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# A Converse of Minkowski's Type Inequalities

*Journal of Inequalities and Applications*
**volume 2010**, Article number: 461215 (2010)

## Abstract

We formulate and prove a converse for a generalization of the classical Minkowski's inequality. The case when is also considered. Applying the same technique, we obtain an analog converse theorem for integral Minkowski's type inequality.

## 1. Introduction

If , , and () are real numbers, then by the classical Minkowski's inequality

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 "".

Many extensions and generalizations of Minkowski's inequality can be found in [2, 3]. We want to point out the following inequality:

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.

An extension of inequality (1.2) was formulated by Ingham and Jessen (see [2, pages 31-32]). In 1948, Tôyama [4] published a converse of the inequality of Ingham and Jessen (see also a recent paper [5] for a weighted version of Tôyama's inequality). Namely, Tôyama showed that if and () are real numbers, then

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.

Let , , and be real numbers. Then for we have

where is a positive constant given by

If , then

where is a positive constant given by

Inequality (1.4) with and inequality (1.6) with are sharp for all and , and they are attained for , , . If , then inequality (1.4) is sharp in the cases when and . In both cases the equalities are attained for

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

Remark 1.2.

Note that, proceeding as in the proof of Theorem 1.1, we can prove similar inequalities to (1.4) and (1.6) with instead of on the left-hand side of these inequalities. For example, such an inequality concerning the case when (i.e., (1.4)) is

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.

Let , , and let , be real numbers. Then for

If , then

Remark 1.4.

It is well known that Minkowski's inequality is also true for complex sequences as well. More precisely, if and , () are arbitrary complex numbers, then

Note that the above inequality with and , , for each , becomes

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]).

If are nonnegative real numbers and , then

Proof of Theorem 1.1.

In our proof we often use the well-known fact that the scale of power means is nondecreasing (see [2]). More precisely, if are nonnegative integers and , then

In all the cases, for each , we denote that

We will consider all the six cases related to the inequalities (1.4) and (1.6).

Case 1 ().

The inequality between power means of orders and 1 for positive numbers , , states that

whence for any fixed , after substitution of , , we obtain

whence after summation over we find that

Because , the inequality between power means of orders and 1 implies that

The above inequality and (2.6) immediately yield

Case 2 ().

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

Now suppose that . By the inequality for power means of orders and 1, we obtain

Next, by the inequality for power means (of orders and 1), we obtain

For any fixed the inequality (2.1) of Lemma 2.1 with implies that

Obviously, inequalities (2.9), (2.10), and (2.11) immediately yield

which is actually inequality (1.4) with the constant .

Case 3 ().

By inequality (2.1) with and , for each , we obtain

whence after summation over , we have

By the inequality for power means (of orders and 1), we get

or equivalently

The above inequality and (2.14) immediately yield

as desired.

Case 4 ().

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

Case 5 ().

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.

Case 6 ().

For any fixed , inequality (2.1) of Lemma 2.1 with and gives

whence after summation over , we get

As , for positive integers , there holds

whence for any fixed , after substitution of , , we obtain

The above inequality and (2.19) immediately yield

and the proof is completed.

## 3. The Integral Analogue of Theorem 1.1

Let be a measure space with a positive Borel measure . For any let denote the usual Lebesgue space consisting of all -measurable complex-valued functions such that

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.

For given let be arbitrary functions in . Then, if , we have

If , then

*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.

The proof of each inequality is completely similar to the corresponding one given in Theorem 1.1 with a fixed . For clarity, we give here only a proof related to the case when . Applying the inequality between power means of orders and 1 to the functions (), we have

Integrating the above relation, we obtain

which can be written in the form

Obviously, the above inequality yields (3.2) for .

Corollary 3.2.

Let , and let be a complex function in . Then there holds the sharp inequality

## References

Minkowski H:

*Geometrie der Zahlen*. Teubner, Leipzig, Germany; 1910.Hardy GH, Littlewood JE, Pólya G:

*Inequalities*. Cambridge Univerity Press, Cambridge, UK; 1952:xii+324.Beckenbach EF, Bellman R:

*Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete*.*Volume 30*. Springer, Berlin, Germany; 1961:xii+198.Tôyama H: On the inequality of Ingham and Jessen.

*Proceedings of the Japan Academy*1948, 24(9):10–12. 10.3792/pja/1195572073Alzer H, Ruscheweyh S: A converse of Minkowski's inequality.

*Discrete Mathematics*2000, 216(1–3):253–256.

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Meštrović, R., Kalaj, D. A Converse of Minkowski's Type Inequalities.
*J Inequal Appl* **2010**, 461215 (2010). https://doi.org/10.1155/2010/461215

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