Let || · || be the uniform norm in the unit disk. We study the quantities Mn (α) := inf (||zP(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 infα>0Mn (α) = 1/n. We find the exact values of Mn (α) and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.
@article{bwmeta1.element.doi-10_2478_v10062-011-0022-5,
author = {Stephan Ruscheweyh and Magdalena Wo\l oszkiewicz},
title = {Estimates for polynomials in the unit disk with varying constant terms},
journal = {Annales UMCS, Mathematica},
volume = {65},
year = {2011},
pages = {169-178},
zbl = {1253.30015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10062-011-0022-5}
}
Stephan Ruscheweyh; Magdalena Wołoszkiewicz. Estimates for polynomials in the unit disk with varying constant terms. Annales UMCS, Mathematica, Tome 65 (2011) pp. 169-178. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10062-011-0022-5/
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