Volume 10 (2009), Issue 3, Article 73, 9 pp.
A VARIANT OF A GENERAL INEQUALITY OF THE HARDY-KNOPP TYPE
DAH-CHIN LUOR
D EPARTMENT OF A PPLIED M ATHEMATICS
I-S HOU U NIVERSITY
TA -H SU , K AOHSIUNG 84008, TAIWAN
dclour@isu.edu.tw
Received 21 June, 2008; accepted 28 August, 2009
Communicated by S.S. Dragomir
A BSTRACT. 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.
Key words and phrases: Inequalities, Hardy’s inequality, Pólya-Knopp’s inequality, Multidimensional inequalities, Convolution inequalities.
2000 Mathematics Subject Classification. 26D10, 26D15.
1. I NTRODUCTION
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
The constants (p/(p − 1))p 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
f (t)dt
≤
φ(f (x)) ,
(1.3)
φ
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-Knopp-type inequalities and their multidimensional
This research is supported by the National Science Council, Taipei, R. O. C., under Grant NSC 96-2115-M-214-003.
184-08
2
DAH -C HIN L UOR
forms. In [4, Theorem 1 & Theorem 2], Čižmešija et al. obtained a strengthened HardyKnopp 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-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
k(z, x)
dz
v(x) = x
u(z) .
z
x K(z)
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
1q Z
p1
p
p
1−p
q
(1.5)
|Tk f (x)| dx
≤
|f (t)| (Hw(t)) q w(t) dt ,
X
T
where 1 < p ≤ q < ∞, X and T are measure spaces, Tk f (x) =
measurable function on T , and
Z
q− pq
Z
m
m
k(x, t)
k(x, y) w(y)dy
(1.6)
Hw(t) =
dx,
X
R
T
k(x, t)f (t)dt, w is a positive
m=
T
pq
.
pq + p − q
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
(1.8)
Tk f (x) =
k(x, t)f (t)dλ(t),
x ∈ X.
T
The purpose of this paper is to establish a modular inequality of the form
p1
Z
1q Z
p
q
p
1−sp
≤
φ (f (t))(Hs w(t)) q w(t)
dλ(t)
(1.9)
φ (Tk f (x))dµ(x)
X
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 Φ+
s (I) denotes the class of all nonnegative
1/s
functions φ on I ⊆ (0, ∞) such that φ is convex on I and φ takes its limiting values, finite
+
+
or
at the ends of I. Note that Φ+
s (I) ⊂ Φr (I) for 0 < r < s and we denote Φ∞ (I) =
T infinite,
+
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.
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 73, 9 pp.
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A VARIANT OF A G ENERAL I NEQUALITY OF THE H ARDY-K NOPP T YPE
3
2. M AIN R ESULTS
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
sq− pq
Z
Z
m
m
(2.1)
Hs w(t) =
k(x, t)
k(x, y) w(y)dλ(y)
dµ(x),
X
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
T
Proof. Since φ1/s is convex, φ(Tk f (x)) ≤ {Tk (φ1/s (f ))(x)}s 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
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)
T
Z
1
1
∗
∗
=
k(x, t)1−m/(sp) +m/(sp) φ1/s (f (t))w(t) (sp)∗ − (sp)∗ dλ(t)
T
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
Z
(2.5)
1q
φ (Tk f (x))dµ(x)
q
X
(
Z Z
≤
(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
dµ(x)
k(x, y)m w(y)dλ(y)
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 PólyaKnopp-type inequalities. These are discussed in Section 3. In the following corollary, we
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 73, 9 pp.
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4
DAH -C HIN L UOR
consider the norm inequality
1q
Z
p1
Z
q
p
φ (Tk f (x))dµ(x)
φ (f (t))dλ(t) .
(2.6)
≤C
X
T
+
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 :
Hr w(t) ≤ Ar w(t)(r−1/p)q
(2.7)
for λ-a.e. t ∈ T,
then we have (2.6) where the best constant C satisfies
1
C ≤ Arq .
(2.8)
(ii) If w satisfies (2.7) for each 1/p ≤ r ≤ s, then we have (2.6) with
1
C≤
(2.9)
inf
1/p≤r≤s
Arq .
(iii) If φ ∈ Φ+
∞ (I) and w satisfies (2.7) for each 1/p ≤ r < ∞, then we have (2.6) with
1
C≤
(2.10)
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].
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 h be a nonnegative function
R
on G such that G hdµ = 1. For a nonnegative function f on G, define the convolution operator
Z
h(xt−1 )f (t)dµ(t),
h ∗ f (x) =
(2.11)
x ∈ G.
G
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
Z
sq− pq
Z
−1 m
−1 m
(2.12)
dµ(x)
Hs w(t) =
h(xt )
h(xy ) dµ(y)
G
G
Z
=
sq
m
h(x) dµ(x)
.
m
G
We then obtain the following result:
Corollary 2.3. Let 0 < Rp ≤ q < ∞, 1/pR ≤ s < ∞, and φ ∈ Φ+
s (I). Let h be a nonnegative
function on G such that G hdµ = 1 and G hm dµ < ∞, where m = spq/(spq + p − q). Let f
be given as in Theorem 2.1. Then we have
ms Z
p1
Z
1q Z
q
m
p
≤
h(x) dµ(x)
φ (f (t))dµ(t) .
(2.13)
φ (h ∗ f (x))dµ(x)
G
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 73, 9 pp.
G
G
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A VARIANT OF A G ENERAL I NEQUALITY OF THE H ARDY-K NOPP T YPE
R
Moreover, if p < q, φ ∈ Φ+
∞ (I) and
G
5
hr dµ < ∞ for some r > 1, then
1q
φ (h ∗ f (x))dµ(x)
Z
q
(2.14)
G
≤
p1 − 1q Z
p1
p
h(x) log h(x)dµ(x)
φ (f (t))dµ(t) .
Z
exp
G
G
Inequality (2.14) can be
R obtained by 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(x) dµ(x)
f (t) dµ(t) ,
(h ∗ f (x)) dµ(x)
q
G
G
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 < ∞,
Z q
1q
h(xt ) log f (t)dµ(t)
dµ(x)
Z
−1
exp
(2.16)
G
G
≤
p1
p1 − 1q Z
p
f (t) dµ(t) .
h(x) log h(x)dµ(x)
Z
exp
G
G
Let G = Rn under addition and µ be the Lebesgue measure. Then (2.15) can be reduced to
q 1q Z
h(x − t)f (t)dt dx
≤
Z
Z
(2.17)
Rn
Moreover, if
R
Rn
Z
Rn
h(x)dx = 1 and
Rn
m1 Z
R
Rn
p1
f (t) dt .
p
h(x) dx
Rn
Rn
h(x)r dx < ∞ for some r > 1, then by (2.16),
q 1q
h(x − t) log f (t)dt
dx
Z
exp
(2.18)
m
Rn
p1 − 1q Z
Z
≤ exp
h(x) log h(x)dx
Rn
p1
f (t) dt .
p
Rn
Let G = (0, ∞) under multiplication and dµ = x−1 dx. Then by (2.15),
∞
Z
dt
h(x/t)f (t)
t
q
h(x)x−1 dx = 1 and
R∞
∞
Z
(2.19)
0
Moreover, if
reduced to
Z
0
R∞
0
∞
Z
exp
(2.20)
0
0
∞
0
dx
x
1q
Z
≤
0
∞
dx
h(x)m
x
m1 Z
0
∞
dt
f (t)p
t
p1
.
h(x)r x−1 dx < ∞ for some r > 1, then (2.16) can be
q 1q
dt
dx
h(x/t) log f (t)
t
x
Z ∞
p1 − 1q Z ∞
p1
dx
p dt
≤ exp
h(x) log h(x)
f (t)
.
x
t
0
0
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 73, 9 pp.
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6
DAH -C HIN L UOR
There are multidimensional cases corresponding to (2.19) and (2.20). For example, the 2dimensional analogue of (2.19) is
q
1
Z ∞ Z ∞ Z ∞ Z ∞ ds dt dx dy q
x y
(2.21)
h
,
f (s, t)
s t
s t
x y
0
0
0
0
p1
Z ∞ Z ∞
m1 Z ∞ Z ∞
p ds dt
m dx dy
f (s, t)
,
≤
h(x, y)
s t
x y
0
0
0
0
and we can also obtain similar results to (2.20).
3. M ULTIDIMENSIONAL H ARDY AND P ÓLYA -K NOPP -T YPE I NEQUALITIES
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}.
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 <R ∞, 1/p ≤ s < ∞, and φ ∈ Φ+
s (I). Let g be a nonnegative
N
N
function on R × R 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
m
m
(3.1)
Hs w(t) =
g(x, t)
g(x, y) w(y)dy
u(x)χSx (t)dx,
E
Sx
where m = spq/(spq + p − q), is finite for almost all t ∈ E. Then we have
Z
Z
1q Z
p1
p
q
p
1−sp
q
(3.2)
φ
g(x, t)f (t)dt u(x)dx
dt .
≤
φ (f (t))(Hs w(t)) w(t)
E
Sx
E
N
Proof. Let X = T = R , 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
Z
Z
(3.3)
φ
g(x, t)f (t)dt u(x)dx ≤
φ(f (t))
g(x, t)u(x)χSx (t)dx dt.
E
Sx
E
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
Inequality (3.4) was also obtained in [6, Theorem 4.1].
−1
Now we consider (3.2) with u(x) = |Sx |a and g(x,
R 1 t) = |Sx | h(|St |/|Sx |), where a ∈
R, h is a nonnegative function defined on [0, 1) and 0 h(x)dx = 1. By (3.1) with w(y) =
q
|Sy |m( p −a−2)/(sq) , we have
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A VARIANT OF A G ENERAL I NEQUALITY OF THE H ARDY-K NOPP T YPE
Z
(3.5) Hs w(t) =
1
h(ξ)m ξ m(q/p−a−2)/(sq) dξ
0
sq− pq
7
|St |−1+m(a+2−q/p)/(sq)
Z 1
×
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:
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
1q
Z
1
|St |
q
a
(3.6)
φ
h
f (t)dt |Sx | dx
|Sx | Sx
|Sx |
E
Z 1
s− p1 Z
p1
p
(a+1) pq −1
m m(q/p−a−2)/(sq)
p
≤
h(ξ) ξ
dξ
φ (f (t))|St |
v(t) q dt ,
0
E
where
Z
1
h(ξ)m ξ m(q/p−a−2)/(sq) dξ.
v(t) =
(|t|/b)N
By (3.6), we see that
1q
Z
p1
Z
Z
|St |
1
(a+1) pq −1
a
p
q
(3.7)
h
f (t)dt |Sx | dx
≤C
φ (f (t))|St |
dt ,
φ
|Sx | Sx
|Sx |
E
E
where C satisfies
Z
C≤
(3.8)
1
q
m m( p −a−2)/(sq)
h(ξ) ξ
s− p1 + 1q
dξ
.
0
Moreover, if φ ∈ Φ+
∞ (I) and p < q, then the estimation given in (3.8) can be replaced by
Z 1
p1 − 1q
(q−(a+2)p)/(q−p)
(3.9)
C ≤ exp
h(ξ) log[h(ξ)ξ
]dξ
.
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 Z
Z 1
(−a−1)/(sp)
p
(−a−1)/(sp)
≤
h(ξ)ξ
dξ
φ (f (t))
h(ξ)ξ
dξ |St |a dt.
0
(|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
|S
|
|S
|
x
x
E
Sx
Z 1
−a−1 Z
Z
p
≤ exp
h(ξ) log ξdξ
φ (f (t))
0
E
1
h(ξ)dξ |St |a 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, and f be
given as in Theorem 3.1. Then we have
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 73, 9 pp.
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8
DAH -C HIN L UOR
Z
p
φ
(3.12)
E
α
|Sx |α
Z
α−1
|St |
Sx
≤
f (t)dt |Sx |a dx
αsp
αsp − a − 1
sp Z
E
φp (f (t)) 1 −
|t|
b
N (αsp−a−1)/(sp) !
|St |a dt.
Moreover, if φ ∈ Φ+
∞ (I), then for a ∈ R, we have
Z
Z
α
p
α−1
(3.13)
φ
|St | f (t)dt |Sx |a dx
α
|Sx | Sx
E
N α !
Z
|t|
|St |a dt.
≤ e(a+1)/α
φp (f (t)) 1 −
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.
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
p
α−1
(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
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)
φ
(|Sx | − |St |) f (t)dt |Sx |a dx
α
|S
|
x
E
Sx
Z 1
−a−1 Z
N !α
|t|
≤ exp
α(1 − ξ)α−1 log ξdξ
φp (f (t)) 1 −
|St |a dt.
b
0
E
In the following, we consider the dual result of Theorem 3.1. Let 0 ≤ b < ∞ and
Ẽ = {x ∈ RN : x = sρ, b ≤ s < ∞, ρ ∈ A}.
For x ∈ Ẽ, we define
S̃x = {y ∈ RN : y = sρ, |x| ≤ s < ∞, ρ ∈ A}.
Let u be a nonnegative function on RN , dµ = u(x)χẼ (x)dx, dλ = χẼR(t)dt, and k(x, t) =
g(x, t)χS̃x (t), where g is a nonnegative function on RN × RN such that S̃x g(x, t)dt = 1 for
almost all x ∈ Ẽ. Suppose that w is a positive function on Ẽ. 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
S̃x
We have the following theorem.
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 73, 9 pp.
http://jipam.vu.edu.au/
A VARIANT OF A G ENERAL I NEQUALITY OF THE H ARDY-K NOPP T YPE
9
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
1q
p1
Z
Z
Z
p
q
p
1−sp
(3.17)
φ
g(x, t)f (t)dt u(x)dx
≤
φ (f (t))(Hs w(t)) q w(t)
dt .
Ẽ
S̃x
Ẽ
In the case p = q = s = 1, then m = 1 and we have
Z
Z
Z
Z
(3.18)
φ
g(x, t)f (t)dt u(x)dx ≤
φ(f (t))
g(x, t)u(x)χS̃x (t)dx dt.
Ẽ
S̃x
Ẽ
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 t
Z ∞ Z ∞
Z ∞
u(x)
u(x)
dx ≤
φ(f (t))
g(x, t)
dx dt.
(3.19)
φ
g(x, t)f (t)dt
x
x
b
b
b
x
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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J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 73, 9 pp.
http://jipam.vu.edu.au/