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Further Strengthening of Rolle’s Theorem for Complex Polynomials

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Abstract

A domain \(\Theta _n\) is called a Rolle’s domain if every complex polynomial p of degree n, satisfying \(p(i)=p(-i)\), has at least one critical point in it. In this paper, we find the smallest possible Rolle’s domain made up of two closed disks that are symmetric with respect to the real and the imaginary axes. This is a strengthening of the main result in Sendov and Sendov (Proc Am Math Soc 146(8):3367–3380, 2018).

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References

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Acknowledgements

We are grateful for the advice of two anonymous referees who helped improve the manuscript. We are thankful to Aletta Jooste for her comments on Proposition 1.

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Authors and Affiliations

  1. Bulgarian Academy of Sciences, Institute of Information and Communication Technologies, Acad. G. Bonchev Str., bl. 25A, 1113, Sofia, Bulgaria

    Blagovest Sendov

  2. Department of Statistical and Actuarial Sciences, Western University, 1151 Richmond Str., London, ON, N6A 5B7, Canada

    Hristo Sendov

Authors
  1. Blagovest Sendov
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  2. Hristo Sendov
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Corresponding author

Correspondence to Hristo Sendov.

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Communicated by Doron S. Lubinsky.

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Blagovest Sendov—deceased.

B. Sendov was partially supported by Bulgarian National Science Fund #DTK 02/44. H. Sendov was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada.

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Sendov, B., Sendov, H. Further Strengthening of Rolle’s Theorem for Complex Polynomials. Constr Approx 52, 341–356 (2020). https://doi.org/10.1007/s00365-019-09483-0

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  • DOI: https://doi.org/10.1007/s00365-019-09483-0

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