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

Abstract

We study the extensions of the classical Eneström-Kakeya theorem and its various generalizations regarding the distribution of zeros of polynomials from the complex to the quaternionic setting. We aim to build upon the previous work by various authors and derive zero-free regions of some special regular functions of a quaternionic variable with restricted coefficients, namely quaternionic coefficients whose real and imaginary components or moduli of the coefficients satisfy suitable inequalities. The obtained results for this subclass of polynomials and slice regular functions produce generalizations of a number of results known in the literature on this subject.