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The inequality of Milne and its converse II

Abstract

We prove the following let, and be real numbers, and let be positive real numbers with. The inequalities hold for all real numbers if and only if and. Furthermore, we provide a matrix version. The first inequality (with and) is a discrete counterpart of an integral inequality published by E. A. Milne in 1925.

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References

  1. 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 

  2. Horn RA, Johnson CR: Matrix Analysis. Cambridge University Press, Cambridge; 1985:xiii+561.

    Book  MATH  Google Scholar 

  3. Horn RA, Johnson CR: Topics in Matrix Analysis. Cambridge University Press, Cambridge; 1991:viii+607.

    Book  MATH  Google Scholar 

  4. Leach EB, Sholander MC: Extended mean values. The American Mathematical Monthly 1978,85(2):84–90. 10.2307/2321783

    Article  MATH  MathSciNet  Google Scholar 

  5. Leach EB, Sholander MC: Extended mean values. II. Journal of Mathematical Analysis and Applications 1983,92(1):207–223. 10.1016/0022-247X(83)90280-9

    Article  MATH  MathSciNet  Google Scholar 

  6. Leach EB, Sholander MC: Multi-variable extended mean values. Journal of Mathematical Analysis and Applications 1984,104(2):390–407. 10.1016/0022-247X(84)90003-9

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  8. Rao C: Statistical proofs of some matrix inequalities. Linear Algebra and its Applications 2000,321(1–3):307–320. 10.1016/S0024-3795(99)00276-1

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Horst Alzer.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Alzer, H., Kovačec, A. The inequality of Milne and its converse II. J Inequal Appl 2006, 21572 (2006). https://doi.org/10.1155/JIA/2006/21572

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  • DOI: https://doi.org/10.1155/JIA/2006/21572

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