An Eneström–Kakeya Theorem with Monotonicity Conditions on the Even- and Odd-Indexed Coefficients of a Polynomial

Gardner, Robert; Herrell, Luke

doi:10.3390/axioms14090651

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An Eneström–Kakeya Theorem with Monotonicity Conditions on the Even- and Odd-Indexed Coefficients of a Polynomial

by

Robert Gardner

^* and

Luke Herrell

Department of Mathematics and Statistics, East Tennessee State University, Johnson City, TN 37614, USA

^*

Author to whom correspondence should be addressed.

Axioms 2025, 14(9), 651; https://doi.org/10.3390/axioms14090651

Submission received: 21 July 2025 / Revised: 20 August 2025 / Accepted: 20 August 2025 / Published: 22 August 2025

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Abstract

The classical Eneström–Kakeya theorem states that an n-degree polynomial

p (z) = \sum_{k = 0}^{n} a_{k} z^{k}

with real coefficients satisfying

0 \leq a_{0} \leq a_{1} \leq \dots \leq a_{n}

has all of its zeros in

| z | \leq 1

in the complex plane. Numerous generalizations of this result exist, many of them weakening the condition on the coefficients in order to be applicable to a larger class of polynomials. In this paper, a monotonicity condition on the real and imaginary parts of the even- and odd-indexed coefficients is imposed and bounds on the location of the zeros are established.

Keywords: complex polynomials; location of zeros; monotone coefficients

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MDPI and ACS Style

Gardner, R.; Herrell, L. An Eneström–Kakeya Theorem with Monotonicity Conditions on the Even- and Odd-Indexed Coefficients of a Polynomial. Axioms 2025, 14, 651. https://doi.org/10.3390/axioms14090651

AMA Style

Gardner R, Herrell L. An Eneström–Kakeya Theorem with Monotonicity Conditions on the Even- and Odd-Indexed Coefficients of a Polynomial. Axioms. 2025; 14(9):651. https://doi.org/10.3390/axioms14090651

Chicago/Turabian Style

Gardner, Robert, and Luke Herrell. 2025. "An Eneström–Kakeya Theorem with Monotonicity Conditions on the Even- and Odd-Indexed Coefficients of a Polynomial" Axioms 14, no. 9: 651. https://doi.org/10.3390/axioms14090651

APA Style

Gardner, R., & Herrell, L. (2025). An Eneström–Kakeya Theorem with Monotonicity Conditions on the Even- and Odd-Indexed Coefficients of a Polynomial. Axioms, 14(9), 651. https://doi.org/10.3390/axioms14090651

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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An Eneström–Kakeya Theorem with Monotonicity Conditions on the Even- and Odd-Indexed Coefficients of a Polynomial

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