A Type of Eneström-Kakeya Theorem for Quaternionic Polynomials Involving Monotonicity with a Reversal

Robert B. Gardner; Matthew Gladin

doi:10.52737/18291163-2025.17.4-1-10

Authors

Robert B. Gardner East Tennessee State University
Matthew Gladin East Tennessee State University

DOI:

https://doi.org/10.52737/18291163-2025.17.4-1-10

Keywords:

Location of Zeros of a Polynomial, Quaternionic Polynomial, Monotone Coefficients

Abstract

The Eneström-Kakeya theorem states that if $P(z)=\sum_{\ell =0}^n a_\ell z^\ell$ is a polynomial of degree $n$ with real coefficients satisfying $0\leq a_0\leq a_1\leq \cdots\leq a_n$, then all zeros of $P$ lie in $|z|\leq 1$ in the complex plane. Motivated by recent results concerning an Eneström-Kakeya "type" condition on the real and imaginary parts of complex coefficients, we give similar results with hypotheses concerning the real and imaginary parts of the coefficients of a quaternionic polynomial. We give bounds on the moduli of quaternionic zeros of such polynomials.

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