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On the minimum moduli of normalized polynomials with two prescribed values

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Abstract

WithP n denoting the set of complex polynomials of degree at mostn (n≥1), define, for any complex numberμ, the subset

$$P_n (\mu ): = \{ p_n (z) \in P_n :p_n (0) = 1 and p_n (1) = \mu \} .$$

In this paper, we determine exactly the nonnegative quantity

$$S_n (\mu ): = \mathop {\sup \{ \min |p_n (z)|\} }\limits_{p_n \in P_n (\mu )|z| \leqslant 1} ,$$

as a function ofn andμ. For fixedn≥2, the three-dimensional surface, generated by the points (Reμ, Imμ,S n(μ)) for all complex numbersμ, has the interesting shape of a volcano.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Würzburg, Würzburg, Germany

    Stephan Ruscheweyh

  2. Institute for Computational Mathematics, Kent State University, 44242, Kent, Ohio, USA

    Richard S. Varga

Authors
  1. Stephan Ruscheweyh
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  2. Richard S. Varga
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Communicated by Peter Henrici.

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Ruscheweyh, S., Varga, R.S. On the minimum moduli of normalized polynomials with two prescribed values. Constr. Approx 2, 349–368 (1986). https://doi.org/10.1007/BF01893437

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  • DOI: https://doi.org/10.1007/BF01893437

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