On the Equivalence and Generalization of Results Related to the Enestr\(\ddot{o}\)m-Kakeya Theorem
Cherng-tiao Perng *
Norfolk State University, 700 Park Avenue, Norfolk, VA 23504, USA.
Evelyn Thomas
Norfolk State University, 700 Park Avenue, Norfolk, VA 23504, USA.
*Author to whom correspondence should be addressed.
Abstract
This note explores a variant of the Enestr\(\ddot{o}\)m-Kakeya Theorem inspired by a question from Seiichi Kirikami. We provide two proofs for the result and establish its equivalence to the strict Enestr\(\ddot{o}\)m-Kakeya Theorem (Theorem 2.1). This equivalence suggests that many existing generalizations of the theorem may be expressed in various alternative forms, potentially leading to a wider range of related results.
Keywords: Polynomials, zeros, Enestr\(\ddot{o}\)m-Kakeya Theorem, Rouche’s Theorem