Bank-Laine functions with preassigned number of zeros

Abstract

A Bank--Laine function $E$ is written as $E=f_1f_2$ for two normalized solutions $f_1$ and $f_2$ of the second order differential equation $f''+Af=0$, 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 $E=f_1f_2$ 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 $\lambda_1\leq \lambda_2$, there exists an entire function $A$ of order $\rho(A)=n$ such that $E=f_1f_2$ satisfies $\lambda(f_1)=\lambda_1$, $\lambda(f_2)=\lambda_2$ and $\lambda(E)=\lambda_2\leq \rho(E)=n$; (2) for every number $\rho\in(n/2,n)$ and $\lambda\in[0,\infty)$, there exists an entire function $A$ of order $\rho(A)=\rho$ such that $E=f_1f_2$ satisfies $\lambda(f_1)=\lambda$, $\lambda(f_2)=\infty$ and, moreover, $E_c=f_1(cf_1+f_2)$ satisfies $\lambda(E_c)=\infty$ 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 $E=f_1f_2$ satisfies $\lambda(f_1)=\lambda$, $\lambda(f_2)=\infty$ and, moreover, $E_c=f_1(cf_1+f_2)$ satisfies $\lambda(E_c)=\infty$ 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.