Abstract
Let \(\| \cdot\|\) be the uniform norm in the unit disk. We study the quantities \(M_n(\alpha) := \inf(\|zP(z) + \alpha\|-\alpha)\) where the infimum is taken over all polynomials \(P\) of degree \(n-1\) with \(\|P(z)\| = 1\) and \(\alpha> 0\). In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that \(\inf_{\alpha> 0} M_n(\alpha) = 1/n\). We find the exact values of \(M_n(\alpha)\) and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.
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