Estimates for polynomials in the unit disk with varying constant terms

• Volume: 65, Issue: 2, page 169-178
• ISSN: 2083-7402

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Abstract

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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.

How to cite

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Stephan Ruscheweyh, and Magdalena Wołoszkiewicz. "Estimates for polynomials in the unit disk with varying constant terms." Annales UMCS, Mathematica 65.2 (2011): 169-178. <http://eudml.org/doc/268161>.

@article{StephanRuscheweyh2011,
abstract = {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.},
author = {Stephan Ruscheweyh, Magdalena Wołoszkiewicz},
journal = {Annales UMCS, Mathematica},
keywords = {Bernstein-type inequalities for complex polynomials; maximal ranges for polynomials},
language = {eng},
number = {2},
pages = {169-178},
title = {Estimates for polynomials in the unit disk with varying constant terms},
url = {http://eudml.org/doc/268161},
volume = {65},
year = {2011},
}

TY - JOUR
AU - Stephan Ruscheweyh
AU - Magdalena Wołoszkiewicz
TI - Estimates for polynomials in the unit disk with varying constant terms
JO - Annales UMCS, Mathematica
PY - 2011
VL - 65
IS - 2
SP - 169
EP - 178
AB - 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.
LA - eng
KW - Bernstein-type inequalities for complex polynomials; maximal ranges for polynomials
UR - http://eudml.org/doc/268161
ER -

References

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1. Andrievskii, V., Ruscheweyh, S., Complex polynomials and maximal ranges: back-ground and applications, Recent progress in inequalities (Niš, 1996), Math. Appl., 430, Kluwer Acad. Publ., Dordrecht, 1998, 31-54. Zbl0898.30002
2. Córdova, A., Ruscheweyh, S., On maximal polynomial ranges in circular domains, Complex Variables Theory Appl. 10 (1988), 295-309. Zbl0658.30003
3. Córdova, A., Ruscheweyh, S., On maximal ranges of polynomial spaces in the unit disk, Constr. Approx. 5 (1989), 309-327. Zbl0675.30004
4. Fournier, R., Letac, G. and Ruscheweyh, S., Estimates for the uniform norm of complex polynomials in the unit disk, Math. Nachr. 283 (2010), 193-199.[WoS] Zbl1184.30034
5. Ruscheweyh, S., Varga, R., On the minimum moduli of normalized polynomials with two prescribed values, Constr. Approx. 2 (1986), 349-368. Zbl0602.30008

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