A note on a problem of R. P. Boas (Q1094549)

Let p(z) be a polynomial of degree n such that p'(z)\(\neq 0\) for \(| z| <k\) \((0<k\leq 1)\). In this paper the following inequality is proved \[ \max_{| z| \leq 1}| p'(z)| \leq (n/(1+k^ n))\max_{| z| \leq 1}| p(z)|. \] This result together with the result of \textit{M. A. Malik} [J. Lond. Math. Soc. 1, 57-60 (1969; Zbl 0179.379)] gives the solution of a certain problem of R. P. Boas. R. P. Boas asked how large can \[ \max_{| z| \leq 1}| p'(z)| /\max_{| z| \leq 1}| p(z)| \] be if p(z)\(\neq 0\) for \(| z| <k\), where k is a given positive number?