# A Converse of Minkowski's Type Inequalities

- Romeo Meštrović
^{1}Email author and - David Kalaj
^{2}

**2010**:461215

https://doi.org/10.1155/2010/461215

© R. Meštrović and David Kalaj. 2010

**Received: **6 August 2010

**Accepted: **20 October 2010

**Published: **24 October 2010

## Abstract

## Keywords

## 1. Introduction

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 [2]) 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.

Theorem 1.1.

When , the equalities in (1.6) concerned with and are also attained for previously defined values .

Remark 1.2.

The above inequality is sharp if , but it is not in spirit of a converse of Minkowski's type inequality.

The following consequence of Theorem 1.1 for and can be viewed as a converse of Minkowski's inequality (1.1).

Corollary 1.3.

Remark 1.4.

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).

If , then in (1.4), and a related proof is the same as that for the following case when .

which is actually inequality (1.4) with the constant .

as desired.

The proof can be obtained from those of Case 1, by replacing " " with " " in each related inequality.

If , then the proof is the same as that for Case 6. If , then the proof can be obtained from those of Case 2, by replacing " " with " " in each related inequality.

and the proof is completed.

## 3. The Integral Analogue of Theorem 1.1

Recall that the usual norm of is defined as if ; if .

The following result is the integral analogue of Theorem 1.1.

Theorem 3.1.

*Both inequalities are sharp*

*For *
* the equality in* (3.2) and (3.3) is attained if
a.e. on
. If
or
, then the equality is attained for
, where
are
-measurable sets with
, such that
and
whenever
.

Proof.

Obviously, the above inequality yields (3.2) for .

Corollary 3.2.

## Authors’ Affiliations

## References

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*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.
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## Copyright

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