Generalization of an Eneström-Kakeya type theorem to the quaternions

Abstract

The well-known Eneström-Kakeya theorem states that polynomial $p(z)=\sum_{\nu =0}^n a_\nu z^\nu$, where $0\leq a_0\leq a_1\leq \cdots\leq a_n$, has all of its (complex) zeros in $|z|\leq 1$. Many generalizations of this result exist in the literature. In this paper, we extend one such result to the quaternionic setting and state one of the possible corollaries.

On the Eneström–Kakeya theorem and its various forms in the quaternionic setting

“…Very recently, Gardner and Taylor [6] established the following more general result giving a ring-shaped region containing all the zeros of a quaternionic polynomial with restricted real and imaginary components. As a consequence, it gives Theorem 1.8 and many related results as special cases.…”

“…The author would like thank the anonymous referee for carefully reading the manuscript, pointing out reference [6] and making certain useful suggestions.…”