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Some extensions of Enestrom-Kakeya theorem for quaternionic polynomials

Abdul Liman|Shahbaz Mir
강원경기수학회 (The Kangweon-Kyungki Mathematical Society)|2022.12|한국수학논문집 (Korean Journal of Mathematics)|제30권 제4호|pp. 615-628

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키워드

Enestrom-Kakeya theorem.Quaternionic Polynomial
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초록

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.

Abstract

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.

참고문헌 (13)

  1. Brand, L., "The Roots of a Quaternion", Amer. Math. Monthly, 49(8), pp.519~520 (1942)
  2. Carney, N., "The Eestrom-Kakeya theorem of a quaternionic variable", J. Appl.Theory, 250 (2020)
  3. Dinesh, D., "A note on Enestr¨om-Kakeya theorem for a polynomial with quaternionic variable", Arab. J. Math, 9, pp.707~714 (2020)
  4. Enestr¨om, G., "Remarque sur un th´eor´eme relatif aux racines de 10´equation anx n +. . . + a0 = 0o´u tous les coefficientes a sont r´eels et positifs", Tˆohoku Math. J, 18, pp.34~36 (1920)
  5. Gauss, C. F., "Beitrage Zur Theorie der algebraischen Gleichungen", Abh.Ges.Wiss.Gottingen, 4, pp.73~102 (1850)
  6. Gentili, G., "Zeros of regular functions and polynomials of a quaternionic variable", Michigan Math. J, 56, pp.655~667 (2008)
  7. Gentili, G., "A new theory of regular functions of a quaternionic variable", Adv. Math, 216, pp.279~301 (2007)
  8. Govil, N. K., "On the Enestr¨om–Kakeya theorem", Tohoku Math. J, 20(2), pp.126~136 (1968)
  9. Hurwitz, A., "Uber einen Satz des Harrn Kakeya", Tohoku Math. J. First Ser, 4, pp.626~631 (1913)
  10. Joyal, A., "On the location of zeros of polynomials", Can. Math. Bull, 10(1), pp.53~63 (1967)
  11. Kakeya, S., "On the limits of the roots of an algebraic equation with positive coefficient", Tohoku Math. J, 2, pp.140~142 (1912)
  12. Liman, A., "A Note on a Generalisation of Enestr¨om–Kakeya Theorem for Quaternionic Polynomials", Mediterranean Journal of Mathematics, 19(4), pp.1~10 (2022)
  13. Marden, M., "Geometry of Polynomials", Amer. Math. Soc

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