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

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.