On Bank-Laine type functions (Q2860878)

A Bank-Laine function is an entire function that satisfies NEWLINE\[NEWLINE f\left( z\right) =0\Longrightarrow f^{\prime }\left( z\right) \in \left\{ -1,1\right\} . NEWLINE\]NEWLINE The paper under review deals with the study of the meromorphic functions \(f\) on the plane \(\mathbb{C}\) that satisfy NEWLINE\[NEWLINE f\left( z\right) =0\Longleftrightarrow f^{\prime }\left( z\right) \in \left\{ a,b\right\} , \tag{1}NEWLINE\]NEWLINE where \(a\neq 0\), \(b\neq 0\) are two distinct values. Such functions are called Bank-Laine type functions. The first main result of this paper states as follows.NEWLINENEWLINETheorem 1. For two distinct nonzero values \(a\) and \(b\) that satisfy \( \dfrac{a}{b}\in\mathbb{N}\), there is no transcendental meromorphic function that satisfies \(\left( 1\right)\).NEWLINENEWLINE\noindent In order to prove Theorem 1, the authors study the quasi-normality of the family \(\mathcal{F}_{a,b}\left( D\right) \) which consists of all meromorphic functions \(f\) in a plane domain \(D\subset\mathbb{C}\) that satisfy \(\left( 1\right)\). Recall that a family \(\mathcal{F}\) of functions meromorphic on a plane domain \(D\subset\mathbb{C}\) is said to be quasi-normal on \(D\) if from each sequence \(\left\{ f_{n}\right\} \subset \mathcal{F}\) one can extract a subsequence \(\left\{ f_{n_{k}}\right\} \) which converges locally uniformly with respect to the spherical metric on \(D\backslash E\), where the set \(E\) (which may depend on \( \left\{ f_{n_{k}}\right\} \)) has no accumulation point in \(D\). If \(E\) can always be chosen to satisfy \(\left| E\right| \leq \alpha \), \( \mathcal{F}\) is said to be quasi-normal of order \(\alpha \) on \(D\). The second main result of this paper states as follows. NEWLINENEWLINENEWLINE Theorem 2. For two distinct nonzero values \(a\) and \(b\) that satisfy \(\dfrac{a}{b}\in\mathbb{N},\) the family \(\mathcal{F}_{a,b}\left( D\right) \) is quasi-normal on \(D\) of order 1.NEWLINENEWLINEThe results obtained are an extension of previous results obtained by the authors in [Tohoku Math. J. (2) 63, No. 2, 149--162 (2011; Zbl 1285.30018)].