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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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ1_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ4_HTML.gif)
where is a positive constant given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ5_HTML.gif)
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ6_HTML.gif)
where is a positive constant given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ10_HTML.gif)
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ12_HTML.gif)
Note that the above inequality with and
,
, for each
, becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ13_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ14_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ15_HTML.gif)
In all the cases, for each , we denote that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ16_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ17_HTML.gif)
whence for any fixed , after substitution of
,
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ18_HTML.gif)
whence after summation over we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ19_HTML.gif)
Because , the inequality between power means of orders
and 1 implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ20_HTML.gif)
The above inequality and (2.6) immediately yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ21_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ22_HTML.gif)
Next, by the inequality for power means (of orders and 1), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ23_HTML.gif)
For any fixed the inequality (2.1) of Lemma 2.1 with
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ24_HTML.gif)
Obviously, inequalities (2.9), (2.10), and (2.11) immediately yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ25_HTML.gif)
which is actually inequality (1.4) with the constant .
Case 3 ().
By inequality (2.1) with and
, for each
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ26_HTML.gif)
whence after summation over , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ27_HTML.gif)
By the inequality for power means (of orders and 1), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ28_HTML.gif)
or equivalently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ29_HTML.gif)
The above inequality and (2.14) immediately yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ30_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ31_HTML.gif)
whence after summation over , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ32_HTML.gif)
As , for positive integers
, there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ33_HTML.gif)
whence for any fixed , after substitution of
,
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ34_HTML.gif)
The above inequality and (2.19) immediately yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ35_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ36_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ37_HTML.gif)
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ38_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ39_HTML.gif)
Integrating the above relation, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ40_HTML.gif)
which can be written in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ41_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F461215/MediaObjects/13660_2010_Article_2156_Equ42_HTML.gif)
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/1195572073
Alzer 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