Estimates for polynomials in the unit disk with varying constant terms

ArticleOriginal scientific text

Title

Authors ,

Let

‖ \cdot ‖

be the uniform norm in the unit disk. We study the quantities

M_{n} (α) := inf (‖ z P (z) + α ‖ - α)

where the infimum is taken over all polynomials

P

of degree

n - 1

with

‖ P (z) ‖ = 1

and

α > 0

. In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that

inf_{α > 0} M_{n} (α) = 1 / n

. We find the exact values of

M_{n} (α)

and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.

Bernstein-type inequalities for complex polynomials, maximal ranges for polynomials

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