Second order differential equations with transcendental coefficients

Abstract:

Let ${w_1}$ and ${w_2}$ be two linearly independent solutions to $w'' + Aw = 0$, where $A$ is a transcendental entire function of order $\rho (A) < 1$. We show that the exponent of convergence $\lambda (E)$ of the zeros of $E = {w_1}{w_2}$ is either infinite or satisfies $\rho {(A)^{ - 1}} + \lambda {(E)^{ - 1}} \leq 2$. For $\rho (A) = \tfrac {1}{2}$, this answers a question of Bank.