Valency of Certain Complex-valued Functions

Abstract

The valence of a function f at a point $z_0$ is the number of distinct, finite solutions to $f(z) = z_0.$ In this paper, we bound the valence of complex-valued harmonic polynomials in the plane for some special harmonic polynomials of the form $f(z) =p(z)\overline{q(z)},$ where $p(z)$ is an analytic polynomial of degree $n$ and $q(z)$ is an analytic polynomial of degree $m,$ and $q(z) \neq \alpha p(z)$ for some constant $\alpha.$ Using techniques of complex dynamics used in the work Sheil-Small and Wilmshurst on the valence of harmonic polynomials, we prove that the harmonic polynomial $f(z) = p(z)q(z)$ has the valency of $m + n.$