  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.
