Location of the Zeros of Certain Complex-Valued Harmonic Polynomials

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.

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Finding the regions containing all the zeros of complex-valued harmonic polynomials is a relatively new research area. In this article, we provide the inclusion regions of all the zeros of these polynomials in general, and specifically, we bound the zeros of certain harmonic trinomials within a certain annular region.