Journal of Inequalities in Pure and
Applied Mathematics
FACTORIZATION OF INEQUALITIES
L. LEINDLER
Bolyai Institute
University of Szeged
Aradi Vértanúk tere 1.
6720 Szeged, Hungary.
EMail: leindler@math.u-szeged.hu
volume 5, issue 1, article 12,
2004.
Received 13 October, 2003;
accepted 13 December, 2003.
Communicated by: H. Bor
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
145-03
Quit
Abstract
We give a Bennett-type factorization of the space ces(pn ) for monotone nonincreasing sequence {pn }. If the sequence {pn } is nondecreasing and bounded
then certain converse assertion is proved.
2000 Mathematics Subject Classification: 26015, 40A05, 40A99.
Key words: Inequalities for sums, factorization, space `(pn ).
Factorization of Inequalities
This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant T 029080.
L. Leindler
Title Page
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
The Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
Contents
3
6
7
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 13
J. Ineq. Pure and Appl. Math. 5(1) Art. 12, 2004
http://jipam.vu.edu.au
1.
Introduction
G. Bennett raised the interesting problem of the factorization of inequalities,
and in his basic work [1] he gave a systematic treatment of the factorization of
several classic and latest inequalities. In his essay we can also find the precise
definition of factorization of inequalities and an explanation of its benefits.
In some previous papers we also studied such problems (see e.g. [3] – [7]).
Now we recall only one sample result.
It is well known that the classical Hardy inequality, in its crudest form, asserts that
Factorization of Inequalities
L. Leindler
lp ⊆ ces(p),
(1.1)
p > 1,
Title Page
where
(
lp :=
x:
(
ces(p) :=
x:
∞
X
n=1
∞
X
n=1
)
|xn |p < ∞ ,
!p
)
n
1X
|xk | < ∞ ,
n k=1
Contents
JJ
J
II
I
Go Back
Close
and x := {xn } is a sequence of real numbers.
G. Bennett [1] gave the factorization of (1.1) as follows:
Theorem 1.1. Let p > 1. A sequence x belongs to ces(p) if and only if it admits
a factorization
Quit
Page 3 of 13
J. Ineq. Pure and Appl. Math. 5(1) Art. 12, 2004
http://jipam.vu.edu.au
(1.2)
x = y · z (xn = yn · zn )
with
(1.3)
y ∈ lp
and
n
X
∗
|zk |p = O(n),
p∗ :=
k=1
p
.
p−1
This theorem may be stated succinctly as:
ces(p) = lp · g(p∗ )
(p > 1),
Factorization of Inequalities
if
(
g(p) :=
x:
n
X
)
|xk |p = O(n) .
L. Leindler
k=1
It is clear that Theorem 1.1 contains Hardy’s inequality, namely if x ∈ lp ,
then x may be factorized as in (1.2) such that y and z satisfy (1.3) by taking
y = x and z = 1 = (1, 1, . . .); and by Theorem 1.1 then x ∈ ces(p).
In [7] we considered the problem of factorization of the set
(
!
)
∞
n
X
X
λ(ϕ) := x :
λn ϕ
|xk | < ∞ ,
n=1
k=1
where {λn } is a sequence of nonnegative terms having infinitely many positive
ones, and ϕ is a nonnegative function on [0, ∞), ϕ(0) = 0, ϕ(x)x−p is nondecreasing. If ϕ(x) = xp and λn = n−p then, clearly λ(ϕ) ≡ ces(p). In this
theme several open problems still remain.
A great number of mathematicians have investigated the following generalizations of the spaces lp and ces(p):
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 13
J. Ineq. Pure and Appl. Math. 5(1) Art. 12, 2004
http://jipam.vu.edu.au
(
l(pn ) :=
x:
∞
X
)
|xn |pn < ∞
n=1
and
(
ces(pn ) :=
x:
∞
X
n=1
!pn
)
n
1X
|xk |
<∞ ,
n k=1
where p : = {pn } is a sequence of positive numbers.
A good survey and some new results of this type can be found in the paper
[2] by P.D. Johnson and R.N. Mohapatra.
The aim of the present paper is to find certain factorization of the space
ces(pn ). Unfortunately we can do this only if the sequence p := {pn } is monotone decreasing. In the case if p is monotone increasing we can give sufficient
conditions for the sequences y and z such that their product sequence x = y· z
should belong to ces(pn ). Hence it is clear that if pn = p for all n, then we get
a necessary and sufficient condition as in Theorem 1.1.
Naturally many similar problems can be raised (and solved if you have
enough stoicism and calculation ability) if you want to give the analogies of
the results given in the special case pn = p regarding all the factorizations of
inequalities (see e.g. only [1], [6] and [7]).
To present our theorem easily we need one more definition:
(
)
n
X
|xk |pk ≤ K pn −1 n ,
g(pn ) := x :
Factorization of Inequalities
L. Leindler
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 13
k=1
where K = K( p) ≥ 1 is a constant depending only on the sequence p.
J. Ineq. Pure and Appl. Math. 5(1) Art. 12, 2004
http://jipam.vu.edu.au
2.
The Results
Our theorem reads as follows:
Theorem 2.1. (i) If p := {pn } is a nonincreasing sequence of numbers, all
pn > 1; and x ∈ ces(pn ), then x admits a factorization (1.2) with
z ∈ g(p∗n ).
y ∈ l(pn ) and
(2.1)
(ii) Conversely, if p is a nondecreasing and bounded sequence of numbers,
p0 > 1, furthermore (2.1) holds, then the product sequence x = y · z ∈
ces(pn ).
Factorization of Inequalities
L. Leindler
Part (ii) of our theorem clearly implies that
l(pn ) ⊆ ces(pn ) (1 < p0 ≤ . . . ≤ pn ),
Title Page
since if x ∈ l(pn ) then x can be factorized as in (1.2) with (2.1) by taking
y = x and z = 1 = (1, 1, . . .), and thus we get that x ∈ ces(pn ).
In order to prove our theorem we require the following lemma.
Contents
Lemma 2.2. Let u, v, w be sequences with nonnegative terms and suppose that
wk decreases with k. If
n
n
X
X
uk ≤
vk (n = 1, 2, . . .),
k=1
then
n
X
k=1
uk wk ≤
II
I
Go Back
Close
Quit
k=1
n
X
JJ
J
Page 6 of 13
vk wk
(n = 1, 2, . . .).
k=1
This result is well known, see e.g. Lemma 3.6 in [1].
J. Ineq. Pure and Appl. Math. 5(1) Art. 12, 2004
http://jipam.vu.edu.au
3.
Proof of Theorem 2.1
(i) We assume that x 6= 0 := (0, 0, . . .), otherwise the statement is trivial.
If x ∈ ces(pn ), we set
(3.1)
bn :=
∞
X
k=n
k −pk
k
X
!pk −1
|xi |
,
i=1
and we note that bn monotonically tends to zero. Indeed, using the wellknown inequality
∗
ap b p
+ ∗ , p > 1,
ab ≤
p
p
we have that
!pk −1
k
∞
X
1 1X
|xi |
bn =
k
k
i=1
k=n
!pk
∞
∞
k
X 1
X
1 1X
(3.2)
≤
+
|xi |
,
∗
p
p
p
k
kk k
k
i=1
k=n
k=n
where the second sum clearly tends to zero becausePx ∈ ces(pn ) and
pk > 1. The first sums also tends to zero, namely if ∞
i=1 |xi | ≤ 1, then,
by pk ≤ p0 , the inequality
!pk
!p0
∞
k
∞
k
X
X
X
X
1
1
|xi |
≥
|xi |
k i=1
k pk i=1
k=1
k=1
Factorization of Inequalities
L. Leindler
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 7 of 13
J. Ineq. Pure and Appl. Math. 5(1) Art. 12, 2004
http://jipam.vu.edu.au
and x 6= 0 imply this assertion.
Now we factorize x as follows
x=y·z
(xn = yn zn ),
where
yn := (|xn |bn )1/pn sign(xn )
(3.3)
Factorization of Inequalities
and
L. Leindler
zn :=
(3.4)
∗
|xn |1/pn bn−1/pn (≥
0).
Title Page
Thus, by (3.1) and (3.3), we have
∞
X
|yn |pn =
n=1
∞
X
|xn |
∞
X
n=1
=
=
∞
X
k −pk
k
−pk
k
X
i=1
∞
X
k
X
k −pk
!pk −1
!pk −1
|xi |
k
X
n=1
!pk
|xi |
,
i=1
thus the condition x ∈ ces(pn ) implies y ∈ l(pn ).
Contents
JJ
J
|xi |
i=1
k=n
k=1
k=1
k
X
|xn |
II
I
Go Back
Close
Quit
Page 8 of 13
J. Ineq. Pure and Appl. Math. 5(1) Art. 12, 2004
http://jipam.vu.edu.au
On the other hand, by Hölder’s inequality,
!pm
!pm
m
m
X
X
∗ /p
1
1
−p
p∗k
+
zk
=
|xk | p∗m pm bk k k
k=1
k=1
≤
(3.5)
m
X
k=1
! pm
∗
pm
|xk |
m
X
−
|xk |bk
p m p∗
k
pk
!
.
k=1
Consequently, for m = 1, 2, . . ., we have by (3.1), (3.4) and (3.5),
!pm
!pm −1 m
m
∞
m
∞
p m p∗
X
X
X
X
− p k
1 X p∗k
−pm
zk
≤
n
|xk |
|xk |bk k
n
n=m
n=m
k=1
k=1
k=1
!pm −1
m
∞
m
pm p∗
X
X
− p k X
−p
=
|xk |bk k
(3.6)
n m
|xk |
.
n=m
k=1
k=1
Since x ∈ ces(pn ) implies that
n
1X
|xk | ≤ K1 (p)
n k=1
holds for all n; where and later on Ki (p) is written instead of Ki ( p), thus,
by (3.2),
!pm −1
∞
m
X
X
n−pm
|xk |
n=m
k=1
Factorization of Inequalities
L. Leindler
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 13
J. Ineq. Pure and Appl. Math. 5(1) Art. 12, 2004
http://jipam.vu.edu.au
=
∞
X
m
X
n−pm
n=m
(3.7)
!pn −1
m
X
npn −pm
|xk |
≤ K2 (p)
|xk |
k=1
k=1
∞
X
!pm −pn
n
X
n−pn
n=m
!pn −1
|xk |
= K2 (p)bm .
k=1
By (3.2) we also have that the sequence {bk } is bounded, whence
pk −pm
pk −1
Factorization of Inequalities
≤ K3 (p)
bk
L. Leindler
also hold for any k and m ≥ k. Hence we get that
m
X
−
|xk |bk
pm p∗
k
pk
≡
m
X
p∗
− pk
|xk |bk
k
b
pk −pm
−1
pk −1
Contents
k=1
k=1
≤ K3 (p)b−1
m
(3.8)
m
X
p∗
zk k .
k=1
Collecting the estimations (3.6), (3.7) and (3.8) we have that
!pm −1
∞
m
X
X
∗
p
n−pm
zk k
≤ K4 (p).
n=m
k=1
k=1
p∗
1
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 13
Hence an easy computation gives that
m
X
Title Page
∗
zk k ≤ K(p) pm −1 m = K(p)pm −1 m;
J. Ineq. Pure and Appl. Math. 5(1) Art. 12, 2004
http://jipam.vu.edu.au
and herewith we have proved that z ∈ g(p∗n ).
This completes the proof of part (i).
(ii) The assumption z ∈ g(p∗n ) yields that
n
X
p∗k
|zk |
≤ K(p)
n
X
1
pn −1
k=1
1.
k=1
This and Lemma 2.2 with wk := k −1/2 imply
n
X
(3.9)
1
∗
Factorization of Inequalities
|zk |pk k −1/2 ≤ K(p) pn −1
k=1
n
X
k −1/2 .
L. Leindler
k=1
On the other hand, if x = y · z, applying Hölder’s inequality, we have
!pn
!pn
n
n
X
X
1/2p∗n −1/2p∗n
|xk |
=
k
|zk |
|yk |k
k=1
k=1
≤
n
X
!
|yk |pn k
pn −1
2
n
X
!pn −1
p∗n
k −1/2 |zk |
k=1
k=1
(3.10)
n=1
JJ
J
II
I
Go Back
Quit
!pn
n
1X
|xk |
n k=1
≤ K5 (p)
Contents
Close
This and (3.9) exhibit that
∞
X
.
Title Page
∞
X
n=1
n
Page 11 of 13
−pn
n
X
k=1
|yk |pn k
pn −1
2
n
pn −1
2
J. Ineq. Pure and Appl. Math. 5(1) Art. 12, 2004
http://jipam.vu.edu.au
= K5 (p)
∞
X
∞ X
k
pn
|yk |
k=1
n=k
pn −1
2
n
1
=: S1 , say.
n
Since y ∈ l(pn ), the terms yn are bounded, using this, furthermore the
boundedness and the monotonicity of the sequence p, we know that
S1 ≤ K6 (p)
≤ K6 (p)
∞
X
k=1
∞
X
pk
|yk |
∞ X
k
n=k
|yk |pk k
k=1
(3.11)
≤ K7 (p)
∞
X
pk −1
2
n
∞
X
pk −1
2
1
n
1
n−1− 2 (pk −1)
Factorization of Inequalities
L. Leindler
n=k
|yk |pk .
k=1
The assumption y ∈ l(pn ), (3.10) and (3.11) plainly prove that x ∈
ces(pn ), as stated.
Herewith we have completed the proof of the theorem.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 12 of 13
J. Ineq. Pure and Appl. Math. 5(1) Art. 12, 2004
http://jipam.vu.edu.au
References
[1] G. BENNETT, Factorizing the Classical Inequalities, Memoirs of Amer.
Math. Soc., 120 (1996) No 576, 1–130.
[2] P.D. JOHNSON JR. AND R.N. MOHAPATRA, On an analogue of Hardy’s
inequality, Arch. Math., 60 (1993), 157–163.
[3] L. LEINDLER, A theorem of Hardy–Bennett-type, Acta Math. Hungar., 78
(1998), 315–325.
Factorization of Inequalities
[4] L. LEINDLER, Two theorems of Hardy–Bennett-type, Acta Math. Hungar.,
78 (1998), 365–375.
L. Leindler
[5] L. LEINDLER, Hardy–Bennett-type theorems, Math. Ineq. and Applications, 1 (1998), 517–526.
Title Page
[6] L. LEINDLER, Two Hardy–Bennett-type theorems, Acta Math. Hungar.,
85 (1999), 265–276.
[7] L. LEINDLER, An addition to factorization of inequalities, Math. Pannonica, 10 (1999), 61–65.
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 13 of 13
J. Ineq. Pure and Appl. Math. 5(1) Art. 12, 2004
http://jipam.vu.edu.au