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.

On the Bank–Laine conjecture

“…It was shown independently by Rossi [19] and Shen [20] that this actually holds for ρ(A) ≤ 1 2 . Bank and Laine also showed that in the case of noninteger ρ(A) we always have max{λ(w 1 ), λ(w 2 )} ≥ ρ(A), (1.2) and they gave examples of functions A of integer order for which there are solutions w 1 and w 2 both without zeros.…”

Quasiconformal surgery and linear differential equations

“…This shows that the inequality (1.5) is best possible when ρ(A) ≥ 1. Rossi [38] showed that if 1/2 ≤ ρ(A) < 1, then (1.5) can be improved to the inequality 1 ρ(A)…”

“…The main idea used in [38,41,45] is that (2.2) implies that when A is large, then E is small, except possibly in the set where E ′′ /E or E ′ /E is large, but the latter set is small by the lemma on the logarithmic derivative. We shall also use this idea, but we will need that every unbounded component of the set {z ∈ C : |A(z)| > K|z| p } actually contains a path where E tends to zero.…”

The Second Derivative of a Meromorphic Function

“…To prove that each u n has lower order at least 1/(2 − 1/ρ), assume without loss of generality that a 1 = a 2 and, for n = 1, 2 and t > 0, let θ * n (t) be the angular measure of the intersection of U n with the circle S(0, t). Proceeding as in [28,Lemma 3]…”