August 22, 2025Open Access

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

Key Points

Zeros of polynomials with monotonic conditions on coefficients are bounded within the unit circle.
The study establishes new properties for even-indexed and odd-indexed coefficients in relation to zeros.
This analysis involves examining the real and imaginary parts of polynomial coefficients to derive results.
The findings suggest that the conditions placed may expand the applicability of classical polynomial results.

Abstract

The classical Eneström–Kakeya theorem states that an n-degree polynomial p(z)=∑k=0nakzk with real coefficients satisfying 0≤a0≤a1≤…≤an has all of its zeros in |z|≤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.

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

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Abstract

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