Zero Distribution of Finite Order Bank–Laine Functions

It is known that a Bank–Laine function E is a product of two normalized solutions of the second order differential equation \(f''+Af=0\) $f^{\prime \prime }+Af=0$ \((\dag )\) $(†)$ , where \(A=A(z)\) $A=A(z)$ 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 \(\lambda \in [1,\infty )\) $\lambda \in [1,\infty )$ and each \(\delta \in [0,1]\) $\delta \in [0,1]$ , there exists a Bank–Laine function E such that \(E=f_1f_2\) $E=f_1{f}_2$ with \(f_1\) $f_1$ and \(f_2\) $f_2$ being two entire functions such that \(\lambda (f_1)=\delta \lambda \) $\lambda (f_1)=\delta \lambda$ and \(\lambda (f_2)=\lambda \) $\lambda (f_2)=\lambda$ , respectively. We actually provide a complete construction of the Bank–Laine functions given by Bergweiler and Eremenko.