Abstract
Let G ∈ ℂ be a Jordan domain and P a polynomial of degree at most n, satisfying ¦P(z)¦ ≤1 for z ∈ G and P(z(in1)) = 0, where z 1 ∈ ∂G. If G is bounded by a quasiconformal curve we asymptotically estimate ¦P(z 0)¦, where z 0 ∈ G. In case z 1 is on a sufficiently smooth portion of ∂G, our results correspond to the previous ones by Halász for the special case G = D, the unit disk. We also obtain complete results in case z 1 corresponds to a corner of ∂G.
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Dedicated to Professor Walter Hayman, on the occasion of his 80th birthday
The work of V.A. was supported in part by NSF grant DMS-0554344 and was conducted while visiting the Würzburg University. S.R. received partial support from FONDECYT (grants 1040366 and 7040069), DGIP-UTFSM (grant 240104), and from GIF (grant G-809-234.6/2003).
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Andrievskii, V.V., Ruscheweyh, S. On Polynomials with a Prescribed Zero on a Quasicircle. Comput. Methods Funct. Theory 8, 243–259 (2008). https://doi.org/10.1007/BF03321686
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DOI: https://doi.org/10.1007/BF03321686