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Variants of Čebyšev's inequality with applications


Several variants of Čebyšev's inequality for two monotonic-tuples and also nonnegative-tuples monotonic in the same direction are presented. Immediately after that their refinements of Ostrowski's type are given. Obtained results are used to prove generalizations of discrete Milne's inequality and its converse in which weights satisfy conditions as in the Jensen-Steffensen inequality.



  1. Abramovich S, Klaričić Bakula M, Matić M, Pečarić J: A variant of Jensen-Steffensen's inequality and quasi-arithmetic means. Journal of Mathematical Analysis and Applications 2005,307(1):370–386. 10.1016/j.jmaa.2004.10.027

    MATH  MathSciNet  Article  Google Scholar 

  2. Alzer H, Kovačec A: The inequality of Milne and its converse. Journal of Inequalities and Applications 2002,7(4):603–611. 10.1155/S1025583402000292

    MATH  MathSciNet  Google Scholar 

  3. Lupaş A: On an inequality. Publikacije Elektrotehnickog Fakulteta Univerziteta U Beogradu. Serija Matematika i Fizika 1981, (716–734):32–34.

  4. Milne EA: Note on Rosseland's integral for the stellar absorption coefficient. Monthly Notices of the Royal Astronomical Society 1925, 85: 979–984.

    MATH  Article  Google Scholar 

  5. Mercer AMcD: A variant of Jensen's inequality. Journal of Inequalities in Pure and Applied Mathematics 2003,4(4):1–2. article 73 article 73

    MathSciNet  Google Scholar 

  6. Pečarić J: On the Ostrowski generalization of Čebyšev's inequality. Journal of Mathematical Analysis and Applications 1984,102(2):479–487. 10.1016/0022-247X(84)90187-2

    MATH  MathSciNet  Article  Google Scholar 

  7. Pečarić J: On the Čebyšev inequality. Buletinul Ştiinţific şi Tehnic Institutului Politehnic "Traian Vuia" Timişoara 1980,25(39)(1):5–9 (1981).

    Google Scholar 

  8. Pečarić J, Proschan F, Tong YL: Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering. Volume 187. Academic Press, Massachusetts; 1992:xiv+467.

    Google Scholar 

  9. Rao CR: Statistical proofs of some matrix inequalities. Linear Algebra and Its Applications 2000,321(1–3):307–320.

    MATH  MathSciNet  Google Scholar 

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Correspondence to M. Klaričić Bakula.

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Bakula, M.K., Matković, A. & Pečarić, J. Variants of Čebyšev's inequality with applications. J Inequal Appl 2006, 39692 (2006).

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  • Jensen-Steffensen Inequality
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