A Converse of Minkowski's Type Inequalities

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

(1.1)

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:

(1.2)

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

(1.3)

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

(1.4)

where is a positive constant given by

(1.5)

If , then

(1.6)

where is a positive constant given by

(1.7)

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

(1.8)

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

(1.9)

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

(1.10)

If , then

(1.11)

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

(1.12)

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

(1.13)

We see that the first inequality of Corollary 1.3 may be actually regarded as a converse of the previous inequality.

Lemma 2.1 (see [2, page 26]).

If are nonnegative real numbers and , then

(2.1)

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

(2.2)

In all the cases, for each , we denote that

(2.3)

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

(2.4)

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

(2.5)

whence after summation over we find that

(2.6)

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

(2.7)

The above inequality and (2.6) immediately yield

(2.8)

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

(2.9)

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

(2.10)

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

(2.11)

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

(2.12)

which is actually inequality (1.4) with the constant .

Case 3 ( ).

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

(2.13)

whence after summation over , we have

(2.14)

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

(2.15)

or equivalently

(2.16)

The above inequality and (2.14) immediately yield

(2.17)

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

(2.18)

whence after summation over , we get

(2.19)

As , for positive integers , there holds

(2.20)

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

(2.21)

The above inequality and (2.19) immediately yield

(2.22)

and the proof is completed.

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

(3.1)

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.

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

(3.2)

If , then

(3.3)

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

(3.4)

Integrating the above relation, we obtain

(3.5)

which can be written in the form

(3.6)

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

(3.7)