A VARIANT OF A GENERAL INEQUALITY OF THE
HARDY-KNOPP TYPE
Inequality of Hardy-Knopp Type
DAH-CHIN LUOR
Department of Applied Mathematics
I-Shou University
Ta-Hsu, Kaohsiung 84008, Taiwan
EMail: dclour@isu.edu.tw
Dah-Chin Luor
vol. 10, iss. 3, art. 73, 2009
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Contents
Received:
21 June, 2008
Accepted:
28 August, 2009
Communicated by:
JJ
II
S.S. Dragomir
J
I
2000 AMS Sub. Class.:
26D10, 26D15.
Page 1 of 20
Key words:
Inequalities, Hardy’s inequality, Pólya-Knopp’s inequality, Multidimensional inequalities, Convolution inequalities.
Abstract:
In this paper, we prove a variant of a general Hardy-Knopp type inequality. We
also formulate a convolution inequality in the language of topological groups.
By our main results we obtain a general form of multidimensional strengthened
Hardy and Pólya-Knopp-type inequalities.
Acknowledgements:
This research is supported by the National Science Council, Taipei, R. O. C.,
under Grant NSC 96-2115-M-214-003.
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Contents
1
Introduction
3
2
Main Results
6
3
Multidimensional Hardy and Pólya-Knopp-Type Inequalities
13
Inequality of Hardy-Knopp Type
Dah-Chin Luor
vol. 10, iss. 3, art. 73, 2009
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1.
Introduction
The well-known Hardy’s inequality is stated below (cf. [5, Theorem 327]):
p
p Z ∞
Z ∞ Z x
1
p
(1.1)
f (t)dt dx ≤
f (x)p dx, p > 1, f ≥ 0.
x
p
−
1
0
0
0
1
By replacing f with f p in (1.1) and letting p → ∞, we have the Pólya-Knopp
inequality (cf. [5, Theorem 335]):
Z x
Z ∞
Z ∞
1
(1.2)
exp
log f (t)dt dx ≤ e
f (x)dx.
x 0
0
0
Inequality of Hardy-Knopp Type
Dah-Chin Luor
vol. 10, iss. 3, art. 73, 2009
Title Page
p
The constants (p/(p − 1)) and e in (1.1) and (1.2), respectively, are the best
possible. On the other hand, the following Hardy-Knopp type inequality (1.3) was
proved (cf. [1, Eq.(4.3)] and [7, Theorem 4.1]):
Z ∞ Z x
Z ∞
1
dx
dx
(1.3)
φ
f (t)dt
≤
φ(f (x)) ,
x 0
x
x
0
0
where φ is a convex function on (0, ∞). In [7], S. Kaijser et al. also pointed out
that (1.1) and (1.2) can be obtained from (1.3). Furthermore, in [2] and [3],
Čižmešija and Pečarić proved the so-called strengthened Hardy and Pólya-Knopptype inequalities and their multidimensional forms. In [4, Theorem 1 & Theorem 2],
Čižmešija et al. obtained a strengthened Hardy-Knopp type inequality and its dual
result. With suitable substitutions, they also showed that the strengthened Hardy
and Pólya-Knopp-type inequalities given in the paper [2] are special cases of their
results. In the paper [6], Kaijser et al. proved some multidimensional Hardy-type
inequalities. They also proved the following generalization of the Hardy and Pólya-
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Knopp-type inequality:
Z b Z x
Z b
1
dx
dx
(1.4)
≤
φ
k(x, t)f (t)dt u(x)
φ(f (x))v(x) ,
K(x) 0
x
x
0
0
Rx
where 0 < b ≤ ∞, k(x, t) ≥ 0, K(x) = 0 k(x, t)dt, u(x) ≥ 0, and
Z b
dz
k(z, x)
v(x) = x
u(z) .
z
x K(z)
Inequality of Hardy-Knopp Type
Dah-Chin Luor
vol. 10, iss. 3, art. 73, 2009
A dual inequality to (1.4) was also given. Inequality (1.4) can be obtained by using
Jensen’s inequality and the Fubini theorem. It is elementary but powerful. On the
other hand, in the proof of [8, Lemma 3.1], for proving a variant of Schur’s lemma,
Sinnamon obtained an inequality of the form
Z
(1.5)
|Tk f (x)|q dx
X
1q
Z
≤
p1
p
p
1−p
q
|f (t)| (Hw(t)) w(t) dt ,
T
R
where 1 < p ≤ q < ∞, X and T are measure spaces, Tk f (x) = T k(x, t)f (t)dt, w
is a positive measurable function on T , and
q− pq
Z
Z
pq
(1.6) Hw(t) =
k(x, t)m
k(x, y)m w(y)dy
dx, m =
.
pq + p − q
X
T
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In this paper, let (X, µ) and (T, λ) be two σ-finite measure spaces. Let k be a
nonnegative measurable function on X × T such that
Z
(1.7)
k(x, t)dλ(t) = 1 for µ−a.e. x ∈ X.
T
For a nonnegative measurable function f on (T, λ), define
Z
k(x, t)f (t)dλ(t), x ∈ X.
(1.8)
Tk f (x) =
T
The purpose of this paper is to establish a modular inequality of the form
Z
(1.9)
X
1q Z
p1
p
p
1−sp
φ (Tk f (x))dµ(x)
φ (f (t))(Hs w(t)) q w(t)
≤
dλ(t)
q
T
for 0 < p ≤ q < ∞, φ ∈ Φ+
s (I), s ≥ 1/p, and Hs w(t) is defined by (2.1). As
applications, we prove a convolution inequality in the language of integration on a
locally compact Abelian group. We also show that by suitable choices of w, we can
obtain many forms of strengthened Hardy and Pólya-Knopp-type inequalities. Here
1/s
Φ+
s (I) denotes the class of all nonnegative functions φ on I ⊆ (0, ∞) such that φ
is convex on I and φ takes its limiting values, finite or infinite,Tat the ends of I. Note
+
+
+
that Φ+
s (I) ⊂ Φr (I) for 0 < r < s and we denote Φ∞ (I) =
s>0 Φs (I).
The functions involved in this paper are all measurable on their domains. We
work under the convention that 00 = ∞0 = 1 and ∞/∞ = 0 · ∞ = 0.
Inequality of Hardy-Knopp Type
Dah-Chin Luor
vol. 10, iss. 3, art. 73, 2009
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2.
Main Results
The following theorem is based on Jensen’s inequality and [8, Lemma 3.1]. For the
convenience of readers, we give a complete proof here.
Theorem 2.1. Let 0 < p ≤ q < ∞, 1/p ≤ s < ∞, and φ ∈ Φ+
s (I). Let f be a
nonnegative function on (T, λ) and the range of values of f lie in the closure of I.
Suppose that w is a positive function on (T, λ) such that the function
Z
sq− pq
Z
m
m
dµ(x),
(2.1)
Hs w(t) =
k(x, t)
k(x, y) w(y)dλ(y)
X
T
X
1/s
1/s
s
Proof. Since φ is convex, φ(Tk f (x)) ≤ {Tk (φ (f ))(x)} for µ−a.e. x ∈ X and
hence
sq
Z
Z Z
q
1/s
(2.3)
φ (Tk f (x))dµ(x) ≤
k(x, t)φ (f (t))dλ(t)
dµ(x).
X
T
Let m = spq/(spq + p − q) and w be a positive function on (T, λ) such that Hs w(t)
defined by (2.1) is finite for λ−a.e. t ∈ T . By Hölder’s inequality with indices sp
and (sp)∗ , we have
Z
(2.4)
k(x, t)φ1/s (f (t))dλ(t)
TZ
1
1
∗
∗
k(x, t)1−m/(sp) +m/(sp) φ1/s (f (t))w(t) (sp)∗ − (sp)∗ dλ(t)
=
T
Dah-Chin Luor
vol. 10, iss. 3, art. 73, 2009
T
where m = spq/(spq + p − q), is finite for λ−a.e. t ∈ T . Then we have
p1
Z
1q Z
p
q
p
1−sp
(2.2)
φ (Tk f (x))dµ(x)
≤
φ (f (t))(Hs w(t)) q w(t)
dλ(t) .
X
Inequality of Hardy-Knopp Type
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1
(sp)
∗
k(x, y) w(y)dλ(y)
Z
m
≤
T
Z
(1−m/(sp)∗ )sp p
×
k(x, t)
−sp/(sp)∗
φ (f (t))w(t)
1
(sp)
dλ(t)
T
and this implies
Inequality of Hardy-Knopp Type
Z
(2.5)
1q
q
φ (Tk f (x))dµ(x)
X
(
Z Z
≤
Dah-Chin Luor
vol. 10, iss. 3, art. 73, 2009
(1−m/(sp)∗ )sp p
k(x, t)
X
−sp/(sp)∗
φ (f (t))w(t)
pq
dλ(t)
T
Z
×
) 1q
(sp−1) pq
k(x, y)m w(y)dλ(y)
dµ(x)
T
Z
p
≤
p
q
1−sp
φ (f (t))(Hs w(t)) w(t)
p1
dλ(t) .
T
The last inequality is based on the Minkowski’s integral inequality with index pq .
This completes the proof.
We can apply Theorem 2.1 to obtain some multidimensional strengthened Hardy
and Pólya-Knopp-type inequalities. These are discussed in Section 3. In the following corollary, we consider the norm inequality
Z
(2.6)
X
1q
Z
p1
p
φ (Tk f (x))dµ(x)
≤C
φ (f (t))dλ(t) .
q
T
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The results of Corollary 2.2 can be obtained by Theorem 2.1 and the fact that
+
Φ+
s (I) ⊂ Φr (I) for 0 < r < s.
Corollary 2.2. Let 0 < p ≤ q < ∞, 1/p ≤ s < ∞, and φ ∈ Φ+
s (I). Let f be given
as in Theorem 2.1.
(i) If there exists a positive function w on (T, λ) such that the following condition (2.7) holds for some 1/p ≤ r ≤ s and for some positive constant Ar :
Inequality of Hardy-Knopp Type
Dah-Chin Luor
(2.7)
Hr w(t) ≤ Ar w(t)(r−1/p)q
for λ-a.e. t ∈ T,
then we have (2.6) where the best constant C satisfies
(2.8)
Contents
C ≤ Ar .
C≤
inf
1/p≤r≤s
1
q
Ar .
(iii) If φ ∈ Φ+
∞ (I) and w satisfies (2.7) for each 1/p ≤ r < ∞, then we have (2.6)
with
1
(2.10)
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1
q
(ii) If w satisfies (2.7) for each 1/p ≤ r ≤ s, then we have (2.6) with
(2.9)
vol. 10, iss. 3, art. 73, 2009
C≤
inf
1/p≤r<∞
Arq .
In the case 1 < p ≤ q < ∞ and φ(x) = x, choose s = r = 1 and then
Corollary 2.2 can be reduced to [8, Lemma 3.1].
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In the following, we consider the particular case X = T = G, where G is a
locally compact Abelian group (written multiplicatively),
with Haar measure µ. Let
R
h be a nonnegative function on G such that G hdµ = 1. For a nonnegative function
f on G, define the convolution operator
Z
(2.11)
h ∗ f (x) =
h(xt−1 )f (t)dµ(t), x ∈ G.
G
Inequality of Hardy-Knopp Type
R
Moreover, if G hm dµ is also finite, where m is given in Theorem 2.1, then by (2.1)
with k(x, y) = h(xy −1 ) and w ≡ 1, we have
sq− pq
Z
Z
−1 m
−1 m
(2.12)
Hs w(t) =
h(xt )
h(xy ) dµ(y)
dµ(x)
G
=
sq
m
h(x) dµ(x)
.
Contents
m
G
We then obtain the following result:
+
Corollary 2.3. Let 0 < p ≤ q < ∞, 1/p
R ≤ s < ∞, and
R φ m∈ Φs (I). Let h
be a nonnegative function on G such that G hdµ = 1 and G h dµ < ∞, where
m = spq/(spq + p − q). Let f be given as in Theorem 2.1. Then we have
(2.13)
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G
Z
Z
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vol. 10, iss. 3, art. 73, 2009
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G
≤
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1q
q
φ (h ∗ f (x))dµ(x)
Z
JJ
ms Z
p1
p
h(x) dµ(x)
φ (f (t))dµ(t) .
m
G
Moreover, if p < q, φ ∈ Φ+
∞ (I) and
R
G
hr dµ < ∞ for some r > 1, then
1q
φ (h ∗ f (x))dµ(x)
Z
q
(2.14)
G
≤
Z
exp
p1 − 1q Z
p1
p
h(x) log h(x)dµ(x)
φ (f (t))dµ(t) .
G
G
Inequality of Hardy-Knopp Type
Inequality (2.14) can be obtained by
R letting s → ∞ in (2.13). In the case φ(x) =
x and s = 1 in (2.13), the condition G hdµ = 1 is not necessary and (2.13) can be
reduced to Young’s inequality:
Z
(2.15)
1q Z
m1 Z
p1
m
p
(h ∗ f (x)) dµ(x)
≤
h(x) dµ(x)
f (t) dµ(t) ,
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q
G
G
Z Z
q
1q
exp
h(xt−1 ) log f (t)dµ(t)
dµ(x)
G
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G
Z
p1 − 1q Z
p1
p
≤ exp
h(x) log h(x)dµ(x)
f (t) dµ(t) .
G
n
Contents
G
where 1 ≤ p ≤ q < ∞ and m = pq/(pq + p − q). If φ(x) = ex and f is replaced by
log f in (2.14), then for 0 < p < q < ∞,
(2.16)
Dah-Chin Luor
vol. 10, iss. 3, art. 73, 2009
G
Let G = R under addition and µ be the Lebesgue measure. Then (2.15) can be
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reduced to
q 1q
h(x − t)f (t)dt dx
Z
Z
(2.17)
Rn
Rn
Z
≤
m1 Z
m
h(x) dx
Rn
Moreover, if
R
Rn
h(x)dx = 1 and
R
p1
f (t) dt .
p
Rn
Inequality of Hardy-Knopp Type
r
Rn
h(x) dx < ∞ for some r > 1, then by (2.16),
Dah-Chin Luor
vol. 10, iss. 3, art. 73, 2009
Z
(2.18)
q 1q
h(x − t) log f (t)dt
dx
Z
exp
Rn
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Rn
≤
p1 − 1q Z
Z
exp
h(x) log h(x)dx
Rn
p1
f (t)p dt .
Rn
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−1
Let G = (0, ∞) under multiplication and dµ = x dx. Then by (2.15),
1
dx q
(2.19)
x
0
0
m1 Z ∞
p1
Z ∞
p dt
m dx
.
≤
h(x)
f (t)
x
t
0
0
R∞
R∞
Moreover, if 0 h(x)x−1 dx = 1 and 0 h(x)r x−1 dx < ∞ for some r > 1,
Z
∞
Z
∞
dt
h(x/t)f (t)
t
q
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then (2.16) can be reduced to
Z
∞
(2.20)
q 1q
dt
dx
h(x/t) log f (t)
t
x
0
Z ∞
p1 − 1q Z ∞
p1
dx
p dt
≤ exp
h(x) log h(x)
f (t)
.
t
x
0
0
Z
exp
0
∞
There are multidimensional cases corresponding to (2.19) and (2.20). For example,
the 2-dimensional analogue of (2.19) is
Z
∞
Z
(2.21)
0
0
q
1
x y ds dt dx dy q
,
f (s, t)
h
s t
s t
x y
0
0
Z ∞ Z ∞
m1 Z ∞ Z ∞
p1
p ds dt
m dx dy
,
≤
h(x, y)
f (s, t)
x y
s t
0
0
0
0
∞
Z
∞
Z
Inequality of Hardy-Knopp Type
Dah-Chin Luor
vol. 10, iss. 3, art. 73, 2009
∞
and we can also obtain similar results to (2.20).
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3.
Multidimensional Hardy and Pólya-Knopp-Type Inequalities
In this section, we apply our main results to the case X = T = RN and obtain some
multidimensional forms of the strengthened Hardy and Pólya-Knopp-type inequalities. Let ΣN −1 be the unit sphere in RN , that is, ΣN −1 = {x ∈ RN : |x| = 1},
where |x| denotes the Euclidean norm of x. Let A be a Lebesgue measurable subset
of ΣN −1 , 0 < b ≤ ∞, and define
E = {x ∈ RN : x = sρ, 0 ≤ s < b, ρ ∈ A}.
Inequality of Hardy-Knopp Type
Dah-Chin Luor
vol. 10, iss. 3, art. 73, 2009
For x ∈ E, we define
Sx = {y ∈ RN : y = sρ, 0 ≤ s ≤ |x|, ρ ∈ A},
and denote by |Sx | the Lebesgue measure of Sx . We have the following result:
Theorem 3.1. Let 0 < p ≤ q < ∞, 1/p ≤Rs < ∞, and φ ∈ Φ+
s (I). Let g be a
nonnegative function on RN × RN such that Sx g(x, t)dt = 1 for almost all x ∈ E
and let f be a nonnegative function on RN and the range of values of f lie in the
closure of I. Suppose that u is a nonnegative function on RN and w is a positive
function on E such that the function
Z
sq− pq
Z
(3.1)
Hs w(t) =
g(x, t)m
g(x, y)m w(y)dy
u(x)χSx (t)dx,
E
Sx
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where m = spq/(spq + p − q), is finite for almost all t ∈ E. Then we have
Z
Z
1q
q
(3.2)
φ
g(x, t)f (t)dt u(x)dx
E
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Sx
Z
≤
p
p
q
1−sp
φ (f (t))(Hs w(t)) w(t)
E
p1
dt .
Close
Proof. Let X = T = RN , dµ = u(x)χE (x)dx, dλ = χE (x)dx, and k(x, t) =
g(x, t)χSx (t) in Theorem 2.1. Then Hs w defined by (2.1) can be reduced to (3.1)
and we have (3.2) by Theorem 2.1.
In the case p = q = s = 1, then m = 1 and we have
Z
Z
g(x, t)f (t)dt u(x)dx
(3.3)
φ
E
Sx
Z
Z
≤
φ(f (t))
g(x, t)u(x)χSx (t)dx dt.
E
Inequality (3.4) was also obtained in [6, Theorem 4.1].
Now we consider (3.2) with u(x) = |Sx |a and g(x, t) = |SxR|−1 h(|St |/|Sx |),
1
where a ∈ R, h is a nonnegative function defined on [0, 1) and 0 h(x)dx = 1.
q
By (3.1) with w(y) = |Sy |m( p −a−2)/(sq) , we have
(3.5)
Hs w(t) =
0
Dah-Chin Luor
vol. 10, iss. 3, art. 73, 2009
E
In particular, if N = 1, E = [0, b), Sx = [0, x), and u(x) is replaced by u(x)/x, then
(3.3) can be reduced to
Z b
Z b Z x
Z b
u(x)
u(x)
(3.4)
φ
g(x, t)f (t)dt
dx ≤
φ(f (t))
g(x, t)
dx dt.
x
x
0
0
0
t
Z
Inequality of Hardy-Knopp Type
1
h(ξ)m ξ m(q/p−a−2)/(sq) dξ
Z 1
×
sq− pq
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|St |−1+m(a+2−q/p)/(sq)
h(ξ)m ξ m(q/p−a−2)/(sq) dξ.
(|t|/b)N
As a consequence of Theorem 3.1, we have the following result:
Close
Corollary 3.2. Let 0 < p ≤ q < ∞, 1/p ≤ s < ∞,R φ ∈ Φ+
s (I), and f be given as in
1
m m(q/p−a−2)/(sq)
Theorem 3.1. Let a ∈ R, h be given as above, and 0 h(ξ) ξ
dξ < ∞,
where m = spq/(spq + p − q). Then we have
Z
q
φ
(3.6)
E
1
|Sx |
h
|St |
|Sx |
Z
1
Z
Sx
1q
a
f (t)dt |Sx | dx
m m(q/p−a−2)/(sq)
≤
h(ξ) ξ
s− p1
Inequality of Hardy-Knopp Type
dξ
Dah-Chin Luor
0
vol. 10, iss. 3, art. 73, 2009
Z
×
(a+1) pq −1
p
φ (f (t))|St |
p1
v(t) dt ,
p
q
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E
where
Z
Contents
1
m m(q/p−a−2)/(sq)
v(t) =
h(ξ) ξ
dξ.
(|t|/b)N
By (3.6), we see that
Z
q
φ
(3.7)
E
1
|Sx |
Z
h
Sx
|St |
|Sx |
≤C
(a+1) pq −1
p
φ (f (t))|St |
where C satisfies
(3.8)
C≤
1
q
m m( p −a−2)/(sq)
h(ξ) ξ
0
s− p1 + 1q
dξ
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Z
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1q
a
f (t)dt |Sx | dx
Z
JJ
.
p1
dt ,
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Moreover, if φ ∈ Φ+
∞ (I) and p < q, then the estimation given in (3.8) can be
replaced by
(3.9)
Z
C ≤ exp
1
h(ξ) log[h(ξ)ξ (q−(a+2)p)/(q−p) ]dξ
p1 − 1q
.
0
In the following, we consider the particular case p = q. In this case, m = 1 and
(3.6) can be reduced to
Z
Z
1
|St |
p
(3.10)
φ
h
f (t)dt |Sx |a dx
|Sx | Sx
|Sx |
E
Z 1
sp−1
(−a−1)/(sp)
≤
h(ξ)ξ
dξ
0
Z 1
Z
p
(−a−1)/(sp)
×
φ (f (t))
h(ξ)ξ
dξ |St |a dt.
(|t|/b)N
E
In the case φ ∈ Φ+
∞ (I), by letting s → ∞ in (3.10), we have
Z
Z
1
|St |
p
(3.11)
φ
h
f (t)dt |Sx |a dx
|Sx | Sx
|Sx |
E
Z 1
−a−1 Z
Z 1
p
h(ξ) log ξdξ
φ (f (t))
≤ exp
0
E
Inequality of Hardy-Knopp Type
Dah-Chin Luor
vol. 10, iss. 3, art. 73, 2009
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a
h(ξ)dξ |St | dt.
(|t|/b)N
If h(ξ) = αξ α−1 , α > 0, then we have the following corollary.
Corollary 3.3. Let 0 < p < ∞, 1/p ≤ s < ∞, φ ∈ Φ+
s (I), α > 0, a + 1 < αsp,
Close
and f be given as in Theorem 3.1. Then we have
Z
Z
α
α−1
p
|St | f (t)dt |Sx |a dx
(3.12)
φ
α
|S
|
x
Sx
E
sp Z
N (αsp−a−1)/(sp) !
|t|
αsp
φp (f (t)) 1 −
|St |a dt.
≤
αsp − a − 1
b
E
Inequality of Hardy-Knopp Type
Φ+
∞ (I),
Moreover, if φ ∈
then for a ∈ R, we have
Z
Z
α
α−1
p
(3.13)
φ
|St | f (t)dt |Sx |a dx
α
|Sx | Sx
E
N α !
Z
|t|
≤ e(a+1)/α
φp (f (t)) 1 −
|St |a dt.
b
E
Inequality (3.12) was obtained in [3, Theorem 1(i)] for the case φ(x) = x, p > 1,
s = 1, a < p − 1, α = 1, and E is the ball in RN centered at the origin and of
radius b. If φ(x) = ex , p = 1, and f is replaced by log f in (3.13), then we have [3,
Theorem 2(i)]. If h(ξ) = α(1 − ξ)α−1 , α > 0, then we have the following corollary.
Dah-Chin Luor
vol. 10, iss. 3, art. 73, 2009
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Corollary 3.4. Let 0 < p < ∞, 1/p ≤ s < ∞, φ ∈ Φ+
s (I), α > 0, a + 1 < sp, and
f be given as in Theorem 3.1. Then we have
Z
Z
α
α−1
p
(3.14)
φ
(|Sx | − |St |) f (t)dt |Sx |a dx
α
|S
|
x
E
Sx
sp−1 Z
sp − a − 1
≤ αB
,α
φp (f (t))|St |a v(t)dt,
sp
E
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where B(δ, η) is the Beta function and
Z 1
v(t) =
α(1 − ξ)α−1 ξ (−a−1)/(sp) dξ.
(|t|/b)N
Moreover, if φ ∈ Φ+
∞ (I), then for a ∈ R we have
Z
Z
α
p
α−1
(3.15)
(|Sx | − |St |) f (t)dt |Sx |a dx
φ
α
|S
|
x
E
Sx
Z 1
−a−1
α−1
≤ exp
α(1 − ξ)
log ξdξ
Inequality of Hardy-Knopp Type
Dah-Chin Luor
vol. 10, iss. 3, art. 73, 2009
0
Z
×
E
φp (f (t)) 1 −
|t|
b
N !α
|St |a dt.
In the following, we consider the dual result of Theorem 3.1. Let 0 ≤ b < ∞ and
N
Ẽ = {x ∈ R : x = sρ, b ≤ s < ∞, ρ ∈ A}.
For x ∈ Ẽ, we define
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N
S̃x = {y ∈ R : y = sρ, |x| ≤ s < ∞, ρ ∈ A}.
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N
Let u be a nonnegative function on R , dµ = u(x)χẼ (x)dx, dλ = χẼ (t)dt, and
k(x, t) = g(x, t)χS̃x (t), where g is a nonnegative function on RN × RN such that
R
g(x, t)dt = 1 for almost all x ∈ Ẽ. Suppose that w is a positive function on Ẽ.
S̃x
Then Hs w defined by (2.1) can be reduced to
Z
sq− pq
Z
m
m
(3.16)
Hs w(t) =
g(x, t)
g(x, y) w(y)dy
u(x)χS̃x (t)dx.
Ẽ
We have the following theorem.
S̃x
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Theorem 3.5. Let 0 < p ≤ q < ∞, 1/p ≤ s < ∞, φ ∈ Φ+
s (I), and g, u, w be given
as above. Let f be given as in Theorem 3.1. Suppose that Hs w(t) given in (3.16) is
finite for almost all t ∈ Ẽ. Then we have
Z
q
Z
φ
(3.17)
Ẽ
1q
g(x, t)f (t)dt u(x)dx
S̃x
Z
≤
p
q
p
1−sp
φ (f (t))(Hs w(t)) w(t)
p1
dt .
Ẽ
Dah-Chin Luor
vol. 10, iss. 3, art. 73, 2009
In the case p = q = s = 1, then m = 1 and we have
Z
Z
(3.18)
φ
g(x, t)f (t)dt u(x)dx
Ẽ
S̃x
Z
Z
≤
φ(f (t))
g(x, t)u(x)χS̃x (t)dx dt.
Ẽ
Inequality of Hardy-Knopp Type
Ẽ
In particular, if N = 1, Ẽ = [b, ∞), S̃x = [x, ∞), and u(x) is replaced by u(x)/x,
then by (3.18) we have
Z ∞ Z ∞
u(x)
(3.19)
φ
g(x, t)f (t)dt
dx
x
b
x
Z t
Z ∞
u(x)
≤
φ(f (t))
g(x, t)
dx dt.
x
b
b
Inequality (3.19) was also obtained in [6, Theorem 4.3]. Using a similar method,
we can also obtain companion results of (3.6) – (3.15). We omit the details.
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References
[1] R.P. BOAS AND C.O. IMORU, Elementary convolution inequalities, SIAM J.
Math. Anal., 6 (1975), 457–471.
[2] A. ČIŽMEŠIJA AND J. PEČARIĆ, Some new generalisations of inequalities of
Hardy and Levin-Cochran-Lee, Bull. Austral. Math. Soc., 63 (2001), 105–113.
[3] A. ČIŽMEŠIJA AND J. PEČARIĆ, New generalizations of inequalities of Hardy
and Levin-Cochran-Lee type for multidimensional balls, Math. Inequal. Appl., 5
(2002), 625–632.
[4] A. ČIŽMEŠIJA, J. PEČARIĆ AND L.-E. PERSSON, On strengthened Hardy
and Pólya-Knopp’s inequalities, J. Approx. Theory, 125 (2003), 74–84.
[5] G.H. HARDY, J.E. LITTLEWOOD, AND G. PÓLYA, Inequalities, 2nd edition.
Cambridge University Press, 1952.
[6] S. KAIJSER, L. NIKOLOVA, L.-E. PERSSON AND A. WEDESTIG, Hardytype inequalities via convexity, Math. Inequal. Appl., 8(3) (2005), 403–417.
[7] S. KAIJSER, L.-E. PERSSON AND A. ÖBERG, On Carleman and Knopp’s
inequalities, J. Approx. Theory, 117 (2002), 140–151.
[8] G. SINNAMON, From Nörlund matrices to Laplace representations, Proc. Amer.
Math. Soc., 128 (1999), 1055–1062.
Inequality of Hardy-Knopp Type
Dah-Chin Luor
vol. 10, iss. 3, art. 73, 2009
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