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Inequalities for Polynomial Zeros

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Survey on Classical Inequalities

Part of the book series: Mathematics and Its Applications ((MAIA,volume 517))

Abstract

This survey paper is devoted to inequalities for zeros of algebraic polynomials. We consider the various bounds for the moduli of the zeros, some related inequalities, as well as the location of the zeros of a polynomial, with a special emphasis on the zeros in a strip in the complex plane.

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References

  1. A. Abian, The precise annulus containing all the zeros of a polynomial, Period. Math. Hungarica 14 (1983), 107–109.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, Dover, New York, 1972.

    MATH  Google Scholar 

  3. H.E. Bell, Gershgorin’s theorem and the zeros of polynomials, Amer. Math. Monthly 72 (1965), 292–295.

    Article  MathSciNet  MATH  Google Scholar 

  4. L. Berwald, Elementare Sätze über die Abgrenzung der Wurzeln einer algebraischen Gleichung, Acta Litt. Sci. (Szeged) 6 (1934), 209–221.

    Google Scholar 

  5. B. Bojanov, On an estimation of the roots of algebraic equations, Zastos. Mat. 11 (1970), 195–205.

    MathSciNet  MATH  Google Scholar 

  6. N.G. de Bruijn, Inequalities concerning polynomials in the complex domain, Nederl. Akad. Wetensch. Proc. 50 (1947), 1265–1272 [= Indag. Math. 9 (1947), 591-598].

    MathSciNet  MATH  Google Scholar 

  7. M. G. De Bruin, Polynomials orthogonal on a circular arc, J. Comput. Appl. Math. 31 (1990), 253–266.

    Article  MathSciNet  MATH  Google Scholar 

  8. N. Bourbaki, Topologie générale, 2nd ed., Hermann, Paris 1955.

    Google Scholar 

  9. F. Calio’, M. Frontini, and G. V. Milovanović, Numerical differentiation of analytic functions using quadratures on the semicircle, Comput. Math. Appl. 22 (1991), 99–106.

    Article  MathSciNet  MATH  Google Scholar 

  10. R. D. Carmichael, and T.E. Mason, Note on the roots of algebraic equations, Bull. Amer. Math. Soc. 21 (1914), 14–22.

    Article  MathSciNet  MATH  Google Scholar 

  11. A.L. Cauchy, Exercises de mathématique, In: Euvres (2) Vol. 9 (1829), p. 122.

    Google Scholar 

  12. M.-T. Chien, On the numerical range of tridiagonal operators, Linear Algebra Appl. 246 (1996), 203–214.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Cohn, Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise, Math. Z. 14 (1922), 110–148.

    Article  MathSciNet  MATH  Google Scholar 

  14. B. Datt and N. K. Govil, On the location of the zeros of a polynomial, J. Approx. Theory 24 (1978), 78–82.

    Article  MathSciNet  MATH  Google Scholar 

  15. K.K. Dewan, On the location of zeros of polynomials, Math. Student 50 (1982), 170–175.

    MathSciNet  Google Scholar 

  16. J. Dieudonné, La théorie analytique des polynomes d’une variable (à coefficients quelcon-ques), Mémor. Sci. Math. 93, Gauthier-Villars, Paris, 1938.

    Google Scholar 

  17. M. Fujii and F. Kubo, Buzano’s inequality and bounds for roots of algebraic equations, Proc. Amer. Math. Soc. 117 (1993), 359–361.

    MathSciNet  MATH  Google Scholar 

  18. M. Fujiwara, Ueber die Wurzeln der algebraischen Gleichungen, Tohoku Math. J. 8 (1915), 78–85.

    MATH  Google Scholar 

  19. W. Gautschi, On the zeros of polynomials orthogonal on the semicircle, SIAM J. Math. Anal. 20 (1989), 738–743.

    Article  MathSciNet  MATH  Google Scholar 

  20. W. Gautschi, H. Landau, and G. V. Milovanović, Polynomials orthogonal on the semicircle, II, Constr. Approx. 3(1987), 389–404.

    Article  MathSciNet  MATH  Google Scholar 

  21. W. Gautschi and G. V. Milovanović, On a class of complex polynomials having all zeros in a half disc, In: Numerical Methods and Approximation Theory (Niš, 1984) (G.V. Milovanović, ed.), Univ. Niš, Niš, 1984, pp. 49–53.

    Google Scholar 

  22. W. Gautschi and G.V. Milovanović, Polynomials orthogonal on the semicircle, Rend. Sem. Mat. Univ. Politec. Torino 1985, 179–185 [Special Issue (Special Functions: Theory and Computation)].

    Google Scholar 

  23. W. Gautschi and G.V. Milovanović, Polynomials orthogonal on the semicircle, J. Approx. Theory 46(1986), 230–250.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Giroux, Estimates for the imaginary parts of the zeros of a polynomial, Proc. Amer. Math. Soc. 44 (1974), 61–67.

    Article  MathSciNet  MATH  Google Scholar 

  25. C. D. Godsil and I. Gutman, On the theory of the matching polynomial, J. Graph Theory 5 (1981) 137–144.

    Article  MathSciNet  MATH  Google Scholar 

  26. E. M. Gol’berg and V. N. Malozemov, Estimates for roots of some polynomials, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1973, no 7, 18–24 (Russian).

    MathSciNet  Google Scholar 

  27. A. Hurwitz, Ueber die Nullstellen der Bessel’schen Function, Math. Ann. 33 (1889), 246–266.

    Article  MATH  Google Scholar 

  28. A. Joyal, G. Labelle, and Q.I. Rahman, On the location of zeros of polynomials, Canad. Math. Bull. 10 (1967), 53–63.

    Article  MathSciNet  MATH  Google Scholar 

  29. S.B. Kelleher, Des limites des zéros d’un polynome, J. Math. Pures Appl. 2 (1916), 169–171.

    Google Scholar 

  30. F. Kittaneh, Singular values of companion matrices and bounds on zeros of polynomials, SIAM J. Matrix Anal. Appl. 16 (1995), 333–340.

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Kuniyeda, Note on the roots of algebraic equations, Tohoku Math. J. 9 (1916), 167–173.

    MATH  Google Scholar 

  32. M. Kuniyeda, Note on the roots of algebraic equations, Tôhoku Math. J. 10 (1916), 185–188.

    Google Scholar 

  33. E. Landau, Über eine Aufgabe aus der Funktionentheorie, Tôhoku Math. J. 5 (1914), 97–116.

    MATH  Google Scholar 

  34. X. Li, I. Gutman, and G.V. Milovanović, The β-polynomials of complete graphs are real, Publ. Inst. Math. (Beograd) (N.S.) (to appear).

    Google Scholar 

  35. H. Linden, Bounds for the zeros of polynomials from eigenvalues and singular values of some companion matrices, Linear Algebra Appl. 271 (1998), 41–82.

    Article  MathSciNet  MATH  Google Scholar 

  36. H. Linden, Numerical radii of some companion matrices and bounds for the zeros of polynomials, In: Analytic and Geometric Inequalities (Th.M. Rassias and H.M. Srivastava, eds.), Kluwer, Dordrecht, 1999, pp. 205–229.

    Chapter  Google Scholar 

  37. S. Lipka, Über die Lage der Wurzeln von algebraischen Gleichungen, Monatsh. Math. Phys. 50 (1944), 125–127.

    MathSciNet  Google Scholar 

  38. M. Marden, A refinement of Pellet’s theorem, Bull. Amer. Math. Soc. 54 (1948), 550–557.

    Article  MathSciNet  MATH  Google Scholar 

  39. M. Marden, Geometry of Polynomials, Math. Surveys 3, Amer. Math. Soc. Providence, R. I., 1966.

    Google Scholar 

  40. D. Marković, Sur la limite inférieure des modules des zéros d’un polynôme, Acad. Serbe Sci. Publ. Inst. Math. 2 (1948), 236–242.

    MathSciNet  Google Scholar 

  41. G. V. Milovanović, Some applications of the polynomials orthogonal on the semicircle, In: Numerical Methods (Miskolc, 1986), Colloq. Math. Soc. János Bolyai, Vol. 50, North-Holland, Amsterdam — New York, 1987, pp. 625–634.

    Google Scholar 

  42. G.V. Milovanović, Complex orthogonality on the semicircle with respect to Gegenbauer weight: Theory and applications, In: Topics in Mathematical Analysis (Th. M. Rassias, ed.), World Scientific Publishing Company, Singapore, 1989, pp. 695–722.

    Google Scholar 

  43. G.V. Milovanović, Numerical Analysis, I, 3rd ed., Naucna Knjiga, Belgrade, 1991 (Serbian).

    Google Scholar 

  44. G.V. Milovanović, On polynomials orthogonal on the semicircle and applications, J. Cornput. Appl. Math. 49 (1993), 193–199.

    Article  MATH  Google Scholar 

  45. G.V. Milovanović, D.S. Mitrinović, and Th.M. Rassias, Topic in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific, Singapore-New Jersey-London-Hong Kong, 1994.

    Google Scholar 

  46. G.V. Milovanović and P.M. Rajković, Geronimus concept of orthogonality for polynomials orthogonal on a circular arc, Rend, di Matematica, Serie VII (Roma) 10 (1990), 383–390.

    MATH  Google Scholar 

  47. G.V. Milovanović and P.M. Rajković, On polynomials orthogonal on a circular arc, J. Comput. Appl. Math. 51 (1994), 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  48. D.S. Mitrinović, Analytic Inequalities, Springer, Berlin-Heidelberg-New York, 1970.

    MATH  Google Scholar 

  49. P. Montel, Sur la limite supérieure des modules des zéros des polynomes, C.R. Acad. Sci. Paris 193 (1931), 974–976.

    Google Scholar 

  50. P. Montel, Sur la limite supérieure du module des racines d’une équation algébrique, C. R. Soc. Sci. Varsovie 24 (1932), 317–326.

    MATH  Google Scholar 

  51. A.M. Ostrowski, On a theorem by J. L. Walsh concerning the moduli of roots of algebraic equations, Bull. Amer. Math. Soc. 47 (1941), 742–746.

    Article  MathSciNet  MATH  Google Scholar 

  52. M. Parodi, La localisation des valeurs caractéristiques des matrices et ses applications, Gauther-Villars, Paris, 1959.

    Google Scholar 

  53. M. A. Pellet, Sur une mode de séparation des racines des équations et la formule de Lagrange, Bull. Sci. Math. 5 (1881), 393–395.

    Google Scholar 

  54. S.B. Prešić Sur un théoreme de S. Zervos Publ. Inst. Math. Beograd N.S. 1024 1970, 51–5

    MathSciNet  Google Scholar 

  55. Q.I. Rahman, A bound for the moduli of the zeros of polynomials, Canad. Math. Bull. 13(4) (1970), 541–542.

    Article  MathSciNet  MATH  Google Scholar 

  56. Th. M. Rassias, On certain properties of polynomials and their derivative, In: Topics in Mathematical Analysis (Th. M. Rassias, ed.), World Scientific Publishing Company, Singapore, 1989, pp. 758–802.

    Google Scholar 

  57. Th.M. Rassias, Zeros of polynomials and their derivative, Rev. Roumaine Math. Pures Appl. 36 (1991), 441–478.

    MathSciNet  MATH  Google Scholar 

  58. Th.M. Rassias and H.M. Srivastava, Some recent advances in the theory of the zeros and critical points of a polynomial, In: Topics in Polynomials of One and Several Variables and Their Applications: A Mathematical Legacy of P. L. Chebyshev (1821-1894) (Th.M. Rassias, H. Srivastava, and A. Yanushauskas, eds.), World Scientific Publishing Company, Singapore, 1993, pp. 463–481.

    Google Scholar 

  59. R. C. Riddell, Location of the zeros of a polynomial relative to certain disk, Trans. Amer. Math. Soc. 205 (1975), 37–45.

    Article  MathSciNet  MATH  Google Scholar 

  60. E. Rouché, Mémoire sur la série de Lagrange, J. École Polytech. 22 (1862), 217–218.

    Google Scholar 

  61. J. Rudnicki, Remarque sur un théorème de Mr. Walsh, Mathematica (Cluj) 8 (1934), 136–138.

    MATH  Google Scholar 

  62. G. Schmeisser, Inequalities for the zeros of an orthogonal expansion of a polynomial, In: Recent Progress in Inequalities (G.V. Milovanović, ed.), Kluwer, Dordrecht, 1998, pp. 381–396.

    Google Scholar 

  63. D.M. Simeunović, Sur les limites des modules des zéros des polynômes, Mat. Vesnik 419) (1967), 293–298.

    MathSciNet  Google Scholar 

  64. W. Specht, Die Lage der Nullstellen eines Polynoms, Math. Nachr. 15 (1956), 353–374.

    Article  MathSciNet  Google Scholar 

  65. W. Specht, Die Lage der Nullstellen eines Polynoms. II, Math. Nachr. 16 (1957), 257–263.

    Article  MathSciNet  Google Scholar 

  66. W. Specht, Die Lage der Nullstellen eines Polynoms. III, Math. Nachr. 16 (1957), 369–389.

    Article  MathSciNet  Google Scholar 

  67. W. Specht, Die Lage der Nullstellen eines Polynoms. IV, Math. Nachr. 21 (1960), 201–222.

    Article  MathSciNet  MATH  Google Scholar 

  68. S.B. Steckhin, On zero distribution of polynomials, Mat. Zametki 1 (1967), 425–434 (Russian).

    MathSciNet  Google Scholar 

  69. G. Szegő, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, 4th ed., Amer. Math. Soc., Providence, R. I., 1975.

    Google Scholar 

  70. M. R. Taskovic, Remark on some results of S. Presić and S. Zervos, Publ. Inst. Math. (Beograd) (N.S.) 11(25) (1971), 121–122.

    MathSciNet  Google Scholar 

  71. M.R. Tasković, Minimax theorems on posets with applications, Math. Japonica 35 (1990), 805–815.

    MATH  Google Scholar 

  72. T.T. Tonkov, Estimate of the moduli of the roots of polynomials, Zh. Vychisl. Mat. i Mat. Fiz. 4 (1964), 748–749.

    MathSciNet  Google Scholar 

  73. T. Tôya, Some remarks on MonteVs paper concerning upper limits of absolute values of roots of algebraic equations, Sci. Rep. Tokyo Bunrika Daigaku A 1 (1933), 275–282.

    MATH  Google Scholar 

  74. P. Turán, Hermite-expansions and strips for zeros of polynomials, Arch. Math. 5 (1954), 148–152.

    Article  MathSciNet  MATH  Google Scholar 

  75. P. Turán, To the analytic theory of algebraic equations, Izvestija Mat. Inst. Bulg. Akad. Nauk. 3 (1959), 123–137.

    Google Scholar 

  76. R. Vermes On the zeros of a linear combination of polynomials, Pacific J. Math. 19 (1966), 553–559.

    MathSciNet  MATH  Google Scholar 

  77. J.L. Walsh, On the location of the roots of certain types of polynomials, Trans. Amer. Math. Soc. 24 (1922), 163–180.

    Article  MathSciNet  Google Scholar 

  78. J.L. Walsh, An inequality for the roots of an algebraic equation, Ann. of Math. (2) 25 (1924), 285–286.

    Article  MathSciNet  MATH  Google Scholar 

  79. J.L. Walsh, On Pellet’s theorem concerning the roots of a polynomial, Ann. of Math. (2) 26 (1924), 59–64.

    Article  MathSciNet  MATH  Google Scholar 

  80. E. C. Westerfield, A new bound for the zeros of polynomials, Amer. Math. Monthly 40 (1933), 18–23.

    Article  MathSciNet  MATH  Google Scholar 

  81. K. P. Williams, Note concerning the roots of an equation, Bull. Amer. Math. Soc. 28 (1922), 394–396.

    Article  MathSciNet  MATH  Google Scholar 

  82. S.P. Zervos, Aspects modernes de la localisation des zéros des polynômes d’une variable, Ann. Sci. École Norm. Sup. (3) 77 (1960), 303–410.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Electronic Engineering, P.O. Box 73, 18000, Niš, Yugoslavia

    Gradimir V. Milovanović

  2. Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780, Athens, Greece

    Themistocles M. Rassias

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  1. Gradimir V. Milovanović
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  2. Themistocles M. Rassias
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Milovanović, G.V., Rassias, T.M. (2000). Inequalities for Polynomial Zeros. In: Survey on Classical Inequalities. Mathematics and Its Applications, vol 517. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4339-4_6

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  • DOI: https://doi.org/10.1007/978-94-011-4339-4_6

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-5868-1

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