Über eine Differentialungleichung m-ter Ordnung im Komplexen. (On a differential equation of m-th order in the complex domain)
Let A denote the class of holomorphic functions in \(| z| <1\) and \(A_ 0\) the subclass of \(f\in A\) such that \(f(0)=0\). Denote further for \(m\in {\mathbb{N}}\) and \(p_ j\in {\mathbb{R}}\) by \[ (Lf)(z)=p_ mz^ mf^{(m)}(z)+...+p_ 1f'(z)+p_ 0f(z) \] an operator \(L: A\to A\), resp. \(L: A_ 0\to A_ 0\). The main result of this article is the following theorem: Let \(p_ j>0\) for \(j=0,1,...,m\), let \(f\in A_ 0\) and \(p_ 0\leq 2p_ 2\), if \(m\geq 3\). Denoting \(\lambda_ 1=p_ 0+p_ 1\) we get \[ | (Lf)(z)| <1\quad for\quad | z| <1\Rightarrow \lambda_ 1| f(z)| <1\quad for\quad | z| <1. \] This result solves in a sharpened form a problem proposed by Miller, see Problem 5.61 in [\textit{D. M. Campbell, J. G. Clunie} and \textit{W. K. Hayman} Aspects of contemporary complex analysis, Proc. instr. Conf., Durham/Engl. 1979, 527-572 (1980; Zbl 0497.30001)]. In the proof of the above theorem, an essential use is made of a result due to Brown and Hewitt, see Theorem A in [\textit{G. Brown} and \textit{E. Hewitt}, Math. Ann. 268, 91-122 (1984; Zbl 0522.42001)].
scientific article; zbMATH DE number 3912695