Counting Zeros of Complex-Valued Harmonic Functions via Rouch\'e's Theorem

Rouch\'e's Theorem is among the most useful results in complex analysis for counting zeros of analytic functions. Rouch\'e'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 Rouch\'e-type argument to determine the total number of zeros of the complex harmonic family given by $f(z) = z^n + az^k + b\overline{z}^k - 1$, where $n>k\geq1$ 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.