Advertisement

Positivity

, Volume 19, Issue 2, pp 251–288 | Cite as

Some pretty simple inequalities I

  • Grahame Bennett
  • Karl-G. Grosse-ErdmannEmail author
Article
  • 342 Downloads

Abstract

We show that a great variety of inequalities are best understood via an extrapolation principle: they hold true simply because they are valid on a small set of test sequences (or functions). The real challenge then is to determine the best constant. This invariably leads to interesting discussions of monotonicities of sequences.

Keywords

Extrapolation principle Cone Sums of powers Boas’s inequality Hölder’s inequality 

Mathematics Subject Classification (2000)

26D15 

References

  1. 1.
    Beesack, P.R.: Inequalities for absolute moments of a distribution: from Laplace to von Mises. J. Math. Anal. Appl. 98, 435–457 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bennett, G.: Lower bounds for matrices. Linear Algebra Appl. 82, 81–98 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bennett, G.: Some elementary inequalities. II. Q. J. Math. Oxf. Ser. (2) 39, 385–400 (1988)Google Scholar
  4. 4.
    Bennett, G.: Factorizing the classical inequalities. Mem. Am. Math. Soc. 120(576) (1996)Google Scholar
  5. 5.
    Bennett, G.: Inequalities complimentary to Hardy. Q. J. Math. Oxf. Ser. (2) 49, 395–432 (1998)Google Scholar
  6. 6.
    Bennett, G.: Sums of powers and the meaning of \(\ell ^p\). Houst. J. Math. 32, 801–831 (2006)zbMATHGoogle Scholar
  7. 7.
    Bennett, G.: Meaningful sequences. Houst. J. Math. 33, 555–580 (2007)zbMATHGoogle Scholar
  8. 8.
    Bennett, G.: Some forms of majorization. Houst. J. Math. 36, 1037–1066 (2010)zbMATHGoogle Scholar
  9. 9.
    Bennett, G., Grosse-Erdmann, K.-G.: On series of positive terms. Houst. J. Math. 31, 541–586 (2005)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Bennett, G., Grosse-Erdmann, K.-G.: Weighted Hardy inequalities for decreasing sequences and functions. Math. Ann. 334, 489–531 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Bennett, G., Grosse-Erdmann, K.-G.: Some pretty simple inequalities II (in preparation)Google Scholar
  12. 12.
    Bennett, G., Jameson, G.: Monotonic averages of convex functions. J. Math. Anal. Appl. 252, 410–430 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Bergh, J.: Hardy’s inequality—a complement. Math. Z. 202, 147–149 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Bergh, J.: A converse inequality of Hölder type. Math. Z. 215, 205–208 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Bergh, J., Burenkov, V., Persson, L.E.: Best constants in reversed Hardy’s inequalities for quasimonotone functions. Acta Sci. Math. (Szeged) 59, 221–239 (1994)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Bergh, J., Löfström, J.: Interpolation spaces. An introduction. Springer, Berlin (1976)zbMATHCrossRefGoogle Scholar
  17. 17.
    Boas, R.P.: Inequalities for monotonic series. J. Math. Anal. Appl. 1, 121–126 (1960)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Faber, G.: Bemerkungen zu Sätzen der Gaussschen theoria combinationis observationum. Sitzungsber. Bayer. Akad. Wiss. 1922, 7–21 (1922)Google Scholar
  19. 19.
    Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952)zbMATHGoogle Scholar
  20. 20.
    Heinig, H., Maligranda, L.: Weighted inequalities for monotone and concave functions. Studi. Math. 116, 133–165 (1995)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Lai, S.: Weighted norm inequalities for general operators on monotone functions. Trans. Am. Math. Soc. 340, 811–836 (1993)zbMATHCrossRefGoogle Scholar
  22. 22.
    van de Lune, J., te Riele, H.J.J.: On some conjectural inequalities and their consequences. CWI Report-Probability, networks and algorithms R0502 (2005)Google Scholar
  23. 23.
    Marshall, A.W., Olkin, I., Proschan, F.: Monotonicity of ratios of means and other applications of majorization. In: Shisha, O. (ed.) Inequalities (Proceedings of a Symposium Wright-Patterson Air Force Base, Ohio, 1965), pp. 177–190. Academic Press, New York (1967)Google Scholar
  24. 24.
    Myasnikov, E.A., Persson, L.E., Stepanov, V.D.: On the best constants in certain integral inequalities for monotone functions. Acta Sci. Math. (Szeged) 59, 613–624 (1994)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Pečarić, J.E., Persson, L.-E.: On Bergh’s inequality for quasi-monotone functions. J. Math. Anal. Appl. 195, 393–400 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Pečarić, J.E., Proschan, F., Tong, Y.L.: Convex functions, partial orderings, and statistical applications. Academic Press, Boston (1992)zbMATHGoogle Scholar
  27. 27.
    Roberts, A.W., Varberg, D.E.: Convex functions. Academic Press, New York (1973)zbMATHGoogle Scholar
  28. 28.
    Vasić, P.M., Milovanović, I.: On the ratio of means. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. no. 577–598, 33–37 (1977)Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA
  2. 2.Département de Mathématique, Institut ComplexysUniversité de MonsMonsBelgium

Personalised recommendations