Zero distribution of finite order Bank--Laine functions

Abstract: It is known that a Bank-Laine function $E$ is a product of two normalized solutions of the second order differential equation $f"+Af=0$ $(\dagger)$, where $A=A(z)$ is an entire function. By using Bergweiler and Eremenko's method of constructing transcendental entire function $A(z)$ by gluing certain meromorphic functions with infinitely many times, we show that, for each $\lambda\in[1,\infty)$ and each $\delta\in[0,1]$, there exists a Bank--Laine function $E$ such that $E=f_1f_2$ with $f_1$ and $f_2$ being two entire functions such that $\lambda(f_1)=\delta\lambda$ and $\lambda(f_2)=\lambda$, respectively. We actually provide a simpler construction of the special Bank--Laine functions given by Bergweiler and Eremenko.