International Journal of Nonlinear Analysis and Applications

Some extensions of the Enestrom-Kakeya theorem for quaternionic polynomials

Articles in Press

Document Type : Research Paper

Authors

Department of Mathematics, National Institute of Technology, Hazratbal Srinagar-190006, Jammu and Kashmir, India

Abstract

In this paper, we are interested in finding regions of some quaternionic polynomials with restricted coefficients. We relax the hypothesis and put less restrictive conditions on the coefficients of the quaternionic polynomial, and thereby generalize some known results by virtue of a maximum modulus theorem. The obtained results for the class of quaternionic polynomials generalize some recently proved results about the distribution of zeros of polynomials.

Keywords

[1] A. Aziz and Q.G. Mohammad, Zero free regions for polynomials and some generalizations of Enestrom-Kakeya theorem, Canad. Math. Bull. 27 (1984), 265–272.
[2] A. Aziz and B.A. Zargar, Some extension of Enestrom-Kakeya theorem, Glasnik. Math. 31 (1996), 239–244.
[3] L. Brand, The roots of a quaternion, Amer. Math. Monthly 49 (1942), 519–520.
[4] N. Carney, R. Gardner, R. Keaton, and A. Powers, The Enestrom-Kakeya theorem of a quaternionic variable, J. Appl. Theory 250 (2020), Article 105325.
[5] G. Enestrom, Remarque sur un theoreme relatif aux racines de l'equation anxn+⋯+a0=0 ou tous les coefficients a sont reels et positifs, Tohoku. Math. J. 18 (1920), 34–36.
[6] C.F. Gauss, Beitrage zur theorie der algebraischen gleichungen, Abh. Ges. Wiss. Gottingen 4 (1850), 73–102.
[7] G. Gentili and C. Stoppato, Zeros of regular functions and polynomials of a quaternionic variable, Michigan. Math. 56 (2008), 655–667.
[8] G. Gentili and C.D. Struppa, A new theory of regular functions of a quaternionic variable, Adv. Math. 216 (2017), 279–301.
[9] N.K. Govil and Q.I. Rahman, On the Enestrom-Kakeya theorem, Tohoku Math. J. 20 (1968), 126–136.
[10] A. Hurwitz, Uber einen Satz des Harrn Kakeya, Tohoku Math. J. First Ser. 4 (1913–1914), 626–631.
[11] A. Joyal, G. Labelle and QI. Rahman, On the location of zeros of polynomials, Can. Math. Bull. 10 (1967), 53–63.
[12] S. Kakeya, On the limits of the roots of an algebraic equation with positive coefficient, Tohoku Math. J. 2 (1912–1913), 140–142.
[13] M. Marden, Geometry of polynomials, Math. Surveys, Amer. Math. Soc. 3 (1949).
[14] D. Tripathi, A note on Enestrom-Kakeya theorem for a polynomial with quaternionic variable, Arab. J. Math. 9 (2020), 707–714.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 12 November 2025
  • Receive Date: 21 August 2024
  • Revise Date: 29 August 2024
  • Accept Date: 19 November 2024