Estimates for polynomials in the unit disk with varying constant terms

Stephan Ruscheweyh; Magdalena Wołoszkiewicz

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2011)

  • Volume: 65, Issue: 2
  • ISSN: 0365-1029

Abstract

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Let ∥ · ∥ 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.

How to cite

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Stephan Ruscheweyh, and Magdalena Wołoszkiewicz. "Estimates for polynomials in the unit disk with varying constant terms." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 65.2 (2011): null. <http://eudml.org/doc/289725>.

@article{StephanRuscheweyh2011,
abstract = {Let $\Vert \cdot \Vert $ be the uniform norm in the unit disk. We study the quantities $M_n(\alpha ) := \inf (\Vert zP(z) + \alpha \Vert -\alpha )$ where the infimum is taken over all polynomials $P$ of degree $n-1$ with $\Vert P(z)\Vert = 1$ 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.},
author = {Stephan Ruscheweyh, Magdalena Wołoszkiewicz},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Bernstein-type inequalities for complex polynomials; maximal ranges for polynomials},
language = {eng},
number = {2},
pages = {null},
title = {Estimates for polynomials in the unit disk with varying constant terms},
url = {http://eudml.org/doc/289725},
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 Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2011
VL - 65
IS - 2
SP - null
AB - Let $\Vert \cdot \Vert $ be the uniform norm in the unit disk. We study the quantities $M_n(\alpha ) := \inf (\Vert zP(z) + \alpha \Vert -\alpha )$ where the infimum is taken over all polynomials $P$ of degree $n-1$ with $\Vert P(z)\Vert = 1$ 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.
LA - eng
KW - Bernstein-type inequalities for complex polynomials; maximal ranges for polynomials
UR - http://eudml.org/doc/289725
ER -

References

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  1. Andrievskii, V., Ruscheweyh, S., Complex polynomials and maximal ranges: background and applications, Recent progress in inequalities (Nis, 1996), Math. Appl., 
  2. 430, Kluwer Acad. Publ., Dordrecht, 1998, 31-54. 
  3. Córdova, A., Ruscheweyh, S., On maximal polynomial ranges in circular domains, Complex Variables Theory Appl. 10 (1988), 295-309. 
  4. Córdova, A., Ruscheweyh, S., On maximal ranges of polynomial spaces in the unit disk, Constr. Approx. 5 (1989), 309-327. 
  5. 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. 
  6. Ruscheweyh, S., Varga, R., On the minimum moduli of normalized polynomials with two prescribed values, Constr. Approx. 2 (1986), 349-368. 

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