Generalization of an Eneström-Kakeya type theorem to the quaternions

Robert B. Gardner; Mariah Taylor

doi:10.52737/18291163-2022.14.9-1-8

Authors

Robert B. Gardner East Tennessee State University
Mariah Taylor East Tennessee State University

DOI:

https://doi.org/10.52737/18291163-2022.14.9-1-8

Keywords:

Location of Zeros of a Polynomial, Eneström-Kakeya Theorem, Quaternionic Polynomial

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.

References

A. Aziz and Q. G. Mohammad, On the zeros of a certain class of polynomials and related analytic functions. J. Math. Anal. Appl., 75 (1980), no. 2, pp. 495-502. https://doi.org/10.1016/0022-247x(80)90097-9

N. Carney, R. Gardner, R. Keaton, A. Powers, The Eneström-Kakeya theorem for polynomials of a quaternionic variable. J. Approx. Theory, 250 (2020), Article 105325, 10 pp. https://doi.org/10.1016/j.jat.2019.105325

G. Eneström, Härledning af en allmän formel för antalet pensionärer, som vid en godtyeklig tidpunkt förefinnas inom en sluten pensionslcassa. Övfers. Vetensk.-Akad. Fórhh., 50 (1893), pp. 405-415.

R. Gardner and N. K. Govil, On the location of the zeros of a polynomial. J. Approx. Theory, 78 (1994), no. 2, pp. 286-292. https://doi.org/10.1006/jath.1994.1078

R. Gardner and N. K. Govil, The Eneström-Kakeya theorem and some of its generalizations, in Current topics in pure and computational complex analysis, eds. S. Joshi, M. Dorff, and I. Lahiri, New Delhi: Springer-Verlag (2014), pp. 171-200. https://doi.org/10.1007/978-81-322-2113-5_8

G. Gentili and C. Stoppato, Zeros of regular functions and polynomials of a quaternionic variable. Michigan Math. J., 56 (2008), no. 3, pp. 655-667. https://doi.org/10.1307/mmj/1231770366

G. Gentili and D. Struppa, A new theory of regular functions of a quaternionic variable, Adv. in Math., 216 (2007), no. 1, pp. 279-301. https://doi.org/10.1016/j.aim.2007.05.010

S. Kakeya, On the limits of the roots of an algebraic equation with positive coefficients. Tôhoku Math. J. First Ser., 2 (1912-1913), pp. 140-142.

T. Tam, A first course in noncommutative rings, Graduate Texts in Mathematics, vol. 123, Springer-Verlag, 1991.

E. C. Titchmarsh, The theory of functions, 2nd ed., Oxford Univ. Press, London, 1939.