A Note on Location of the Zeros of Quaternionic Polynomials
DOI:
https://doi.org/10.52737/10.52737/18291163-2023.15.7-1-12Keywords:
Quaternionic Polynomial, Zeros, Eneström-Kakeya theoremAbstract
The purpose of this paper is to investigate the extensions of the classical Eneström-Kakeya theorem and its various generalizations concerning the distribution of zeros of polynomials from the complex to the quaternionic setting. Using a maximum modulus theorem and the zero set structure in the recently published theory of regular functions and polynomials of a quaternionic variable, we construct new bounds of the Eneström-Kakeya type for the zeros of these polynomials. The obtained results for this subclass of polynomials and slice regular functions give generalizations of a number of results previously reported in the relevant literature.
References
A. Aziz and B.A. Zargar, Some extension of the Eneström-Kakeya theorem. Glasńik Mathematiki, 31 (1996), pp. 239-244.
N. Carney, R. Gardner, R. Keaton and A. Powers, The Eneström-Kakeya theorem of a quaternionic variable. J. Approx. Theory, 250 (2020), pp. 105-325. https://doi.org/10.1016/j.jat.2019.105325
A. Cauchy, Exercises de mathématique. Oeuvres, 9 (1829), pp. 122.
G. Eneström, Remarque sur un théoréme relatif aux racines de I'équation $a_n x^n + … + a_0 = 0$ oú tous les coefficients sont réels et positifs, Tôhoku Math. J., (1920), pp. 34-36.
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. Math., 216 (2007), no. 1, pp. 279-301. https://doi.org/10.1016/j.aim.2007.05.010
N.K. Govil and Q.I. Rahman, On the Eneström-Kakeya theorem, Tôhoku Math. J., 20 (1920), pp. 126-136.
A. Hussain, A Note on Eneström-Kakeya theorem for quaternionic polynomial, Korean J. Math., 30 (2022), pp. 503-513.
A. Hussain, Bounds for the zeros of a quaterionic polynomial with restricted coefficients. Int. J. Non linear. Anal., 14 (2023), pp. 1809-1816.
A. Joyal, G. Labelle and Q.I. Rahman, On the location of zeros of polynomials, Can. Math. Bull., 10 (1967), pp. 53-63.
M. Marden, Geometry of polynomials. Math. Surveys, 3, 1966.
G.V. Milovanović, D.S. Mitrinović and Th.M. Rassias, Topics in polynomials: extremal problems, inequalities, zeros, World Scientific publishing Co., Singapore, 1994. https://doi.org/10.1142/1284
W.M. Shah and A. Liman, On Eneström Kakeya theorem and related analytic functions. Proc. Math. Sci., 117 (2007), pp. 359-370. https://doi.org/10.1007/s12044-007-0031-z
D. Tripathi, A note on Eneström-Kakeya theorem for a polynomial with quaternionic variable. Arab. J. Math., 9 (2020), pp. 707-714. https://doi.org/10.1007/s40065-020-00283-0
I.A. Wani, M.I. Mir and I. Nazir, Location of zeros of Lacunary-type polynomials in annular regions. J. Classical Anal., 20 (2022), no. 2, pp. 103-116. https://doi.org/10.7153/jca-2022-20-08
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