Bank-Laine functions with preassigned number of zeros

A Bank–Laine function $E$ is written as $E=f_1{f}_2$ for two normalized solutions $f_1$ and $f_2$ of the second order differential equation $f^{\mathrm{\prime }\mathrm{\prime }}+Af=0$, where $A$ is an entire function. In this paper, we first complete the construction of Bank–Laine functions by Bergweiler and Eremenko. Then, letting $n\in \mathbb{N}$ be a positive integer, we show the existence of entire functions $A$ for which the associated Bank–Laine functions $E=f_1{f}_2$ have preassigned exponent of convergence of number of zeros $\lambda (E)$ of three types: (1) for every two numbers ${\lambda }_1,{\lambda }_2\in [0,n]$ such that ${\lambda }_1\le {\lambda }_2$, there exists an entire function $A$ of order $\rho (A)=n$ such that $E=f_1{f}_2$ satisfies $\lambda (f_1)={\lambda }_1$, $\lambda (f_2)={\lambda }_2$ and $\lambda (E)={\lambda }_2\le \rho (E)=n$; (2) for every number $\rho \in (n\mathrm{/}2,n)$ and $\lambda \in [0,\mathrm{\infty })$, there exists an entire function $A$ of order $\rho (A)=\rho$ such that $E=f_1{f}_2$ satisfies $\lambda (f_1)=\lambda$, $\lambda (f_2)=\mathrm{\infty }$ and, moreover, $E_c=f_1(c{f}_1+f_2)$ satisfies $\lambda (E_c)=\mathrm{\infty }$ for any constant $c$; (3) for every number $\lambda \in [0,n]$, there exists an entire function $A$ of order $\rho (A)=n$ such that $E=f_1{f}_2$ satisfies $\lambda (f_1)=\lambda$, $\lambda (f_2)=\mathrm{\infty }$ and, moreover, $E_c=f_1(c{f}_1+f_2)$ satisfies $\lambda (E_c)=\mathrm{\infty }$ for any constant $c$. The construction for the three types of Bank–Laine functions requires new developments of the method of quasiconformal surgery by Bergweiler and Eremenko.