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)


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.


Richardson Extrapolation Minimum Modulus Multiple Precision Related Conjecture Multiplier Problem 
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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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