Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 461215, 9 pages
doi:10.1155/2010/461215
Research Article
A Converse of Minkowski’s Type Inequalities
Romeo Meštrović1 and David Kalaj2
1
2
Maritime Faculty, University of Montenegro, Dobrota 36, 85330 Kotor, Montenegro
Faculty of Natural Sciences and Mathematics, University of Montenegro, Džordža Vašingtona BB,
81000 Podgorica, Montenegro
Correspondence should be addressed to Romeo Meštrović, romeo@ac.me
Received 6 August 2010; Accepted 20 October 2010
Academic Editor: Jong Kim
Copyright q 2010 R. Meštrović and D. Kalaj. 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.
We formulate and prove a converse for a generalization of the classical Minkowski’s inequality.
The case when 0 < p < 1 is also considered. Applying the same technique, we obtain an analog
converse theorem for integral Minkowski’s type inequality.
1. Introduction
If p > 1, ai ≥ 0, and bi ≥ 0 i 1, . . . , n are real numbers, then by the classical Minkowski’s
inequality
1/p 1/p 1/p
n
n
n
p
p
p
≤
ai
bi
.
ai bi i1
i1
1.1
i1
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 0 < p < 1 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:
⎛
⎝
n
m
j1
i1
⎛
⎞1/p
p ⎞1/p
m
n
⎝ ap ⎠ ,
aij ⎠ ≤
ij
i1
j1
1.2
2
Journal of Inequalities and Applications
where p > 1 and aij ≥ 0 i 1, . . . , m; j 1, . . . , n are real numbers. Furthermore, if 0 <
p < 1, then the inequality 1.2 is satisfied with “≥” instead of “≤” 2, Theorem 24, page 30.
In both cases, equality holds if and only if all columns a1j , a2j , . . . , amj , j 1, 2, . . . , n, 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 0 < q < p and aij ≥ 0 i 1, . . . , m; j 1, . . . , n are real numbers, then
⎛
⎞q/p ⎞1/q
⎛ p/q ⎞1/p
m
n
n
m
p⎠
q
⎟
⎜ ⎝
1/q−1/p ⎝
⎠ .
aij
aij
⎠ ≤ minm, n
⎝
⎛
i1
j1
j1
1.3
i1
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 p > 0, q > 0, and aij ≥ 0 i 1, . . . , m; j 1, . . . , n be real numbers. Then for
p ≥ 1 we have
m
i1
⎛ ⎛
⎞1/p
p/q ⎞1/p
n
n
m
q
⎝ ap ⎠ ≤ C⎝
⎠ ,
aij
ij
j1
j1
1.4
i1
where C is a positive constant given by
⎧
⎪
m1−1/q
⎪
⎪
⎨
C minm, n1/q−1/p m1−1/q
⎪
⎪
⎪
⎩ 1−1/p
m
if 1 ≤ p ≤ q,
if 1 ≤ q < p,
1.5
if 0 < q ≤ 1 ≤ p.
If 0 < p < 1, then
⎛ ⎛
⎞1/p
p/q ⎞1/p
m
n
n
m
p
q
⎝ a ⎠ ≥ K⎝
⎠ ,
aij
ij
i1
j1
j1
1.6
i1
where K is a positive constant given by
⎧
⎪
m1−1/q
⎪
⎪
⎨
K minm, n1/q−1/p m1−1/q
⎪
⎪
⎪
⎩ 1−1/p
m
if 0 < q ≤ p < 1,
if 0 < p < q < 1,
if 0 < p < 1 ≤ q.
1.7
Journal of Inequalities and Applications
3
Inequality 1.4 with 1 ≤ p ≤ q and inequality 1.6 with 0 < q ≤ p < 1 are sharp for all m and n,
and they are attained for aij a, i 1, . . . , m, j 1, . . . , n. If m ≤ n, then inequality 1.4 is sharp in
the cases when 1 ≤ q < p and 0 < q ≤ 1 ≤ p. In both cases the equalities are attained for
aij ⎧
⎨a,
if i j,
⎩0,
if i / j.
1.8
When m ≤ n, the equalities in 1.6 concerned with 0 < p < q < 1 and 0 < p < 1 ≤ q are also attained
for previously defined values aij .
Remark 1.2. Note that, proceeding as in the proof of Theorem 1.1, we can prove similar
m n
inequalities to 1.4 and 1.6 with nj1 m
i1 instead of
i1 j1 on the left-hand side
of these inequalities. For example, such an inequality concerning the case when 1 ≤ q < p
i.e., 1.4 is
⎛ 1/p
p/q ⎞1/p
n
m
n
m
p
q
⎠ .
aij
≤ n1−1/p ⎝
aij
j1
i1
j1
1.9
i1
The above inequality is sharp if n ≤ m, but it is not in spirit of a converse of Minkowski’s type
inequality.
The following consequence of Theorem 1.1 for m 2 and q 2 can be viewed as a
converse of Minkowski’s inequality 1.1.
Corollary 1.3. Let n ≥ 1, p > 0, and let aj ≥ 0, bj ≥ 0 j 1, . . . , n be real numbers. Then for p ≥ 1
⎞1/p ⎛
⎞1/p
⎛
⎞1/p
n
n
n p/2
⎠ .
⎝ ap ⎠ ⎝ bp ⎠ ≤ 21−min{1/2,1/p} ⎝
a2j bj2
j
j
⎛
j1
j1
1.10
j1
If 0 < p < 1, then
⎞1/p ⎛
⎞1/p
⎛
⎞1/p
n
n
n p/2
⎠ .
⎝ ap ⎠ ⎝ vp ⎠ ≥ 21−1/p ⎝
a2j bj2
j
j
⎛
j1
j1
1.11
j1
Remark 1.4. It is well known that Minkowski’s inequality is also true for complex sequences
as well. More precisely, if p ≥ 1 and ui , vi i 1, . . . , n are arbitrary complex numbers, then
⎞1/p ⎛
⎞1/p ⎛
⎞1/p
n n n p
⎝ uj vj ⎠ ≤ ⎝ uj p ⎠ ⎝ vj p ⎠ .
⎛
j1
j1
j1
1.12
4
Journal of Inequalities and Applications
Note that the above inequality with uj aj ∈ R and vj ibj , bj ∈ R, for each j 1, 2, . . . , n,
becomes
⎛
⎞1/p ⎛
⎞1/p ⎛
⎞1/p
n n
n
p/2
p
p
⎝
⎠ ≤⎝ a ⎠ ⎝ v ⎠ .
a2j bj2
j
j
j1
j1
1.13
j1
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 u1 , u2 , . . . uk , s, r are nonnegative real numbers and 0 < s < r,
then
us1 us2 · · · usk
1/s
1/r
≥ ur1 ur2 · · · urk
.
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 a1 , a2 , . . . , ak are nonnegative integers
and 0 < α ≤ β < ∞, then
k
i1
aαi
⎛
1/α
≤⎝
k
k
i1
β
ai
k
⎞1/β
⎠
.
2.2
In all the cases, for each i 1, 2, . . . , m, we denote that
⎞1/p
n
p
ai : ⎝ aij ⎠ .
⎛
2.3
j1
We will consider all the six cases related to the inequalities 1.4 and 1.6.
Case 1 1 ≤ p ≤ q. The inequality between power means of orders q/p ≥ 1 and 1 for m
positive numbers bi , i 1, 2, . . . , m, states that
⎛
⎝
m
i1
q/p
bi
m
⎞p/q
⎠
≥
m
i1
m
bi
2.4
,
p
whence for any fixed j 1, 2, . . . n, after substitution of bi aij , i 1, 2, . . . m, we obtain
q
q
q
a1j a2j · · · amj
p/q
p
p
p
≥ mp/q−1 a1j a2j · · · amj ,
2.5
Journal of Inequalities and Applications
5
whence after summation over j we find that
n q
q
q
j1
a1j a2j · · · amj
p/q
≥ mp/q−1
m
n p
aij
j1 i1
mp/q−1
m
i1
2.6
p
ai .
Because p ≥ 1, the inequality between power means of orders p and 1 implies that
p
m
m
p
1−p
ai ≥ m
ai .
i1
2.7
i1
The above inequality and 2.6 immediately yield
⎛ ⎛
⎞1/p
p/q ⎞1/p
n
m
m
n
q
⎠ ≥
⎝ ap ⎠ .
m1−1/q ⎝
aij
ij
j1
i1
i1
2.8
j1
Case 2 1 ≤ q < p. If m ≤ n, then C m1−1/p in 1.4, and a related proof is the same as that
for the following case when 0 < q ≤ 1 ≤ p.
Now suppose that m > n. By the inequality for power means of orders p/q ≥ 1 and 1,
we obtain
⎛
⎜
⎝
n
j1
q
q
q
a1j a2j · · · amj
p/q ⎞q/p
⎟
⎠
n
n ≥
j1
q
q
q
a1j a2j · · · amj
m
·
n
n
m q
q
q
ai1 ai2 · · · ain
i1
2.9
m
.
Next, by the inequality for power means of orders q ≥ 1 and 1, we obtain
m i1
q
q
q
ai1 ai2 · · · ain
⎛
⎜
≥⎝
m
m
i1
q
q
q
ai1 ai2 · · · ain
1/q ⎞q
m
⎟
⎠ .
2.10
For any fixed i ∈ {1, 2, . . . , m} the inequality 2.1 of Lemma 2.1 with s p > q r implies
that
q
q
q
ai1 ai2 · · · ain
1/q
p
p
p 1/p
≥ ai1 ai2 · · · ain
.
2.11
6
Journal of Inequalities and Applications
Obviously, inequalities 2.9, 2.10, and 2.11 immediately yield
⎛ ⎛
⎛ ⎞1/p ⎞q
p/q ⎞q/p
n
m
m
n
q
⎟
⎜ p
⎠
aij
≥ ⎝ ⎝ aij ⎠ ⎠ ,
n1−q/p · mq−1 ⎝
j1
i1
i1
2.12
j1
which is actually inequality 1.4 with the constant C n1/q−1/p · m1−1/q .
Case 3 0 < q ≤ 1 ≤ p. By inequality 2.1 with r q and s p, for each j 1, 2, . . . , n, we
obtain
q
q
q
a1j a2j · · · amj
p/q
p
p
p
≥ a1j a2j · · · amj ,
2.13
whence after summation over j, we have
n j1
q
q
q
a1j a2j · · · amj
≥
m
n j1 i1
p
aij
p/q
m p
ai1
i1
p
ai2
··· p
ain
m
i1
2.14
p
ai .
By the inequality for power means of orders p ≥ 1 and 1, we get
m
i1
p 1/p
ai
m
≥
m
i1
ai
2.15
m
or equivalently
m
i1
1/p
p
ai
≥m
1/p−1
m
ai m
1/p−1
i1
m
⎛
⎞1/p
n
p
⎝ a ⎠ .
i1
j1
ij
2.16
The above inequality and 2.14 immediately yield
⎛ ⎛
⎞1/p
p/q ⎞1/p
n
m
m
n
q
p
⎠ ≥
⎝ a ⎠ ,
aij
m1−1/p ⎝
ij
j1
i1
i1
2.17
j1
as desired.
Case 4 0 < q ≤ p < 1. The proof can be obtained from those of Case 1, by replacing “≥” with
“≤” in each related inequality.
Journal of Inequalities and Applications
7
Case 5 0 < p < q < 1. If m ≤ n, then the proof is the same as that for Case 6. If m > n, then
the proof can be obtained from those of Case 2, by replacing “≥” with “≤” in each related
inequality.
Case 6 0 < p < 1 ≤ q. For any fixed j 1, 2, . . . , n, inequality 2.1 of Lemma 2.1 with r q
and s p gives
q
q
q
a1j a2j · · · amj
p/q
p
p
p
≤ a1j a2j · · · amj ,
2.18
whence after summation over j, we get
n j1
q
q
q
a1j a2j · · · amj
p/q
≤
m
n j1 i1
p
aij m
i1
p
ai .
2.19
As 1/p > 1, for positive integers b1 , b2 , . . . , bm , there holds
m
i1
bi
m
⎛
≤⎝
m
i1
1/p
bi
m
⎞p
⎠ ,
2.20
p
whence for any fixed j 1, 2, . . . n, after substitution of bi ai , i 1, 2, . . . m, we obtain
m
i1
1/p
p
ai
≤m
1/p−1
m
ai m
1/p−1
i1
m
⎛
⎞1/p
n
p
⎝ a ⎠ .
i1
j1
ij
2.21
The above inequality and 2.19 immediately yield
⎛ ⎛
⎞1/p
p/q ⎞1/p
n
m
m
n
q
p
⎠ ≤
⎝ a ⎠ ,
aij
m1−1/p ⎝
ij
j1
i1
i1
2.22
j1
and the proof is completed.
3. The Integral Analogue of Theorem 1.1
Let X, Σ, μ be a measure space with a positive Borel measure μ. For any 0 < p < ∞ let
Lp Lp μ denote the usual Lebesgue space consisting of all μ-measurable complex-valued
functions f : X → C such that
p
f dμ < ∞.
X
3.1
8
Journal of Inequalities and Applications
1/p
Recall that the usual norm · p of f ∈ Lp is defined as fp X |f|p dμ if p ≥ 1; fp |f|p dμ if 0 < p < 1.
X
The following result is the integral analogue of Theorem 1.1.
Theorem 3.1. For given 0 < p < ∞ let u1 , u2 , . . . , um be arbitrary functions in Lp . Then, if 1 ≤ p <
∞, we have
u1 p · · · um p ≤ m
2
2
|u1 | · · · |um | .
1−min{1/2,1/p} p
3.2
If 0 < p < 1, then
u1 p · · · um p ≥ m
2
2
|u1 | · · · |um | .
1−1/p p
3.3
Both inequalities are sharp
For 1 < p ≤ 2 the equality in 3.2 and 3.3 is attained if u1 u2 · · · um a.e. on X. If p > 2
or 0 < p < 1, then the equality is attained for ui χEi , where Ei are μ-measurable sets with
j.
i 1, 2, . . . , m, such that μE1 μE2 · · · μEn and Ei ∩ Ej ∅ whenever i /
Proof. The proof of each inequality is completely similar to the corresponding one given in
Theorem 1.1 with a fixed q 2. For clarity, we give here only a proof related to the case when
1 ≤ p ≤ 2. Applying the inequality between power means of orders 2/p ≥ 1 and 1 to the
functions |ui |p i 1, . . . , m, we have
m
p/2
|ui |
2
≥m
p/2−1
m
p
|ui | .
i1
3.4
i1
Integrating the above relation, we obtain
m
X
p/2
|ui |
2
dμ ≥ m
p/2−1
m
i1
i1
p
|ui | dμ ,
3.5
X
which can be written in the form
1/p
m
p
|u1 |2 · · · |um |2 ≥ m1/2−1/p
|ui | dμ
p
i1
≥
√
√
p 1/p
m
m
ui p
m
i1
m
m·
X
i1
ui p
m
Obviously, the above inequality yields 3.2 for 1 < p ≤ 2.
.
3.6
Journal of Inequalities and Applications
9
Corollary 3.2. Let p ≥ 1, and let w u iv be a complex function in Lp . Then there holds the sharp
inequality
up vp ≤ 21−min1/2,1/p u ivp .
3.7
References
1 H. Minkowski, Geometrie der Zahlen, Teubner, Leipzig, Germany, 1910.
2 G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univerity Press, Cambridge, UK,
1952.
3 E. F. Beckenbach and R. Bellman, Inequalities, vol. 30 of Ergebnisse der Mathematik und ihrer Grenzgebiete,
Springer, Berlin, Germany, 1961.
4 H. Tôyama, “On the inequality of Ingham and Jessen,” Proceedings of the Japan Academy, vol. 24, no. 9,
pp. 10–12, 1948.
5 H. Alzer and S. Ruscheweyh, “A converse of Minkowski’s inequality,” Discrete Mathematics, vol. 216,
no. 1–3, pp. 253–256, 2000.