A Generalization of Cauchy–Bunyakovsky Integral Inequality Via Means with Max and Min Values

  • P. AgarwalEmail author
  • A. A. Korenovskii
  • S. M. Sitnik
Part of the Trends in Mathematics book series (TM)


In the paper, we give a brief survey of a method for constructing generalizations of Cauchy–Bunyakovsky integral inequality using abstract mean values. One special inequality of this type is considered in details in terms of min and max functions. Some direct proofs of this inequality are given and application to inequalities for special functions. Also related recent references are briefly considered.


  1. 1.
    G.H. Hardy, Littlewood, G. Pólya, Inequalities. Cambridge Mathematical Library, 2nd edn. (Cambridge University Press, Cambridge, 1952), 324 pGoogle Scholar
  2. 2.
    E.F. Beckenbach, R. Bellman, Inequalities. Second revised printing. Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Band 30.-(Springer, New York, Inc., 1965), xi+198 pGoogle Scholar
  3. 3.
    D.S. Mitrinović, J.E. Pečarić, A.M. Fink, Classical and New Inequalities in Analysis (Kluwer, 1993), 740 pGoogle Scholar
  4. 4.
    A.W. Marshall, I. Olkin, B.C. Arnold, Inequalities: Theory of Majorization and Its Applications, 2nd edn. (Springer, 2011), 909 pGoogle Scholar
  5. 5.
    S.S. Dragomir, A Survey on Cauchy–Buniakowsky–Schwartz Type Discrete Inequalities. RGMIA monographs, (2003), 214 pGoogle Scholar
  6. 6.
    J.M. Steele, The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. (Cambridge University Press, Cambridge, 2004), 306 pGoogle Scholar
  7. 7.
    S.M. Sitnik, Generalized Young and Cauchy–Bunyakowsky Inequalities with Applications: A Survey 2010, (2012). arXiv:1012.3864, 51 p
  8. 8.
    A.L. Cauchy, Cours d’analyse de l’ École Royale Polytechnique. I partie. Analyse algebrique. (Oeuvres complè tes, II serie, III). (Paris, 1821)Google Scholar
  9. 9.
    V. Buniakowski, Sur quelques inégalités concernant les intégrales ordinaires et les intégrales aux différences finies Mémoires de l’ Acad. de St. - Pétersbourg (VII), 9 (1859)Google Scholar
  10. 10.
    H.A. Schwarz, Ueber ein Flachen Kleinsten Flacheninhalts betreffendes Problem der Variationsrechnung Acta. Soc. Sci. Fenn. 15 (1885), pp. 315–362Google Scholar
  11. 11.
    J. Sandór, Applications of the Cauchy–Bouniakowsky Inequality in the Theory of Means. RGMIA Research Report Collection, 17, Article 11, 18 p (2014)Google Scholar
  12. 12.
    J. Sandór, Bouniakowsky and the Logarithmic Mean Inequalities. RGMIA Research Report Collection, 17, Article 5, 5 p (2014)Google Scholar
  13. 13.
    A. Pietsch, History of Banach Spaces and Linear Operators. Birkhäuser, 877 p (2007)Google Scholar
  14. 14.
    P.S. Bullen, D.S. Mitrinović, P.M. Vasić, Means and Their Inequalities, 2nd edn. 2015 (D.Reidel Publishing Company, Dordrecht, 1988), 480 pGoogle Scholar
  15. 15.
    P.S. Bullen, Handbook of Means and Their Inequalities (Kluwer, 2003), 587 pGoogle Scholar
  16. 16.
    T. Radó, On convex functions. Trans. Amer. Math. Soc. 37, 266–285 (1935)MathSciNetCrossRefGoogle Scholar
  17. 17.
    F.W.J. Ed. Olver, NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, 2010), 968 pGoogle Scholar
  18. 18.
    D.B. Karp, S.M. Sitnik, Log–convexity and log–concavity of hypergeometric–like functions. J. Math. Anal. Appl. 364(2), 384–394 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    D.B. Karp, S.M. Sitnik, Inequalities and monotonicity of ratios for generalized hypergeometric function. J. Approximation Theory 161, 337–352 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    S.M. Sitnik, K.h. Mehrez, Proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions. Analysis (De Gruyter) 36(4), 263–268 (2016)Google Scholar
  21. 21.
    Kh. Mehrez, S.M. Sitnik, Results in mathematics (Springer International Publishing). Funct. Inequalities Mittag-Leffler Funct. 72(1–2), 703–714 (2017)Google Scholar
  22. 22.
    J.X. Xiang, A note on the Cauchy–Schwarz inequality. Am. Math. Monthly 120(5), 456–459 (2013)Google Scholar
  23. 23.
    I. Pinelis, On the Hölder and Cauchy–Schwarz Inequalities, 3 p (2015). arXiv:1503.00348v2
  24. 24.
    I. Pinelis, On the Hölder and Cauchy–Schwarz inequalities. Am. Math. Monthly 121, 1 (2015)Google Scholar
  25. 25.
    E. Omey, On Xiang’s observations concerning the Cauchy–Schwarz inequality. Am. Math. Monthly 122(7), 696–698 (2015)Google Scholar
  26. 26.
    M. Masjed-Jamei, E. Omey, Improvement of some classical inequalities. J. Inequalities Spec. Funct. 7(1), 18–28 (2016)MathSciNetGoogle Scholar
  27. 27.
    G. Alpargu, The Kantorovich Inequality, with Some Extensions and with Some Statistical Applications (Thesis, McGill University, Montréale, Canada, 1996)Google Scholar
  28. 28.
    N.S. Barnett, S.S. Dragomir, An additive reverse of the Cauchy-Bunyakovsky-Schwarz integral inequality. Appl. Math. Lett. 21(4), 388–393 (2008)MathSciNetCrossRefGoogle Scholar
  29. 29.
    T.K. Pogány, A new (probabilistic) proof of the Diaz-Metcalf and Pólya-Szegó inequalities and some of their consequences. Theory Probab. Math. Statist. 70, 113–122 (2005)CrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • P. Agarwal
    • 1
    • 2
    Email author
  • A. A. Korenovskii
    • 3
  • S. M. Sitnik
    • 4
  1. 1.Department of MathematicsAnand International College of EngineeringJaipurIndia
  2. 2.International Centre for Basic and Applied SciencesJaipurIndia
  3. 3.Department of Mathematical Analysis, Institute of Mathematics, Economics and MechanicsOdessa I.I. Mechnikov National UniversityOdessaUkraine
  4. 4.Belgorod State National Research UniversityBelgorodRussia

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