Valency of Certain Complex-Valued Functions

Abstract

Oluma Ararso Alemu

The valence of a function f at a point z0 is the number of distinct,finite solutions to f(z) = z0. 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)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) = αp(z) for some constant α. Using techinques of complex dynamics used in the work Sheil-Small and Wilmshurst on the valence of harmonic polynomial, we prove that the harmonic polynomial f(z) = p(z)q(z) has the valency of m + n.

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