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On the minimum moduli of normalized polynomials

  • S. Ruscheweyh
  • R. S. Varga
Approximation And Interpolation Theory
Part of the Lecture Notes in Mathematics book series (LNM, volume 1105)

Abstract

Consider any complex polynomial pn(z)=1+\(\mathop \Sigma \limits_{j = l}^n\) ajzj which satisfies \(\mathop \Sigma \limits_{j = l}^n\)|aj|=1, and let Γn denote the supremum of the minimum moduli on |z|=1 of all such polynomials pn(z). We show that
$$1 - \frac{1}{n} \leqslant \Gamma _n \leqslant \sqrt {1 - \frac{1}{n}} , for all n \geqslant 1.$$
If the coefficients of pn(z) are further restricted to be positive numbers and if \(\tilde \Gamma _n\) denotes the analogous supremum of the minimum modulion |z|=1 of such polynomials, we similarly show that
$$1 - \frac{1}{n} \leqslant \tilde \Gamma _n \leqslant \sqrt {1 - \frac{3}{{\left( {2n + 1} \right)}}} , for all n \geqslant 1.$$

We also include some recent numerical experiments on the behavior of Γn, as well as some related conjectures.

Keywords

Richardson Extrapolation Minimum Modulus Multiple Precision Related Conjecture Multiplier Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • S. Ruscheweyh
    • 1
  • R. S. Varga
    • 1
  1. 1.Mathematisches InstitutUniversität WürzburgWürzburg am HublandGermany

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