On Erdős-Lax theorem
Let \(P(z)=\sum^{n}_{\nu =0}c_{\nu}z^{\nu}\) be a complex polynomial and put \(\| P\|:=\max_{| z| =1}| P(z)|.\) The authors prove Theorem 1. If \[ P(z)=c_ 0+\sum^{n}_{\nu =\mu}c_{\nu}z^{\nu}\neq 0\quad for\quad | z| <K,\quad K\geq 1, \] then \(\| P'\| \leq n(1+K^{\mu})^{- 1}\| P\|\) holds. Here, equality holds for \(P(z)=(z^{\mu}+K^{\mu})^{n/\mu},\) where \(\mu\) divides n. Corollary. If \[ P(z)=\sum^{n-\mu}_{\nu =0}c_{\nu}z^{\nu}+c_ nz^ n\neq 0\quad for\quad | z| >k,\quad k\leq 1, \] then \(\| P'\| \geq n(1+k^{\mu})^{-1}\| P\|\) holds. Here, equality holds for \(P(z)=(z^{\mu}+k^{\mu})^{n/\mu},\) where \(\mu\) divides n. Theorem 2. If \(c_ n=1,P(z)\neq 0\) for \(| z| <1\) and \(\| P\| =2\), then \(P(z)=z^ n+c_ 0\) with \(| c_ 0| =1\).
scientific article; zbMATH DE number 3877480