The inequality of Milne and its converse II
© Alzer and Kovačec 2006
Received: 15 September 2004
Accepted: 19 September 2004
Published: 3 January 2006
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.
- Alzer H, Kovačec A: The inequality of Milne and its converse. Journal of Inequalities and Applications 2002,7(4):603–611. 10.1155/S1025583402000292MATHMathSciNetGoogle Scholar
- Horn RA, Johnson CR: Matrix Analysis. Cambridge University Press, Cambridge; 1985:xiii+561.MATHView ArticleGoogle Scholar
- Horn RA, Johnson CR: Topics in Matrix Analysis. Cambridge University Press, Cambridge; 1991:viii+607.MATHView ArticleGoogle Scholar
- Leach EB, Sholander MC: Extended mean values. The American Mathematical Monthly 1978,85(2):84–90. 10.2307/2321783MATHMathSciNetView ArticleGoogle Scholar
- Leach EB, Sholander MC: Extended mean values. II. Journal of Mathematical Analysis and Applications 1983,92(1):207–223. 10.1016/0022-247X(83)90280-9MATHMathSciNetView ArticleGoogle Scholar
- 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)90003-9MATHMathSciNetView ArticleGoogle Scholar
- Milne E: Note on Rosseland's integral for the stellar absorption coefficient. Monthly Notices of the Royal Astronomical Society 1925, 85: 979–984.MATHView ArticleGoogle Scholar
- Rao C: Statistical proofs of some matrix inequalities. Linear Algebra and its Applications 2000,321(1–3):307–320. 10.1016/S0024-3795(99)00276-1MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.