Nonvanishing derivatives and normal families
Suppose \(f\) is a meromorphic function in the complex plane and \(k\geq 2\). Suppose \(f\) and \(f^{(k)}\) have no zeros. Then \textit{G. Frank} proved for \(k\geq 3\) [Math. Z. 149, 29--36 (1976; Zbl 0312.30032)] and \textit{J. Langley} proved for \(k= 2\) [J. Lond. Math. Soc., II. Ser. 48, No. 3, 500--514 (1993; Zbl 0789.30020)] that \(f(z)= \exp(az + b)\) or \(f(z)= (az+ b)^{-n}\) where \(a,b\in\mathbb{C}\), \(a\neq 0\) and \(n\) is a positive integer. The normal family analogue according to Bloch's principle restricted to entire functions is the following result of \textit{W. Schwick} [J. Anal. Math. 52, 241--289 (1989; Zbl 0667.30028)]: If \(k\geq 2\) and \(F\) is a family of functions holomorphic in a domain \(\Omega\) such that \(f\) and \(f^{(k)}\) have no zeros in \(\Omega\) for all \(f\) in \(F\), then \(S=\{f'/f|f\in F\}\) is a normal family. In the paper under review the authors obtain considerable generalizations of both of these theorems for which, as a corollary, the latter theorem is shown to be true when holomorphic is replaced by meromorphic (the case \(k= 2\) was already known by \textit{W. Bergweiler} [Proc. Am. Math. Soc. 129, No. 1, 121--129 (2001; Zbl 0961.30025)]). The proofs are deep and intricate in form and include careful study of the residues of a meromorphic function \(F\) at a simple pole \(a\), where \(F\) is a function for which \(\Psi_k(F)\) has no zeros and \(\Psi_k\) is defined by \(\Psi_1(y)= y\) and \(\Psi_{k+1}(y)= y\Psi_k(y)+ (d/dz)\Psi_k(y)\).
scientific article; zbMATH DE number 2103501
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