Quaternionic Generalization of the Eneström-Kakeya Theorem

TL;DR

This work extends Eneström–Kakeya–type zero-location results to quaternionic polynomials using the regular-symbol framework and the star-product. It introduces the sparse-exponent class $\mathbb{P}_n$ and proves explicit radius bounds for all zeros in terms of coefficient magnitudes $|a_{n_j}|$, angular constraints on coefficients via $|\arg a_{n_j}-\beta|\le\alpha$, and auxiliary quantities $M_{n_j}$. A further generalization incorporates quaternionic coefficient components, yielding bounds that depend on component-wise monotone patterns, such as $\alpha_n$, $\beta_n$, $\gamma_n$, and $\delta_n$. These results generalize classical EK-type theorems to quaternionic polynomials, leveraging regular-function theory, maximum modulus, and convolution-product zero-set descriptions, with potential applications in areas requiring localization of quaternionic polynomial zeros.

Abstract

{In 2020, Carney et.al. proved the quaternionic version of the Eneström-Kakeya Theorem, which states that a polynomial $p(q)=\sum_{ν=0}^n q^νa_ν$ with non-negative and monotonically increasing coefficients $(0<a_0\le a_1\le \cdots \le a_n)$ has all of its zeros within the unit ball $|q|\le 1$. Numerous generalizations of Eneström-Kakeya Theorem are available in the literatures (\cite{m}-\cite{mmr}). In this paper, we extend some of these generalizations to the quaternionic context and present several potential results.}

Theorems & Definitions (18)

• Theorem 1.1
• Definition 1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• Theorem 2.2
• Lemma 3.1
• proof