A Converse of Minkowski's Type Inequalities
© R. Meštrović and David Kalaj. 2010
Received: 6 August 2010
Accepted: 20 October 2010
Published: 24 October 2010
This inequality was published by Minkowski [1, pages 115–117] hundred years ago in his famous book "Geometrie der Zahlen."
It is also known (see ) that for the above inequality is satisfied with " " instead of " ".
where and ( ) are real numbers. Furthermore, if , then the inequality (1.2) is satisfied with " " instead of " " [2, Theorem , page 30]. In both cases, equality holds if and only if all columns , , are proportional.
The main result of this paper gives a converse of inequality (1.2). On the other hand, our result may be regarded as a nonsymmetric analogue of the above inequality, and it is given as follows.
We see that the first inequality of Corollary 1.3 may be actually regarded as a converse of the previous inequality.
2. Proof of Theorem 1.1
Lemma 2.1 (see [2, page 26]).
Proof of Theorem 1.1.
We will consider all the six cases related to the inequalities (1.4) and (1.6).
and the proof is completed.
3. The Integral Analogue of Theorem 1.1
The following result is the integral analogue of Theorem 1.1.
Both inequalities are sharp
- Minkowski H: Geometrie der Zahlen. Teubner, Leipzig, Germany; 1910.MATHGoogle Scholar
- Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge Univerity Press, Cambridge, UK; 1952:xii+324.MATHGoogle Scholar
- Beckenbach EF, Bellman R: Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete. Volume 30. Springer, Berlin, Germany; 1961:xii+198.Google Scholar
- Tôyama H: On the inequality of Ingham and Jessen. Proceedings of the Japan Academy 1948, 24(9):10–12. 10.3792/pja/1195572073MathSciNetView ArticleMATHGoogle Scholar
- Alzer H, Ruscheweyh S: A converse of Minkowski's inequality. Discrete Mathematics 2000, 216(1–3):253–256.MathSciNetView ArticleMATHGoogle 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.