Positivity

, Volume 15, Issue 1, pp 17–48

Hausdorff means and moment sequences

Article

Abstract

We prove Schur’s theorem on the complete symmetric functions by using Hausdorff means. This approach leads to a more general result and to some new properties of moment sequences.

Keywords

Moment sequence Hausdorff mean Complete symmetric function 

Mathematics Subject Classification (2000)

40G05 30E05 05E05 26E60 

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References

  1. 1.
    Ahiezer, N.I., Krein, M.: Some Questions in the Theory of Moments. Am. Math. Soc. Translations, Providence (1962)Google Scholar
  2. 2.
    Belbachir H.: A multinomial extension of an inequality of Haber. J. Ineq. Pure Appl. Math. 9, 1–5 (2008)MathSciNetGoogle Scholar
  3. 3.
    Bennett, G.: Mercer’s inequality and totally monotonic sequences. (to appear)Google Scholar
  4. 4.
    Brenti, F.: Unimodal, log-concave and Pólya frequency sequences in combinatorics. Mem. Am. Math. Soc. 81(413), vii + 106 pp (1989)Google Scholar
  5. 5.
    Callan D.: On generating functions involving the square root of a quadratic polynomial. J. Integer Seq. 10, 1–7 (2007)MathSciNetGoogle Scholar
  6. 6.
    Chu W.: Finite differences and determinant identities. Linear Alg. Appl. 430, 215–228 (2009)MATHCrossRefGoogle Scholar
  7. 7.
    Davenport H., Pólya G.: On the product of two power series. Canad. J. Math. 1, 1–5 (1949)MATHCrossRefGoogle Scholar
  8. 8.
    DeTemple D.W., Robertson J.M.: On generalized symmetric means of two variables. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 634–637, 236–238 (1979)MathSciNetGoogle Scholar
  9. 9.
    Flajolet P., Sedgewick R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)MATHGoogle Scholar
  10. 10.
    Graham R.L., Knuth D.E., Patashnik O.: Concrete Mathematics. Addison-Wesley Publ. Co., Reading (1989)MATHGoogle Scholar
  11. 11.
    Haber S.: An elementary inequality. Int. J. Math. Math. Sci. 2, 531–535 (1979)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hardy G.H.: Divergent Series. Oxford University Press, Oxford (1949)MATHGoogle Scholar
  13. 13.
    Hardy G.H., Littlewood J.E., Pólya G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1967)Google Scholar
  14. 14.
    Harper L.H.: Stirling behavior is asymptotically normal. Ann. Math. Statist. 38, 410–414 (1967)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kurtz D.C.: A note on concavity properties of triangular arrays of numbers. J. Combin. Theory Ser. A 13, 135–139 (1972)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Laguerre E.: Oeuvres, vol. I. Gauthier-Villars, Paris (1898)Google Scholar
  17. 17.
    Lieb E.H.: Concavity properties and a generating function for Stirling numbers. J. Combin. Theory 5, 203–206 (1968)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lorentz G.G.: Logarithmic concavity. Am. Math. Monthly 61, 205 (1954)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Menon K.V.: Inequalities for symmetric functions. Duke Math. J. 35, 37–45 (1968)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Menon K.V.: Inequalities for generalized symmetric functions. Canad. Math. Bull. 12, 615–623 (1969)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Menon K.V.: An inequality of Schur and an inequality of Newton. Proc. Am. Math. Soc. 22, 441–449 (1969)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Mercer A.M.D.: A note on a paper by S. Haber. Int. J. Math. Math. Sci. 6, 609–611 (1983)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Milne-Thomson L.M.: The Calculus of Finite Differences. Macmillan, London (1933)Google Scholar
  24. 24.
    Mitrinovic D.S.: Analytic Inequalities. Springer, New York (1970)MATHGoogle Scholar
  25. 25.
    Neuman E.: On generalized symmetric means and Stirling numbers of the second kind. Zastus. Mat. 18, 645–656 (1985)MATHMathSciNetGoogle Scholar
  26. 26.
    Neuman E.: Inequalities involving generalized symmetric means. J. Math. Anal. Appl. 120, 315–320 (1986)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Neuman E.: On complete symmetric functions. SIAM J. Math. Anal. 19, 736–750 (1988)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Pólya G., Szegö G.: Problems and Theorems in Analysis II, 2nd edn. Springer, New York (1976)Google Scholar
  29. 29.
    Radoux C.: The Hankel determinant of exponential polynomials: a very short proof and a new result concerning Euler numbers. Am. Math. Monthly 109, 277–278 (2002)MATHCrossRefGoogle Scholar
  30. 30.
    Shohat J.A., Tamarkin J.D.: The Problem of Moments. Amer. Math. Soc., New York (1943)MATHGoogle Scholar
  31. 31.
    Sylvester J.J.: Mathematical Papers, 2. Cambridge University Press, Cambridge (1908)Google Scholar
  32. 32.
    Szegö G.: Orthogonal Polynomials. Amer. Math. Soc., New York (1939)Google Scholar
  33. 33.
    Szegö G.: On an inequality of P. Turán concerning Legendre polynomials. Bull. Am. Math. Soc. 54, 401–405 (1948)MATHCrossRefGoogle Scholar
  34. 34.
    Whiteley J.N.: A generalization of a theorem of Newton. Proc. Am. Math. Soc. 13, 144–151 (1962)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA

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