Research Article Location of the Zeros of Certain Complex-Valued Harmonic Polynomials Hunduma Legesse Geleta and Oluma Ararso Alemu Department of Mathematics, Addis Ababa University, Addis Ababa, Ethiopia Correspondence should be addressed to Hunduma Legesse Geleta; hunduma.legesse@aau.edu.et Received 28 March 2022; Accepted 18 July 2022; Published 11 August 2022 Academic Editor: Firdous A. Shah Copyright © 2022 Hunduma Legesse Geleta and Oluma Ararso Alemu. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Finding the approximate region containing all the zeros of analytic polynomials is a well-studied problem. But the number of the zeros and regions containing all the zeros of complex-valued harmonic polynomials is relatively a fresh research area. It is well known that all the zeros of analytic trinomials are enclosed in some annular sectors that take into account the magnitude of the coefficients. Following Kennedy and Dehmer, we provide the zero inclusion regions of all the zeros of complex-valued harmonic polynomials in general, and in particular, we bound all the zeros of some families of harmonic trinomials in a certain annular region. 1. Introduction e location of the zeros of analytic polynomials has been studied by many researchers (see Brilleslyper and Schaubroeck [1], Dhemer [2], Frank [3], Gilewicz and Leopold [4], Howell and Kyle [5], Johnson and Tucker [6], Kennedy [7], and Melman [8]). Recently, researching the number of the zeros of general analytic trinomials and the regions in which the zeros are located has become of interest due to their application in other fields. For example, the roots of the trinomial equations can be interpreted as the equilibrium points of unit masses that are located at the vertices of two regular concentric polygons centered at the origin in the complex plane [9]. A useful result in determining the location of the zeros of analytic polynomials is the argument principle. Recall that if f is analytic inside and on positively oriented rectifiable Jordan curve C and f(z) ≠ 0 on C, then the winding number of the image curve f(C) about the origin (1/2π)Δ C argf(z) equals the total number of zeros of f in D, counted according to multiplicity where D is a plane domain bounded by C. In 1940, Kennedy [7] showed that the roots of analytic trinomial equations of the form z n + az k + b  0, where ab ≠ 0, have certain bounds to their respective absolute values. In 2006, Dehmer [2] proved that all the zeros of complex-valued analytic polynomials lie in certain closed disk. In 2012, Melman [8] investigated the regions in which the zeros of analytic trinomials of the formp(z) z n − αz k − 1, with integers n ≥ 3 and 1 ≤ k ≤ n − 1withgcd(k, n) 1, andα ∈ C, lie. He determined the zero inclusion regions for the following cases: (a for any value of |α|; (b) |α| > σ (n, k); and (c |α| < σ n (), k (), where σ (n, k) n/n − k(n − k/k) k/n . In each case, Melman provided useful information on the location of the zeros of p. One area of investigation that has recently become of interest is the number and location of zeros of complex- valued harmonic polynomials. In 1984, Clunie and Sheil- Small [10] introduced the family of complex-valued har- monic functions f  u + iv defined in the unit disk D  z: |z| < 1 { }, where u and v are real harmonic in D.A continuous function f  u + iv defined in a domain G ⊂ C is harmonic in G if u and v are real harmonic in G. In any simply connected subdomain of G, we can decompose f as f  h + g, where g and h are analytic and g denotes the function z↦ g(z) (see Duren [11]). is family of complex- valued harmonic functions is a generalization of analytic mappings studied in geometric function theory, and much research has been done investigating the properties of these harmonic functions. For an overview of the topic, see Duren [12] and Dorff and Rolf [13]. It was shown by Bshouty et al. Hindawi Journal of Mathematics Volume 2022, Article ID 4886522, 5 pages https://doi.org/10.1155/2022/4886522