Zeros of Convex Combinations of Elementary Families of Harmonic Functions

Jennifer Brooks, Megan Dixon, Michael Dorff (), Alexander Lee and Rebekah Ottinger
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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

Mathematics, 2023, vol. 11, issue 19, 1-14

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.

Keywords: harmonic; polynomials; zeros (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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