An extremal differential inequality for analytic functions
Let \(Q(z)=c_ 0+c_ 1z+...+c_ nz^ n\) be a polynomial and let \[ Q_*(z)=c_ 0+\sum^{n}_{k=1}c_ k z(z-1)...(z-k+1). \] Also, let \(Q[f]=\sum^{n}_{k=0}c_ k z^ k f^{(k)}(z)\). The main result of this paper is the following theorem. If m is a positive integer such that \(Q_*(k+m)\neq 0\) for each nonnegative integer k, and if \(f(z)=a_ mz^ m+a_{m+1}z^{m+1}+..\). is analytic in \(| z| <1\), then a necessary and sufficient condition that \[ | Q_*(m)f(z)| \leq | z|^ m \| Q[f]\|_{\infty},\quad | z| <1, \] is that \[ Re(Q_*(m)\quad \sum^{\infty}_{k=0}z^ k/Q_*(k+m))\geq,\quad | z| <1. \] A corollary to this result is that if \(f(0)=0\), \(n\geq 1\), and \[ | f(z)+zf^ 1(z)+...+z^ nf^{(n)}(z)| \leq 1, \] then \(\| f\|_{\infty}\leq\), and this result is sharp. This corollary extends a result of \textit{S. S. Miller} and \textit{P. T. Mocanu} [ibid. 65, 289-305 (1978; Zbl 0367.34005)].
scientific article; zbMATH DE number 3959851
scientific article