2025/08/08 by Carlson, Japheth
Mathematics · #30C15 #31A05 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2508.06721
openalex publication_date 2025/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Rouché's Theorem is among the most useful results in complex analysis for counting zeros of analytic functions. Rouché's Theorem also admits a harmonic analogue for counting zeros of complex harmonic functions. Previously, this analogue has been applied primarily to closed curves of simple geometry, such as circles, to count zeros. We demonstrate that non-circular critical curves can serve as effective contours by applying a harmonic Rouché-type argument to determine the total number of zeros of the complex harmonic family given by f(z) = zn + azk + bzk - 1 , where n>k≥1 and a,b > 0. Under explicit inequalities relating a and b, we determine the total number of zeros is either n or n+2k (counted with multiplicity). We also prove the zeros of f are confined to the union of two explicit annuli in the plane: an inner annulus containing k zeros and an outer annulus containing the remainder.