Zeros of Convex Combinations of Elementary Families of Harmonic Functions

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 Rouche's theorem.

Automated summary

In this paper, the authors investigated the variation of the number of zeros of a one-parameter family of harmonic trinomials with a real parameter. They also explored the number of zeros for convex combinations of members in these families. The harmonic analog of Rouche's theorem served as the main tool to prove these results.