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On Critical Values of Polynomials with Real Critical Points

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Abstract

Let f be a polynomial of degree at least 2 with f(0)=0 and f′(0)=1. Suppose that all the zeros of f′ are real. We show that there is a zero ζ of f′ such that |f(ζ)/ζ|≤2/3, and that this inequality can be taken to be strict unless f is of the form f(z)=z+cz 3.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana–Champaign, 1409 West Green Street, Urbana, IL, 61801, USA

    Aimo Hinkkanen

  2. Institute of Mathematics and Mechanics, Kazan State University, Universitetskaya 17, 420 008, Kazan, Russia

    Ilgiz Kayumov

Authors
  1. Aimo Hinkkanen
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  2. Ilgiz Kayumov
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Corresponding author

Correspondence to Aimo Hinkkanen.

Additional information

Communicated by Edward B. Saff.

This material is based upon work supported by the National Science Foundation under Grant No. 0758226. This research was also supported by the grant 07365 from the Campus Research Board of the University of Illinois at Urbana–Champaign.

I. Kayumov was supported by RFBR 08-01-00381.

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Hinkkanen, A., Kayumov, I. On Critical Values of Polynomials with Real Critical Points. Constr Approx 32, 385–392 (2010). https://doi.org/10.1007/s00365-009-9079-6

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  • DOI: https://doi.org/10.1007/s00365-009-9079-6

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