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Zeros of Convex Combinations of Elementary Families of Harmonic Functions

Author

Listed:
  • Jennifer Brooks

    (Department of Mathematics, Brigham Young University, Provo, UT 84602, USA)

  • Megan Dixon

    (Department of Mathematics, Brigham Young University, Provo, UT 84602, USA)

  • Michael Dorff

    (Department of Mathematics, Brigham Young University, Provo, UT 84602, USA)

  • Alexander Lee

    (Department of Mathematics, Brigham Young University, Provo, UT 84602, USA)

  • Rebekah Ottinger

    (Department of Mathematics, Brigham Young University, Provo, UT 84602, USA)

Abstract

Brilleslyper et al. investigated how the number of zeros of a one-parameter family of harmonic trinomials varies with a real parameter. Brooks and Lee obtained a similar theorem for an analogous family of harmonic trinomials with poles. In this paper, we investigate the number of zeros of convex combinations of members of these families and show that it is possible for a convex combination of two members of a family to have more zeros than either of its constituent parts. Our main tool to prove these results is the harmonic analog of Rouché’s theorem.

Suggested Citation

  • Jennifer Brooks & Megan Dixon & Michael Dorff & Alexander Lee & Rebekah Ottinger, 2023. "Zeros of Convex Combinations of Elementary Families of Harmonic Functions," Mathematics, MDPI, vol. 11(19), pages 1-14, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:19:p:4057-:d:1246923
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    References listed on IDEAS

    as
    1. Hunduma Legesse Geleta & Oluma Ararso Alemu & Firdous A. Shah, 2022. "Location of the Zeros of Certain Complex-Valued Harmonic Polynomials," Journal of Mathematics, Hindawi, vol. 2022, pages 1-5, August.
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      Keywords

      harmonic; polynomials; zeros;
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