On annuli containing all the zeros of a polynomial. (English) Zbl 1205.65167
Summary: We obtain the annuli that contain all the zeros of the polynomial \(p(z)=a_{0}+a_{1}z+a_{2}z^{2}+\dots +a_nz^n\), where \(a_i\)’s are complex coefficients and \(z\) is a complex variable. Our results sharpen some of the recently obtained results in this direction. Also, we develop a MATLAB code to show that for some polynomials the bounds obtained by our results are considerably sharper than the bounds obtainable from the known results.
MSC:
| 65H04 | Numerical computation of roots of polynomial equations |
| 26C10 | Real polynomials: location of zeros |
Software:
MatlabReferences:
| [1] | Marden, M., Geometry of Polynomials, (Amer. Math. Soc. Math. Surveys, vol. 3 (1966), Amer. Math. Soc: Amer. Math. Soc Providence, RI) · JFM 56.0118.08 |
| [2] | Milovanović, G. V.; Mitrinović, D. S.; Rassias, Th. M., Topics in Polynomials: Extremal Problems, Inequalities, Zeros (1994), World Scientific · Zbl 0848.26001 |
| [3] | Rahman, Q. I.; Schmeisser, G., Analytic Theory of Polynomials (2002), Oxford University Press Inc.: Oxford University Press Inc. New York · Zbl 1072.30006 |
| [4] | Cauchy, A. L., Exercises de mathématiques, IV (1829), Année de Bure Fréres: Année de Bure Fréres Paris |
| [5] | Datt, B.; Govil, N. K., On the location of zeros of a polynomial, J. Approx. Theory, 24, 78-82 (1978) · Zbl 0393.41003 |
| [6] | Joyal, A.; Labelle, G.; Rahman, Q. I., On the location of zeros of polynomials, Canad. Math. Bull., 10, 53-63 (1967) · Zbl 0152.06102 |
| [7] | Diaz-Barrero, J. L., An annulus for the zeros of polynomials, J. Math. Anal. Appl., 273, 349-352 (2002) · Zbl 1014.30005 |
| [8] | Sun, Y. J.; Hsieh, J. G., A note on circular bound of polynomial zeros, IEEE Trans. Circuits Syst. I, 43, 476-478 (1996) |
| [9] | Z˘ilović, M. S.; Roytman, L. M.; Combettes, P. L.; Swamy, M. N.S., A bound for the zeros of polynomials, IEEE Trans. Circuits Syst. I, 39, 476-478 (1992) · Zbl 0753.30004 |
| [10] | Jain, V. K., On Cauchy’s bound for zeros of a polynomial, Turkish J. Math., 30, 95-100 (2006) · Zbl 1094.30007 |
| [11] | Affane-Aji, C.; Agarwal, N.; Govil, N. K., Location of zeros of polynomials, Math. Comput. Modeling, 50, 306-313 (2009) · Zbl 1185.65084 |
| [12] | Govil, N. K.; Rahman, Q. I.; Schmeisser, G., On the derivative of a polynomial, Illinois J. Math., 23, 319-329 (1979) · Zbl 0408.30003 |
| [13] | Govil, N. K.; Jain, V. K., On the Eneström-Kakeya Theorem II, J. Approx. Theory, 22, 1-10 (1978) · Zbl 0377.30001 |
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