A Type of Eneström-Kakeya Theorem for Quaternionic Polynomials Involving Monotonicity with a Reversal
DOI:
https://doi.org/10.52737/18291163-2025.17.4-1-10Keywords:
Location of Zeros of a Polynomial, Quaternionic Polynomial, Monotone CoefficientsAbstract
The Eneström-Kakeya theorem states that if $P(z)=\sum_{\ell =0}^n a_\ell z^\ell$ is a polynomial of degree $n$ with real coefficients satisfying $0\leq a_0\leq a_1\leq \cdots\leq a_n$, then all zeros of $P$ lie in $|z|\leq 1$ in the complex plane. Motivated by recent results concerning an Eneström-Kakeya "type" condition on the real and imaginary parts of complex coefficients, we give similar results with hypotheses concerning the real and imaginary parts of the coefficients of a quaternionic polynomial. We give bounds on the moduli of quaternionic zeros of such polynomials.
Downloads
References
A. Aziz and B. A. Zargar, Bounds for the zeros of a polynomial with restricted coefficients. Appl. Math., 3 (2012), no. 1, pp. 30-33. DOI: https://doi.org/10.4236/am.2012.31005
N. Carney, R. Gardner, R. Keaton and A. Powers, The Eneström-Kakeya theorem for polynomials of a quaternionic variable. J. Approx. Theory, 250 (2020), Article 105325, 10 pp. DOI: https://doi.org/10.1016/j.jat.2019.105325
T. Chan and M. Malik, On Erdős-Lax theorem. Proc. Indian Acad. Sci., 92 (1983), no. 3, pp. 191-193. DOI: https://doi.org/10.1007/BF02876763
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 M. Gladin, Generalizations of the Eneström-Kakeya theorem involving weakened hypotheses. Appl. Math, 2 (2022), no. 4, pp. 687-699. DOI: https://doi.org/10.3390/appliedmath2040040
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. DOI: https://doi.org/10.1006/jath.1994.1078
R. Gardner and M. Taylor, Generalization of an Eneström-Kakeya type theorem to the quaternions. Armen. J. Math., 14 (2022), no. 9, pp. 1-8. DOI: https://doi.org/10.52737/18291163-2022.14.9-1-8
G. Gentili and D. Struppa, A new theory of regular functions of a quaternionic variable. Adv. Math., 216 (2007), no. 1, pp. 279-301. DOI: 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.
G.V. Milovanović, A. Mir and A. Ahmad, On the zeros of a quaternionic polynomial with restricted coefficients. Linear Algebra Appl., 653 (2022), pp. 231-245. DOI: https://doi.org/10.1016/j.laa.2022.08.010
A. Mir, On the zeros of a quaternionic polynomial: An extension of the Eneström-Kakeya theorem. Czech. Math. J., 73 (2023), no. 3, pp. 649-662. DOI: https://doi.org/10.21136/CMJ.2023.0097-22
A. Mir and A. Ahmad, On the Eneström-Kakeya theorem for quaternionic polynomials. Comptes Rendus Mathematique, 361 (2023), pp. 1051-1062. DOI: https://doi.org/10.5802/crmath.467
A. Mir and A. Ahmad, Estimation of bounds for the zeros of polynomials and regular functions of a quaternionic variable. Complex Anal. Oper. Theory, 18 (2024), Article number 61, pp. 1-15. DOI: https://doi.org/10.1007/s11785-024-01517-1
T. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 123, Springer-Verlag, 1991. DOI: https://doi.org/10.1007/978-1-4684-0406-7
M. Qazi, On the maximum modulus of polynomials. P. Am. Math. Soc., 115 (1992), no. 2, pp. 337-343. DOI: https://doi.org/10.1090/S0002-9939-1992-1113648-1
M.A. Shah, R. Swroop, H.M. Sofi and I. Nisar, A generalization of Eneström-Kakeya theorem and a zero free region of a polynomial. Journal of Applied Mathematics and Physics, 9 (2021), no. 6, pp. 1271-1277. DOI: https://doi.org/10.4236/jamp.2021.96087
D. Tripathi, A note on Eneström-Kakeya theorem for a polynomial with quaternionic variable. Arab. J. Math., 9 (2020), pp. 707-714. DOI: https://doi.org/10.1007/s40065-020-00283-0
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Armenian Journal of Mathematics
This work is licensed under a Creative Commons Attribution 4.0 International License.
