Korean J. Math. Vol. 30 No. 4 (2022) pp.615-628
DOI: https://doi.org/10.11568/kjm.2022.30.4.615

Some extensions of Enestr¨om-Kakeya theorem for quaternionic polynomials

Main Article Content

Shahbaz Mir
Abdul Liman


In this paper, we will prove some extensions of the Enestr\"{o}m-Kakeya theorem to quaternionic polynomials which were already valid for the classical Enestr\"{o}m-Kakeya theorem to complex polynomials. Our kind of extensions have considerably improved the bounds by relaxing and weakening the hypothesis in some cases.

Article Details


[1] Brand, L., The Roots of a Quaternion, Amer. Math. Monthly, 49 (8) (1942), 519–520. Google Scholar

[2] Carney, N.; Gardner, R.; Keaton, R. and Powers, A., The Eestrom-Kakeya theorem of a quater- nionic variable, J. Appl.Theory, 250, Article 105325 (2020). Google Scholar

[3] Dinesh, D., A note on Enestr ̈om-Kakeya theorem for a polynomial with quaternionic variable, Arab. J. Math. 9 (2020), 707–714. Google Scholar

[4] Enestr ̈om, G., Remarque sur un th ́eor ́eme relatif aux racines de 1′ ́equation anxn + ... + a0 = 0 ou ́ tous les coefficientes a sont r ́eels et positifs, Tˆohoku Math. J. 18 (1920), 34–36. Google Scholar

[5] Gauss, C.F.,Beitrage Zur Theorie der algebraischen Gleichungen,Abh.Ges.Wiss.Gottingen, 4 (1850), 73–102. Google Scholar

[6] Gentili, G.; Stoppato, C., Zeros of regular functions and polynomials of a quaternionic variable, Michigan Math. J. 56 (2008), 655–667. Google Scholar

[7] Gentili, G.; Struppa, D. A new theory of regular functions of a quaternionic variable, Adv. Math. 216 (2007), 279–301. Google Scholar

[8] Govil, N.K.; Rahman, Q.I., On the Enestro ̈m–Kakeya theorem, Tohoku Math. J. 20 (2) (1968), 126–136. Google Scholar

[9] Hurwitz, A., U ̈ber einen Satz des Harrn Kakeya, Tohoku Math. J. First Ser. 4 (1913-1914), 626–631. Google Scholar

[10] Joyal, A.; Labelle, G.; Rahman, Q.I., On the location of zeros of polynomials., Can. Math. Bull. 10 (1) (1967), 53–63. Google Scholar

[11] Kakeya, S., On the limits of the roots of an algebraic equation with positive coefficient, Tohoku Math. J. 2 (1912-1913), 140–142. Google Scholar

[12] Liman, A., Hussain, S., Hussain, I., A Note on a Generalisation of Enestr ̈om–Kakeya Theorem for Quaternionic Polynomials. Mediterranean Journal of Mathematics 19 (4) (2022), 1–10. Google Scholar

[13] Marden, M., Geometry of Polynomials., Math. Surveys, No. 3, Amer. Math. Soc. Google Scholar