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

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Advances in Mathematical Inequalities and Applications

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Abstract

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.

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References

  1. G.H. Hardy, Littlewood, G. Pólya, Inequalities. Cambridge Mathematical Library, 2nd edn. (Cambridge University Press, Cambridge, 1952), 324 p

    Google Scholar 

  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 p

    Google Scholar 

  3. D.S. Mitrinović, J.E. Pečarić, A.M. Fink, Classical and New Inequalities in Analysis (Kluwer, 1993), 740 p

    Google Scholar 

  4. A.W. Marshall, I. Olkin, B.C. Arnold, Inequalities: Theory of Majorization and Its Applications, 2nd edn. (Springer, 2011), 909 p

    Google Scholar 

  5. S.S. Dragomir, A Survey on Cauchy–Buniakowsky–Schwartz Type Discrete Inequalities. RGMIA monographs, (2003), 214 p

    Google Scholar 

  6. J.M. Steele, The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. (Cambridge University Press, Cambridge, 2004), 306 p

    Google Scholar 

  7. S.M. Sitnik, Generalized Young and Cauchy–Bunyakowsky Inequalities with Applications: A Survey 2010, (2012). arXiv:1012.3864, 51 p

  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. 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. H.A. Schwarz, Ueber ein Flachen Kleinsten Flacheninhalts betreffendes Problem der Variationsrechnung Acta. Soc. Sci. Fenn. 15 (1885), pp. 315–362

    Google Scholar 

  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. J. Sandór, Bouniakowsky and the Logarithmic Mean Inequalities. RGMIA Research Report Collection, 17, Article 5, 5 p (2014)

    Google Scholar 

  13. A. Pietsch, History of Banach Spaces and Linear Operators. Birkhäuser, 877 p (2007)

    Google Scholar 

  14. P.S. Bullen, D.S. Mitrinović, P.M. Vasić, Means and Their Inequalities, 2nd edn. 2015 (D.Reidel Publishing Company, Dordrecht, 1988), 480 p

    Google Scholar 

  15. P.S. Bullen, Handbook of Means and Their Inequalities (Kluwer, 2003), 587 p

    Google Scholar 

  16. T. Radó, On convex functions. Trans. Amer. Math. Soc. 37, 266–285 (1935)

    Article  MathSciNet  Google Scholar 

  17. F.W.J. Ed. Olver, NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, 2010), 968 p

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  19. D.B. Karp, S.M. Sitnik, Inequalities and monotonicity of ratios for generalized hypergeometric function. J. Approximation Theory 161, 337–352 (2009)

    Article  MathSciNet  Google Scholar 

  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. 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. J.X. Xiang, A note on the Cauchy–Schwarz inequality. Am. Math. Monthly 120(5), 456–459 (2013)

    Google Scholar 

  23. I. Pinelis, On the Hölder and Cauchy–Schwarz Inequalities, 3 p (2015). arXiv:1503.00348v2

  24. I. Pinelis, On the Hölder and Cauchy–Schwarz inequalities. Am. Math. Monthly 121, 1 (2015)

    Google Scholar 

  25. E. Omey, On Xiang’s observations concerning the Cauchy–Schwarz inequality. Am. Math. Monthly 122(7), 696–698 (2015)

    Google Scholar 

  26. M. Masjed-Jamei, E. Omey, Improvement of some classical inequalities. J. Inequalities Spec. Funct. 7(1), 18–28 (2016)

    MathSciNet  Google Scholar 

  27. G. Alpargu, The Kantorovich Inequality, with Some Extensions and with Some Statistical Applications (Thesis, McGill University, Montréale, Canada, 1996)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Anand International College of Engineering, Near Kanota, Agra Road, Jaipur, 303012, Rajasthan, India

    P. Agarwal

  2. International Centre for Basic and Applied Sciences, Jaipur, India

    P. Agarwal

  3. Department of Mathematical Analysis, Institute of Mathematics, Economics and Mechanics, Odessa I.I. Mechnikov National University, Odessa, Ukraine

    A. A. Korenovskii

  4. Belgorod State National Research University, Belgorod, Russia

    S. M. Sitnik

Authors
  1. P. Agarwal
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  2. A. A. Korenovskii
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  3. S. M. Sitnik
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Corresponding author

Correspondence to P. Agarwal .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Anand International College of Engineering, Jaipur, India

    Praveen Agarwal

  2. Department of Mathematics, Victoria University, Melbourne, VIC, Australia

    Silvestru Sever Dragomir

  3. Department of Mathematics, King Saud University, Riyadh, Saudi Arabia

    Mohamed Jleli

  4. Department of Mathematics, King Saud University, Riyadh, Saudi Arabia

    Bessem Samet

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Agarwal, P., Korenovskii, A.A., Sitnik, S.M. (2018). A Generalization of Cauchy–Bunyakovsky Integral Inequality Via Means with Max and Min Values. In: Agarwal, P., Dragomir, S., Jleli, M., Samet, B. (eds) Advances in Mathematical Inequalities and Applications. Trends in Mathematics. Birkhäuser, Singapore. https://doi.org/10.1007/978-981-13-3013-1_18

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