Bank-Laine functions, the Liouville transformation and the Eremenko-Lyubich class

TL;DR

This paper investigates conditions under which the Bank-Laine conjecture holds for second order linear differential equations, establishing bounds on the order of Bank-Laine functions with specific zero distributions and providing a counterexample.

Contribution

It proves the conjecture is valid when the coefficient's critical and asymptotic values are bounded and constructs a sharp example for zero distribution constraints.

Findings

The conjecture holds if the set of critical and asymptotic values is bounded.

A Bank-Laine function with all positive real zeros has order at least 3/2.

An example with zeros all real and positive is constructed via quasiconformal surgery.

Abstract

The Bank-Laine conjecture concerning the oscillation of solutions of second order homogeneous linear differential equations has recently been disproved by Bergweiler and Eremenko. It is shown here, however, that the conjecture is true if the set of finite critical and asymptotic values of the coefficient function is bounded. It is also shown that a Bank-Laine function with infinitely many zeros, all real and positive, must have order at least $3/2$, and an example is constructed via quasiconformal surgery to demonstrate that this result is sharp.