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Some Comments and a Bibliography on the Laguerre-Samuelson Inequality with Extensions and Applications in Statistics and Matrix Theory

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Analytic and Geometric Inequalities and Applications

Part of the book series: Mathematics and Its Applications ((MAIA,volume 478))

Abstract

We examine an 1880 theorem of Laguerre [50] concerning polynomials with all real roots and a 1968 inequality of Sanmelson [117] for the maximum and minimum deviation from the mean, and establish their equivalence. The bounds provided by Laguerre’s Theorem involve the first three coefficients of an n-th degree polynomial while Samuelson’s Inequality is in terms of the standard deviation (and the mean) of a set of n real numbers (observations). We present eight proofs of this Laguerre-Samuelson inequality and survey the literature; we also give various extensions and applications in statistics and matrix theory. We include some historical and biographical information and present an extensive bibliography with over 100 entries.

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Bibliography

  1. David Abbott, ed., The Biographical Dictionary of Scientists: Mathematicians, Blond Educational (an imprint of Muller, Blond & White Limited), London, England, 1985.

    Google Scholar 

  2. Manzoor Ahmad, On polynomials with real zeros, Canadian Mathematical Bulletin 11 (1968), 237–240 [MR 37:6420, Zbl 165.05501].

    Article  MathSciNet  MATH  Google Scholar 

  3. Barry C. Arnold, Schwarz, regression, and extreme deviance, The American Statistician 28 (1974), 22–23 [MR 49:8223, Zbl 365.62070].

    MathSciNet  MATH  Google Scholar 

  4. Barry C. Arnold and N[arayanaswamy] Balakrishnan, Relations, Bounds and Approximations for Order Statistics. Lecture Notes in Statistics, vol. 53, Springer-Verlag, New York, 1989 [MR 90i:62061, Zbl 703.62064].

    Google Scholar 

  5. Barry C. Arnold and Richard A. Groeneveld, Bounds for deviations between sample and population statistics, Biometrika 61 (1974), 387–389 [MR 51:716, Zbl 281.62012].

    Article  MathSciNet  MATH  Google Scholar 

  6. Richard A. Groeneveld —, Bounds on deviations of estimates arising in finite population regression models, Communications in Statistics-A, Theory and Methods 7 (1978), 1173–1179 [MR 80a:62015, Zbl 398.62056].

    Article  MathSciNet  Google Scholar 

  7. Richard A. Groeneveld — —, Bounds on expectations of linear systematic statistics based on dependent samples, The Annals of Statistics 7 (1979), 220–223 [MR 80c:62061, Zbl 398.62036]. (Correction: 8 (1980), 1401 [Zbl 465.62043].)

    Google Scholar 

  8. Richard A. Groeneveld — —, Maximal deviation between sample and population means in finite populations, Journal of the American Statistical Association 76 (1981), 443–445 [MR 82h:62016, Zbl 462.62012].

    Google Scholar 

  9. Jerzy K. Baksalary and R. Kala, Partial orderings between matrices one of which is of rank one, Bulletin of the Polish Academy of Sciences, Mathematics (Warsaw) 31 (1983), 5–7 [MR 85j:15016, Zbl 535.15006].

    MathSciNet  MATH  Google Scholar 

  10. Diccon Bancroft, Comment (Letter to the Editor about Chaganty [24] and Olkin [96]), The American Statistician 48 (1994), 351.

    Google Scholar 

  11. Vic Barnett and Toby Lewis, Outliers in Statistical Data, Third Edition, Wiley, Chichester, 1994 [Zbl 801.62001].

    Google Scholar 

  12. Edwin F. Beckenbach and Richard Bellman, Inequalities [in English], Fourth [Revised] Printing. Ergebnisse der Mathematik und ihre Grenzgebiete, neue Folge, Heft 30, Springer-Verlag, Berlin, 1983 [MR 33:236, Zbl 513.26003].

    Google Scholar 

  13. Paul R. Beesack, On bounds for the range of ordered variates, Publikacije Elektrotehnickog Fakulteta Univerziteta u Beogradu, Serija Matematika i Fizika (Belgrade) 428 (1973), 93–96 [MR 48:5257, Zbl 274.62031].

    MathSciNet  Google Scholar 

  14. —, On bounds for the range of ordered variates: II, Aequationes Mathematicae 14 (1976), 293–301 [MR 54:2902, Zbl 336.62035].

    Article  MathSciNet  MATH  Google Scholar 

  15. Margherita Benzi, Un “probabilista neoclassico”: Francesco Paolo Cantelli [in Italian with English and German summaries], Historia Mathematica 15 (1988), 53–72 [MR 89k:01022, Zbl 642.01012].

    Article  MathSciNet  MATH  Google Scholar 

  16. J[onathan] M. Borwein, G. P. H. Styan and H. Wolkowicz, Some inequalities involving statistical expressions: Solution to Problem 81-10 (posed by Foster [32]). SIAM Review 24 (1982), 340–342. (Reprinted in Problems in Applied Mathematics: Selections from SIAM Review (Murray S. Klamkin, ed.), SIAM, Philadelphia, 1990, pp. 373-375.).

    Google Scholar 

  17. V[iktor Yakovlevich] Bouniakowsky [Buniakovski, Bunyakovsky], Sur quelques inégalités concernant les intégrales ordinaires et les intégrales aux différences finies [in French], Mémoires de l’Académie Impériale des Sciences de St.-Pétersbourg, Septième Série 1, no. 9 (1859), pp. 1-18. (Cf. pp. 3-4.).

    Google Scholar 

  18. A. V. Boyd, Bounds for order statistics, Publikacije Elektrotehničkog Fakulteta Univerziteta u Beogradu, Serija Matematika i Fizika (Belgrade) 365 (1971), 31–32 [MR 46:9256, Zbl 245.62052].

    Google Scholar 

  19. Alfred Brauer and A. C. Mewborn, The greatest distance between two characteristic roots of a matrix, Duke Mathematical Journal 26 (1959), 653–661 [MR 22:10997, Zbl 095.01202].

    Article  MathSciNet  MATH  Google Scholar 

  20. Claude Brezinski, Edmond Nicolas Laguerre [in French], Polynômes Orthogonaux et Applications: Proceedings of the Laguerre Symposium held at Bar-le-Duc, October 15-18, 1984 (C. Brezinski, A. Draux, A. P. Magnus, P. Maroni and A. Ronveaux, eds.), Lecture Notes in Mathematics, vol. 1171, Springer-Verlag, Berlin, 1985, pp. xxi–xxvi.

    Google Scholar 

  21. K. A. Brownlee, Statistical Theory and Methodology in Science and Engineering, Second Edition, Wiley, New York, 1965 [Zbl 136.39203].

    MATH  Google Scholar 

  22. H. D. Brunk, Note on two papers of K. R. Nair, Journal of the Indian Society of Agricultural Statistics 11 (1959), 186–189.

    Google Scholar 

  23. [Baron] Augustin-Louis Cauchy, Note II: Sur les formules qui résultent de l’emploi du signe > ou <, et sur les moyennes entre plusieurs quantités [in French], Cours d’Analyse de l’École Royale Polytechnique, Première Partie: Analyse Algébrique. L’Imprimerie Royale Chez Debure Frères, Libraires du Roi et de la Bibliothèque du Roi, Paris, 1821, pp. 360-377. (Cf. pp. 373-374. Reprinted in Œuvres Complètes d’Augustin Cauchy, Publiées sous la direction scientifique de l’Académie des Sciences et sous les auspices de M. le Ministre de l’Instruction Publique, Seconde Série: Mémoires Divers et Ouvrages, vol. 3 (Mémoires publiés en corps d’ouvrage), Gauthier-Villars, Paris, 1897, pp. 360–377.)

    Google Scholar 

  24. N. Rao Chaganty, Comment (Letter to the Editor about Olkin [96]), The American Statistician 47 (1993), 158.

    MATH  Google Scholar 

  25. N. Rao Chaganty and Akhil K. Vaish, Response (Letter to the Editor about Bancroft [10] & Neudecker and Liu [91]), The American Statistician 48 (1994), 351–352.

    Google Scholar 

  26. Akhil K. Vaish —, On inequalities for outlier detection in statistical data analysis, Journal of Applied Statistical Science 6 (1997), 235–243 [Zbl 980.21632].

    MathSciNet  MATH  Google Scholar 

  27. Herbert A. David, Order Statistics, Second Edition. Wiley, New York, 1981 [MR 82i:62073, Zbl 553.62046].

    Google Scholar 

  28. —, General hounds and inequalities in order statistics, Communications in Statistics-Theory and Methods 17 (1988), 2119–2134 [MR 89j:62068, Zbl 639.62042].

    Google Scholar 

  29. A. Dey and S. C. Gupta, Singular weighing designs and estimation of total weight, Communications in Statistics-A, Theory and Methods 6 (1977), 289–295 [MR 55:9435, Zbl 362.62034].

    Article  MathSciNet  Google Scholar 

  30. Meyer Dwass, The extreme deviations inequality (Letter to the Editor about Quesenberry [107] and Samuelson [117]), The American Statistician 29 (1975), 108.

    Article  Google Scholar 

  31. Nicholas R. Farnum, An alternate proof of Samuelson’s inequality and its extensions, The American Statistician 43 (1989), 46–47.

    Google Scholar 

  32. L. V. Foster, Some inequalities involving statistical expressions: Problem 81-10, SIAM Review 23 (1981), 256. (Problem solved by Borwein, Styan and Wolkowicz [16].)

    Article  Google Scholar 

  33. Florin Gonzacenco, Inegalitaţi probabiliste ş aplicaţii in statistica [in Romanian with English summary: Some probabilistic inequalities and their statistical applications], Studii şi Cercetari Matematice (Bucharest) 38 (1986), 290–297 [MR 88d:60054, Zbl 639.60025].

    MathSciNet  MATH  Google Scholar 

  34. F[lorin] Gonzacenco and E[ugen] Margaritescu, Best bounds for order statistics [in English], Bulletin Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie, Nouvelle Série (Bucharest) 31, no. 79 (1987), 303–311 [MR 90h:62116, Zbl 634.62043].

    Google Scholar 

  35. Florin Gonzacenco, Eugen Margaritescu and Viorel Gh. Voda, Statistical consequences of some old and new algebraic inequalities [in English], Revue Roumaine de Mathématiques Pures et Appliquées (Bucharest) 37 (1992), 877–886 [MR 93k:60048, Zbl 777.62104].

    Google Scholar 

  36. Richard A. Groeneveld, Best bounds for order statistics and their expectations in range and mean units with applications, Communications in Statistics-Theory and Methods 11 (1982), 1809–1815 [MR 83j:62070, Zbl 506.62031].

    Article  Google Scholar 

  37. Anna Hart, Standard deviation, Encyclopedia of Statistical Sciences, Volume 8: Regressograms-St. Petersburg Paradox (Samuel Kotz, Norman L. Johnson and Campbell B. Read, eds.), Wiley, New York, 1988, pp. 625–629.

    Google Scholar 

  38. H. O. Hartley and H. A. David, Universal bounds for mean range and extreme observation, The Annals of Mathematical Statistics 25 (1954), 85–99 [Zbl 055.12801].

    Article  MathSciNet  MATH  Google Scholar 

  39. Hiroshi Haruki and Themistocles M. Rassias, New integral representations for Bernoulli and Euler polynomials, Journal of Mathematical Analysis and Applications 175 (1993), 81–90 [MR 94e:39016, Zbl 776.11009].

    Google Scholar 

  40. David A. Harville, Matrix Algebra from a Statistician’s Perspective, Springer-Verlag, New York, 1997 [Zbl 881.15001].

    Book  Google Scholar 

  41. Douglas M. Hawkins, On the bounds of the range of order statistics, Journal of the American Statistical Association 66 (1971), 644–645 [Zbl 228.62030].

    Google Scholar 

  42. [Friedrich Robert] Helmert, Die Genauigkeit der Formel von Peters zur Berechnung des wahrscheinlichen Beobachtungsfehlers directer Beobachtungen gleicher Genauigkeit [in German], Astronomische Nachrichten 88 (1876), 115–132.

    Google Scholar 

  43. Constatin Ionescu-Ţiu, Problemei 15356 [in Romanian], Gazeta Matematica, ca. 1983. (Problem solved by Lupaş, [54].).

    Google Scholar 

  44. Shane Tyler Jensen. The Laguerre-Samuelson Inequality, with Extensions and Applications in Statistics and Matrix Theory. M. Sc. thesis, Dept. of Mathematics and Statistics, McGill University, Montréal, 1998, in preparation.

    Google Scholar 

  45. Shane T. Jensen and George P. H. Styan. Some comments and a bibliography on the von Szökefalvi Nagy-Popoviciu and Nair-Thomson inequalities, with extensions and applications in statistics and matrix theory. Dept. of Mathematics and Statistics, McGill University, in preparation for presentation at the “Seventh International Workshop on Matrices and Statistics, in Celebration of T. W. Anderson’s 80th Birthday”: Fort Lauderdale, Florida, December 11–14, 1998.

    Google Scholar 

  46. D. G. Kabe, On extensions of Samuelson’s inequality (Letter to the Editor about Wolkowicz and Styan [135]), The American Statistician 34 (1980), 249.

    Article  Google Scholar 

  47. Oscar Kempthorne, Personal communication, 1973. (Cf. Arnold [3] and Arnold and Balakrishnan [4], pp. 45, 158.).

    Google Scholar 

  48. W. Kirby, Algebraic boundedness of sample statistics, Water Resources Research 10 (1974), 220–222.

    Article  Google Scholar 

  49. Ludvik Kraus, Poznámka k rovnicím, jež mají pouze realné kořeny [in Czech: With a “Dodatek” by Ed. Weyr], Časopis pro Pěestování Matematiky a Fysiky: Část Vědecká (Prague) 15 (1886), 63–64.

    Google Scholar 

  50. [Edmond Nicolas] Laguerre, Sur une méthode pour obtenir par approximation les racines d’une équation algébrique qui a toutes ses racines réelles [in French]. Nouvelles Annales de Mathématiques (Paris), 2e Série 19 (1880), 161–171 & 193-202 [JFM 12:71]. (Reprinted in Œuvres de Laguerre (publiées sous les auspices de l’Académie des Sciences par MM. Ch. Hermite, H. Poincaré et E. Rouché), Gauthier-Villars, Paris, 1 (1898), 87-103.).

    Google Scholar 

  51. H. O. Lancaster, The Helmert matrices, The American Mathematical Monthly 72 (1965), 4–12 [MR 30:1134, Zbl 124.01102].

    Article  MathSciNet  MATH  Google Scholar 

  52. Alexandru Lupaş, Problem 246 [in English], Matematicki Vesnik (Belgrade) 8, no. 23 (1971), 333.

    Google Scholar 

  53. —, Inequalities for the roots of a class of polynomials [in English], Publikacije Elektrotehničkog Fakulteta Univerziteta u Beogradu, Serija Matematika i Fizika (Belgrade) 594 (1977), 79–85 [Zbl 371.26009].

    Google Scholar 

  54. —, Asupra problemei 15356 (posed by Ionescu-Ţiu [43]) [in Romanian], Gazeta Matematica 5, no. 1/2 (1984), 56–60.

    Google Scholar 

  55. B. S. Madhava Rao and B. S. Sastry, On the limits for the roots of a polynomial equation, Journal of the Mysore University, Section B 1 (1940), 5–8 [MR 2:241d].

    MathSciNet  Google Scholar 

  56. C[olin]L. Mallows and Donald Richter, Inequalities of Chebyshev type involving conditional expectations, The Annals of Mathematical Statistics 40 (1979), 1922–1932 [MR 40:6617, Zbl 187.14903].

    Article  MathSciNet  Google Scholar 

  57. E. Malo, Note sur les équations algébriques dont toutes les racines sont réelles [in French], Journal de Mathématiques Spéciales, 4e Série 4 (1894), 7–10 [JFM 26:120].

    Google Scholar 

  58. Eugen Margaritescu, Bornes strictes pour les statistiques d’ordre et inégalités de type Samuelson [in French], Revue Roumaine de Mathématiques Pures et Appliquées (Bucharest) 32 (1987), 343–349 [MR 88m:60043].

    MathSciNet  Google Scholar 

  59. Eugen Margaritescu and Viorel Gh. Voda, O inegalitate algebrica şi cîteva aplicaţii in statistica [in Romanian with English summary: An algebraic inequality and some applications in statistics], Studii şi Cercetari Matematice (Bucharest) 35 (1983), 376–387 [MR 85g:62125, Zbl 518.62059].

    Google Scholar 

  60. Thomas Mathew and Kenneth Nordström, An inequality for a measure of deviation in linear models, The American Statistician 51 (1997), 344–349 [MR 98g:62131].

    MathSciNet  Google Scholar 

  61. A. T. McKay, The distribution of the difference between the extreme observation and the sample mean in samples of n from a normal universe, Biometrika 27 (1935), 466–471 [Zbl 013.03002].

    MATH  Google Scholar 

  62. A. T. McKay and E. S. Pearson, A note on the distribution of range in samples of n, Biometrika 25 (1933), 415–420.

    MATH  Google Scholar 

  63. A. Meir and A. Sharma, Span of derivatives of polynomials, The American Mathematical Monthly 74 (1967), 527–531 [MR 35:5561].

    Article  MathSciNet  MATH  Google Scholar 

  64. A. Sharma —, On zeros of derivatives of polynomials, Canadian Mathematical Bulletin 11 (1968), 443–445 [MR 38:2285, Zbl 164.06405]. (Addendum: 11 (1968), 611 [Zbl 167.33101].)

    Article  MathSciNet  MATH  Google Scholar 

  65. Eugenio Melilli, Cantelli, Francesco Paolo, Leading Personalities in Statistical Sciences: From the Seventeenth Century to the Present (Norman L. Johnson and Samuel Kotz, eds.), Wiley, New York, 1997, pp. 228–232.

    Google Scholar 

  66. Jorma Kaarlo Merikoski, On the trace and the sum of elements of a matrix, Linear Algebra and Its Applications 60 (1984), 177–185 [MR 85h:15012, Zbl 559.15004].

    Article  MathSciNet  MATH  Google Scholar 

  67. Jorma Kaarlo Merikoski, Humberto Sarria and Pablo Tarazaga, Bounds for singular values using traces, Linear Algebra and Its Applications 210 (1994), 227–254 [MR 95h:15030, Zbl 813.15016].

    Article  MathSciNet  MATH  Google Scholar 

  68. Jorma Kaarlo Merikoski, George P. H. Styan and Henry Wolkowicz, Bounds for ratios of eigenvalues using traces, Linear Algebra and Its Applications 55 (1983), 105–124 [MR 85a:15019, Zbl 522:15008].

    Google Scholar 

  69. Jorma Kaarlo Merikoski and Ari Virtanen, Bounds for eigenvalues using the trace and determinant, Linear Algebra and Its Applications 264 (1997), 101–108 [Zbl 885.15011].

    Article  MathSciNet  MATH  Google Scholar 

  70. Jorma Kaarlo Merikoski and Henry Wolkowicz, Improving eigenvalue bounds using extra bounds, Linear Algebra and Its Applications 68 (1985), 93–113 [MR 87e:15038, Zbl 574.15010].

    Article  MathSciNet  MATH  Google Scholar 

  71. N[icolae] Mihaileanu. Istoria Matematicii [in Romanian: History of Mathematics], Editura Ştiintificä şi Enciclopedica, Bucharest, 1981.

    Google Scholar 

  72. G[radimir] V. Milovanovic, D[ragoslav]S. Mitrinović and Th[emistocles] M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific, Singapore, 1994 [MR 95m:30009, Zbl 848.26001].

    Google Scholar 

  73. G[radimir] V. Milovanović and Th[emistocles] M. Rassias, Inequalities connected with trigonometric sums, Constantin Carathéodory: An International Tribute (Themistocles M. Rassias, ed.), vol. 2, World Scientific, Singapore, 1991, pp. 875–941 [MR 93g:42002, Zbl 754.26008].

    Google Scholar 

  74. Th[emistocles] M. Rassias —, On the Markov-Duffin-Schaeffer inequalities, Journal of Natural Geometry 5 (1994), 29–41 [MR 94h:26017, Zbl 788.26011].

    Google Scholar 

  75. L[eon] Mirsky, The spread of a matrix, Mathematika (London) 3 (1956), 127–130 [MR 18:460c, Zbl 073.00903].

    MathSciNet  MATH  Google Scholar 

  76. —, Inequalities for normal and Hermitian matrices, Duke Mathematical Journal 24 (1957), 591–599 [MR 19:832c, Zbl 081.25101].

    Article  MathSciNet  MATH  Google Scholar 

  77. Dragoslav S. Mitrinović, Inégalités impliquées par le système des égalités a+b+c = p, bc + ca + ab = q [in French], Publikacije Elektrotehničkog Fakulteta Univerziteta u Beogradu, Serija Matematika i Fizika (Belgrade) 143-155 (1965), 5-7 [Zbl 184:00085].

    Google Scholar 

  78. —, Analytic Inequalities, In cooperation with Petar M. Vasic, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, vol. 165, Springer-Verlag, New York, 1970 [MR 43:448, Zbl 199.38101]. (Cf. pp. 210-211. Translated into Serbo-Croatian and updated: Mitrinović [79]. Addenda by Mitrinović and Vasić [82] and by Mitrinović Lacković and Stanković [80].).

    Google Scholar 

  79. —, Analiticke Nejednakosti [in Serbo-Croatian: Analytic Inequalities]. In cooperation with Petar M. Vasic, Translation of Mitrinović [78] from the English and updated, University of Belgrade Monographs, vol. 2, Gradevinska Knjiga, Belgrade, 1970.

    Google Scholar 

  80. Dragoslav S. Mitrinović, Ivan B. Lacković and Miomir S. Stanković, Addenda, to the monograph Analytic Inequalities by Mitrinović [78]-II: On some convex sequences connected with N. Ozeki’s results [in English], Publikacije Elektrotehničkog Fakulteta Univerziteta u Beogradu, Serija Matematika i Fizika (Belgrade) 634-677 (1979), 3-24 [MR 82c:26011, Zbl 452.26007].

    Google Scholar 

  81. Dragoslav S. Mitrinović, Josip E. Pečarić and Arlington M. Fink, Classical and New Inequalities in Analysis, Mathematics and Its Applications: East European Series, vol. 61, Kluwer, Dordrecht, 1993 [MR 94c:00004, Zbl 771.26009].

    Google Scholar 

  82. Dragoslav S. Mitrinović and Petar M. Vasić, Addenda, to the monograph Analytic Inequalities by Mitrinović [78]-I [in English], Publikacije Elektrotehničkog Fakulteta Univerziteta u Beogradu, Serija Matematika i Fizika (Belgrade) 577-598 (1977), 3-10 [MR 56:5818, Zbl 369.26010].

    Google Scholar 

  83. Vedula N. Murty, A note on Samuelson’s inequality (Letter to the Editor about Farnum [31],) The American Statistician 44 (1990), 64–65.

    Google Scholar 

  84. Keshavan Raghavan Nair, Ph. D. thesis, University College London, 1947. (Parts published as Nair [85] and [86].).

    Google Scholar 

  85. —, Certain symmetrical properties of unbiased estimates of variance and covariance, Journal of the Indian Society of Agricultural Statistics 1 (1948), 162-172 [MR 11:448a]. (Part of the thesis [84].).

    Google Scholar 

  86. —, The distribution of the extreme deviate from the sample mean and its Studentized form, Biometrika 35 (1948), 118-144 [MR 9:602a]. (Part of the thesis [84].).

    Google Scholar 

  87. —, A note on the estimation of mean rate of change, Journal of the Indian Society of Agricultural Statistics 8 (1956), 122-124.

    Google Scholar 

  88. —, A tail-piece to Brunk’s paper [22], Journal of the Indian Society of Agricultural Statistics 11 (1959), 189-190.

    Google Scholar 

  89. —, On extensions of Samuelson’s inequality (Letter to the Editor about Wolkowicz and Styan [135]), The American Statistician 34 (1980), 249-250.

    Google Scholar 

  90. —, In statistics by design, Glimpses of India’s Statistical Heritage (J. K. Ghosh, S. K. Mitra and K. R. Parthasarathy, eds.), Wiley Eastern, New Delhi, 1992, pp. 101-150 [MR 94c:62001, Zbl 832.01016].

    Google Scholar 

  91. Heinz Neudecker and Shuangzhe Liu, Comment (Letter to the Editor about Chaganty [24] and Olkin [96]), The American Statistician 48 (1994), 351.

    Article  Google Scholar 

  92. Constantin Nicolau, Recherches nouvelles sur les équations algébriques [in French], Annales Scientifiques de l’Université de Jassy (Iaşi) 18 (1933), 15–69 [Zbl 007.39100].

    Google Scholar 

  93. —, Un théorème sur les équations algébriques [in French], Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences (Paris) 208 (1939), 1958–1960 [Zbl 021.19703].

    Google Scholar 

  94. B. Niewenglowski, [Problem No.] 244 [in French], L’Intermédiaire des Mathématiciens (Paris) 1 (1894), 132-133.

    Google Scholar 

  95. John J. O’Connor and Edmund F. Robertson, MacTutor History of Mathematics Archive: http://www-history.mcs.st-andrews.ac.uk/history/Bioglndex.html.

    Google Scholar 

  96. Ingram Olkin, A matrix formulation on how deviant an observation can be, The American Statistician 46 (1992), 205–209.

    MathSciNet  Google Scholar 

  97. Federico J. O’Reilly, On a criterion for extrapolation in normal regression, The Annals of Mathematical Statistics 3 (1975), 219–222 [Zbl 305.62040].

    Google Scholar 

  98. —, The extreme deviations inequality (Letter to the Editor about Arnold [3], Dwass [30] and Quesenberry [107]), The American Statistician 30 (1976), 103.

    Google Scholar 

  99. Jagdish K. Patel, C. H. Kapadia and D[onald] B. Owen, Handbook of Statistical Distributions, Statistics: Textbooks and Monographs, vol. 20, Marcel Dekker, New York, 1976 [MR 56:9771, Zbl 367.62014].

    Google Scholar 

  100. E. S. Pearson, “Student” as statistician. Biometrika 30 (1938), 210–250 [Zbl 020.04005].

    Google Scholar 

  101. E. S. Pearson, assisted by C. Chandra Sekar, The efficiency of statistical tools and a criterion for the rejection of outlying observations, Biometrika 28 (1936), 308–320 [Zbl 015.26205]. (Includes Appendix by Scott [122].)

    MATH  Google Scholar 

  102. Josip E. Pečarić, O nekim nejednakostima za korjene jedne klase polinoma [in Serbo-Croatian with English summary: On some inequalities for the roots of a class of polynomials], Zbornik Fakulteta za Pomorstvo u Kotoru (Kotor) 9/10 (1983/1984), 123-129.

    Google Scholar 

  103. Josip E. Pečarić, Simo Puntanen and George P. H. Styan, Some further matrix extensions of the Cauchy-Schwarz and Kantorovich inequalities, with some statistical applications, Linear Algebra and Its Applications 237/238 (1996), 455-476 [MR 97c:15035, Zbl 860.15021].

    Google Scholar 

  104. Tiberiu Popoviciu, Sur les équations algébriques ayant toutes leurs racines réelles [in French], Mathematica (Cluj) 9 (1935), 129–145 [Zbl 014.10003].

    MATH  Google Scholar 

  105. P. Prescott, Letter quoted in “Editor’s Note” (about Letters to the Editor by Kabe [46], Nair [89], and Wolkowicz and Styan [136]), The American Statistician 34 (1980), 251.

    Google Scholar 

  106. Simo Puntanen, Matriiseja Tilastotieteilijälle [in Finnish: Matrices for the Statistician], Preliminary Edition, Report B47, Dept. of Mathematical Sciences, University of Tampere, Tampere, Finland, 1998.

    Google Scholar 

  107. C. P. Quesenberry, The extreme deviations inequality (Letter to the Editor about Samuelson [117] and Arnold [3]), The American Statistician 28 (1974), 112.

    Google Scholar 

  108. C. P. Quesenberry and H[erbert] A. David, Some tests for outliers, Biometrika 48 (1961), 379–390 [Zbl 108.15604].

    MATH  Google Scholar 

  109. Themistocles M. Rassias, A new inequality for complex-valued polynomial functions, Proceedings of the American Mathematical Society 97 (1986), 296–298 [MR 87h:30005, Zbl 593.30004].

    Google Scholar 

  110. —, On certain properties of polynomials and their derivative, Topics in Mathematical Analysis: A Volume Dedicated to the Memory of A. L. Cauchy (Themistocles M. Rassias, ed.), Series in Pure Mathematics, vol. 11, World Scientific, Singapore, 1989, pp. 758–802 [MR 92m:30009, Zbl 732.30005].

    Google Scholar 

  111. —, On polynomial inequalities and extremal problems, General Inequalities: 6 (Oberwolfach 1990), International Series in Numerical Mathematics, vol. 103, Birkhäuser, Basel, 1992, pp. 161-174. [MR 94i:26008, Zbl 778.30009].

    Google Scholar 

  112. Th[emistocles] M. Rassias and Jaromír Šimša, Finite Sums Decompositions in Mathematical Analysis, Wiley, Chichester, 1995 [MR 96k:26006, Zbl 859.26005].

    Google Scholar 

  113. Th[emistocles] M. Rassias, S. N. Singh and H[ari] M. Srivastava, Some q-generating functions associated with basic multiple hypergeometric series, Computers & Mathematics with Applications 27 (1994), 33–39 [MR 95m:33018, Zbl 792.33013].

    Google Scholar 

  114. Thfemistocles] M. Rassias and H[ari] M. Srivastava, Some general families of generating functions for the Laguerre polynomials, Journal of Mathematical Analysis and Applications 174 (1993), 528–538 [MR 94g:33004, Zbl 787.33006].

    Google Scholar 

  115. Th[emistocles] M. Rassias, H[ari] M. Srivastava, and A. Yanushauskas (eds), Topics in Polynomials of One and Several Variables and their Applications: Dedicated to the Memory of P. L. Chebyshev (1821–1894), World Scientific, Singapore, 1993 [MR 94m:00020, Zbl 849.00029].

    Google Scholar 

  116. Campbell B. Read. Helmert, Friedrich Robert, Leading Personalities in Statistical Sciences: From the Seventeenth Century to the Present (Norman L. Johnson and Samuel Kotz, eds.), Wiley, New York, 1997, pp. 42–43.

    Google Scholar 

  117. Paul A. Samuelson, How deviant can you be? Journal of the American Statistical Association 63 (1968), 1522–1525. (Reprinted in The Collected Scientific Papers of Paul A. Samuelson, vol. 3 (Robert C. Merton, ed.), MIT Press, Cambridge, Mass., 1972, pp. 340-343.)

    Google Scholar 

  118. —, Unpublished letter to R. B. Murphy, 5 February 1980 [mentions “an Australian anticipation”]. (Cited by Wolkowicz and Styan [136].).

    Google Scholar 

  119. —, Economics in my time, Lives of the Laureates: Thirteen Nobel Economists (William Breit and Roger W. Spencer, eds.), Third Edition, MIT Press, Cambridge, Mass., 1995, pp. 59-77.

    Google Scholar 

  120. Issai Schur, Zwei Sätze über algebraische Gleichungen mit lauter reellen Wurzeln [in German], Journal für die reine und angwandte Mathematik 144 (1914), 75–88 [JFM 45:169]. (Reprinted in Issai Schur: Gesammelte Arbeiten, vol. 2 (Alfred Brauer and Hans Rohrbach, eds.), Springer-Verlag, Berlin, 1973, pp. 56-69. Original version has author’s name given as J. Schur.).

    Google Scholar 

  121. [Karl] H[ermann] A[mandus] Schwarz, Ueber ein die Flächen kleinsten Flächeninhalts betreffendes Problem der Variationsrechnung: Festschrift zum Jubelgeburtstage des Herrn Karl Weierstrass [in German], Acta Societatis Scientiarum Fennicæ (Helsinki) 15 (1888), 315–362. (Cf. pp. 343-345. Preface dated 31 October 1885. Reprinted in Gesammelte Mathematische Abhandlungen von H. A. Schwarz, vol. 1, Julius Springer, Berlin, 1890, pp. 223-270; cf. pp. 251-253.).

    Google Scholar 

  122. J. M. C. Scott, Appendix, In Pearson and Chandra Sekar [101], 1936, pp. 319-320.

    Google Scholar 

  123. William P. Smith. Letter quoted in “Editor’s Note” (about Letters to the Editor by Kabe [46], Nair [89], and Wolkowicz and Styan [136]), The American Statistician 34 (1980), 251. (Cf. [4], pp. 46, 161. See also Prescott [105].)

    Google Scholar 

  124. Stephen M. Stigler, The History of Statistics: The Measurement of Uncertainty before 1900, The Belknap Press of Harvard University Press, Cambridge, Mass., and London, England, 1986.

    Google Scholar 

  125. Alan Stuart and J. Keith Ord, Kendall’s Advanced Theory of Statistics, Volume 1: Distribution Theory, Sixth Edition, Edward Arnold, London, and Halsted Press, Wiley, New York, 1994.

    Google Scholar 

  126. Julius von Szökefalvi Nagy [Gyula Szökefalvi-Nagy], Über algebraische Gleichungen mit lauter reellen Wurzeln [in German], Jahresbericht der Deutschen Mathematiker-Vereinigung 27 (1918), 37-43 [JFM 46:125].

    Google Scholar 

  127. —, Review of Nicolau [92], Zentralblatt für Mathematik und ihre Grenzgebiete 7 (1934), 391.

    Google Scholar 

  128. —, Review of Nicolau [93], Zentralblatt für Mathematik und ihre Grenzgebiete 21 (1939), 197.

    Google Scholar 

  129. Pablo Tarazaga, Eigenvalue estimates for symmetric matrices, Linear Algebra and Its Applications 135 (1990), 171–179 [MR 91d:15036, Zbl 701.15012].

    Article  MathSciNet  MATH  Google Scholar 

  130. William R. Thompson, On a criterion for the rejection of observations and the distribution of the ratio of deviation to sample standard deviation, The Annals of Mathematical Statistics 6 (1935), 214–219 [Zbl 012.41102].

    Google Scholar 

  131. G[ötz] Trenkler, Comment (Letter to the Editor about Chaganty [24] and Olkin [96]), The American Statistician 48 (1994), 60.

    Article  Google Scholar 

  132. Rodica-Cristina Voda, Margini pentru amplitudinea Studentizata de selecţie [in Romanian with English summary: Bounds for the sample Studentized range]. Studii şi Cercetari Matematice (Bucharest) 35 (1983), 544–548 [MR 85c:62039, Zbl 549.62020].

    MathSciNet  MATH  Google Scholar 

  133. Geoffrey S. Watson, Gülhan Alpargu and George P. H. Styan, Some comments on six inequalities associated with the inefficiency of ordinary least squares with one regressor, Linear Algebra and Its Applications 264 (1997), 13–53 [MR 98i:15023].

    Google Scholar 

  134. Heinrich Weber, Die Sätze von Laguerre für Gleichungen mit nur reellen Wurzeln [in German], Section 115 in Lehrbuch der Algebra, Erster Band, Third Edition, Chelsea, New York, 1961, pp. 364-371. (Original version: 1895/1896 and Second Edition: 1898, both pub. Friedrich Vieweg und Sohn, Braunschweig.).

    Google Scholar 

  135. Henry Wolkowicz and George P. H. Styan, Extensions of Samuelson’s inequality, The American Statistician 33 (1979), 143–144 [MR 80h:62038].

    Google Scholar 

  136. George P. H. Styan —, Reply (to Letters to the Editor by Kabe [46] and Nair [89] about Wolkowicz and Styan [135]), The American Statistician 34 (1980), 250–251.

    Google Scholar 

  137. George P. H. Styan —, Bounds for eigenvalues using traces, Linear Algebra and Its Applications 29 (1980), 471–506 [MR 81k:15015, Zbl 435:15015 sic].

    Article  MathSciNet  MATH  Google Scholar 

  138. George P. H. Styan —, More bounds for eigenvalues using traces, Linear Algebra and Its Applications 31 (1980), 1–17 [MR 81k:15016, Zbl 434:15003].

    Article  MathSciNet  MATH  Google Scholar 

  139. George P. H. Styan —, Samuelson-Nair inequality, Encyclopedia of Statistical Sciences, Volume 8: Regressograms-St. Petersburg Paradox (Samuel Kotz, Norman L. Johnson and Campbell B. Read, eds.), Wiley, New York, 1988, pp. 258–259 [Zbl 706.62001].

    Google Scholar 

  140. Zentralblatt für Mathematik / Mathematics Abstracts: MATH Database, European Mathematical Society, FIZ Karlsruhe & Springer-Verlag, Berlin. (Three free hits are currently available at http://www.ends.de/cgi-bin/MATH.).

    Google Scholar 

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  1. Department of Mathematics and Statistics, McGill University, 805 rue Sherbrooke ouest, Montréal, Québec, Canada, H3A 2K6

    Shane T. Jensen & George P. H. Styan

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

    Themistocles M. Rassias

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada

    Hari M. Srivastava

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Jensen, S.T., Styan, G.P.H. (1999). Some Comments and a Bibliography on the Laguerre-Samuelson Inequality with Extensions and Applications in Statistics and Matrix Theory. In: Rassias, T.M., Srivastava, H.M. (eds) Analytic and Geometric Inequalities and Applications. Mathematics and Its Applications, vol 478. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4577-0_10

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