Schwarzian derivatives and zeros of solutions to second order linear differential equations

Abstract:

Let $A$ be entire. Suppose that there exists an unbounded quasidisk $D$ such that $A$ is sufficiently small in $D$. We prove that then any nontrivial solution to $y'' + Ay = 0$ has at most one zero in $D$. We show that if $A = Q\exp P$ where $P$ and $Q$ are polynomials, one can usually take $D$ to be an angle of opening $\pi /n$ where $n$ is the degree of $P$.