic measure, while the method in [27] relies on the Carleman integral inequality.…”

“…The case ρ(A) = 1/2 was proved independently by Rossi [24] and Shen [27]. The method in [24] is based on the Beurling-Tsuji estimate for harmonic measure, while the method in [27] relies on the Carleman integral inequality. We note that the BL-conjecture still remains unsolved [18].…”