Journal of Inequalities in Pure and
Applied Mathematics
WEIGHTED GEOMETRIC MEAN INEQUALITIES OVER CONES IN RN
BABITA GUPTA, PANKAJ JAIN, LARS-ERIK PERSSON AND
ANNA WEDESTIG
Department of Mathematics,
Shivaji College (University of Delhi),
Raja Garden,
Delhi-110 027 India.
EMail: babita74@hotmail.com
Department of Mathematics,
Deshbandhu College (University of Delhi),
New Delhi-110019, India.
EMail: pankajkrjain@hotmail.com
Department of Mathematics,
Luleå University of Technology,
SE-971 87 Luleå, Sweden.
EMail: larserik@sm.luth.se
EMail: annaw@sm.luth.se
volume 4, issue 4, article 68,
2003.
Received 7 November, 2002;
accepted 20 March, 2003.
Communicated by: B. Opić
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
118-02
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Abstract
Let 0 < p ≤ q < ∞. Let A be a measurable
sphere in RN , let
subset of the unit
N
N
E = x ∈ R : x = sσ, 0 ≤ s < ∞, σ ∈ A be a cone in R and let Sx be the
part of E with ’radius’ ≤ |x| . A characterization of the weights u and v on E is
given such that the inequality
Z q
1
Z
1
Z
q
p
1
p
exp
f (x)u(x)dx
ln f(y)dy
v(x)dx ≤ C
|Sx | Sx
E
E
holds for all f ≥ 0 and some positive and finite constant C. The inequality is
obtained as a limiting case of a corresponding new Hardy type inequality. Also
the corresponding companion inequalities are proved and the sharpness of the
constant C is discussed.
2000 Mathematics Subject Classification: 26D15, 26D07.
Key words: Inequalities, Multidimensional inequalities, Geometric mean inequalities,
Hardy type inequalities, Cones in RN , Sharp constant.
We thank Professor Alexandra Čižmešija for some valuable advice and the referee
for pointing out an inaccuracy in our original manuscript (see Remark 4.3) and for
several suggestions which have improved the final version of this paper.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Geometric Mean Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4
The Companion Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
References
Weighted Geometric Mean
Inequalities Over Cones in RN
Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
Anna Wedestig
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1.
Introduction
In their paper [2] J.A. Cochran and C.S. Lee proved the inequality
Z ∞
Z x
Z ∞
a+1
−ε
ε−1
a
(1.1)
exp εx
y ln f (y)dy x dx ≤ e ε
xa f (x)dx,
0
0
0
where a, ε are real numbers with ε > 0, f is a positive function defined on
a+1
(0, ∞) and the constant e ε is the best possible. This inequality, in fact, is a
generalization of what sometimes is referred to as Knopp’s inequality 1 , which
is obtained by taking ε = 1 and a = 0 in (1.1). Inequalities of the type (1.1) and
its analogues have further been investigated and generalized by many authors
e.g. see [1], [5] – [11], [14] and [16] – [21].
In particular, very recently A. Čižmešija, J. Pečarić and I. Perić [1, Th. 9,
formula (23)] proved an N − dimensional analogue of (1.1) by replacing the interval (0, ∞) by RN and the means are considered over the balls in RN centered
at the origin. Their inequality reads:
Z
(1.2)
RN
Z
−ε
exp ε |Bx |
Bx
ε−1
|By |
|Bx |a dx
Z
a+1
≤e ε
f (x) |Bx |a dx,
ln f (y)dy
RN
N
Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
Anna Wedestig
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where a ∈ R, ε > 0, f is a positive function on R , Bx is a ball in R with
radius |x| , x ∈ RN , centered at the origin and |Bx | is its volume.
1
Weighted Geometric Mean
Inequalities Over Cones in RN
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Page 3 of 25
See e.g. [15, p. 143–144] and [12]. Note however that according to G.H. Hardy [ 4, p 156]
this inequality was pointed out to him already in 1925 by G. Polya.
J. Ineq. Pure and Appl. Math. 4(4) Art. 68, 2003
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In this paper we prove a more general result, namely we characterize the
weights u and v on RN such that for 0 < p ≤ q < ∞
Z
RN
q
1q
Z
p1
Z
1
p
v(x)dx
f (x)u(x)dx
ln f (y)dy
≤C
exp
|Bx | Bx
RN
holds for some finite positive constant C (See Corollary 3.2). In the case when
v(x) = |Sx |a and u(x) = |Sx |b we obtain a genuine generalization of (1.2) (see
Proposition 3.3 and Remark 3.4).
In this paper we also generalize the results in another direction, namely when
the geometric averages over spheres in RN are replaced by such averages over
spherical cones in RN (see notation below). This means in particular that our
inequalities above and later on also hold when RN is replaced by RN
+ or even
N
more general cones in R .
The paper is organized in the following way. In Section 2 we collect some
preliminaries and prove a new Hardy inequality that averages functions over
the cones in RN (see Theorem 2.1). In Section 3 we present and prove our
main results concerning (the limiting) geometric mean operators (see Theorem
3.1 and Proposition 3.3). Finally, in Section 4 we present the corresponding
companion inequalities (see Theorem 4.1, Corollary 4.2 and Proposition 4.3).
Weighted Geometric Mean
Inequalities Over Cones in RN
Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
Anna Wedestig
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2.
Preliminaries
Let ΣN −1 be the unit sphere in RN , that is, ΣN −1 = {x ∈ RN : |x| = 1}, where
|x| denotes the Euclidean norm of the vector x ∈ RN . Let A be a measurable
subset of ΣN −1 , and let E ⊆ RN be a spherical cone, i.e.,
E = x ∈ RN : x =sσ, 0 ≤ s < ∞, σ ∈ A .
Let Sx , x ∈ RN denote the part of E with ‘radius’ ≤ |x| , i.e.,
Sx = y ∈ RN : y =sσ, 0 ≤ s ≤ |x| , σ ∈ A .
Weighted Geometric Mean
Inequalities Over Cones in RN
Lpw
For 0 < p < ∞ and a non-negative measurable function w on E, by
:=
Lpw (E) we denote the weighted Lebesgue space with the weight function w,
consisting of all measurable functions f on E such that
Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
Anna Wedestig
Title Page
Z
kf kLpw =
p1
p
|f (x)| w(x)dx
<∞,
E
p
and make use of the abbreviations L and kf kLp when w(x) ≡ 1.
Let S = Sx , |x| = 1. The family of regions we shall average over is the
collection of dilations of S. For x ∈ E \ {0} denote by |Sx | the Lebesgue
measure of Sx . Using polar coordinates we obtain (dσ denotes the usual surface
measure on ΣN −1 )
Z
|x| Z
|Sx | =
0
A
N
sN −1 dσds =
|x|
|A|.
N
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Moreover, we say that u is a weight function if it is a positive and measurable
p
function on S. Throughout the paper, for any p > 1 we denote p0 = p−1
.
For later purposes but also of independent interest we now state and prove
our announced Hardy inequality.
Theorem 2.1. Let E be a cone in RN and Sx , A be defined as above. Suppose
that 1 < p ≤ q < ∞ and that u, v are weight functions on E. Then, the
inequality
q
1q
Z
p1
Z Z
p
(2.1)
f (y)dy v(x)dx
≤C
f (x)u(x)dx
E
Sx
E
Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
Anna Wedestig
holds for all f ≥ 0 if and only if
Z
− p1
1−p0
(x)dx
u
(2.2) D := sup
t>0
Weighted Geometric Mean
Inequalities Over Cones in RN
tS
Z
×
Z
v(x)
tS
0
u1−p (y)dy
q
1q
dx
< ∞.
Sx
Moreover, the best constant C in (2.1) can be estimated as follows:
D ≤ C ≤ p0 D.
Remark 2.1. Another weight characterization of (2.1) over balls in RN was
proved by P. Drábek, H.P. Heinig and A. Kufner [3] . This result may be regarded as a generalization of the usual (Muckenhaupt type) characterization in
1-dimension (see e.g. [13]) while our result may be seen as a higher dimensional version of another characterization by V.D. Stepanov and L.E. Persson
(see [19] , [20]).
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Proof. By the duality principle (see e.g. [13]), it can be shown that the inequality (2.1) is equivalent to that the inequality
! 10
1
0
Z
Z Z
p
g(y)dy
(2.3)
E
p
u
1−p0
0
0
g q (x)v 1−q (x)dx
≤C
(x)dx
q0
E
E\Sx
holds for all g ≥ 0 and with the same best constant C. First assume that (2.2)
holds. Using polar coordinates and putting
Z
(2.4)
ge(t) =
g(tσ)tN −1 dσ,
t ∈ (0, ∞)
Weighted Geometric Mean
Inequalities Over Cones in RN
A
Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
Anna Wedestig
and
Z
u
e(t) =
(2.5)
we have
Z Z
u
N −1
(tτ )t
1−p
dτ
,
t ∈ (0, ∞)
A
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p0
g(y)dy
E
1−p0
u
1−p0
(x)dx
E\Sx
Z
∞
Z Z
∞
Z
=
g(sσ)s
0
Z
A
t
∞ Z
=
0
t
N −1
p0
dσds
0
u1−p (tτ )tN −1 dτ dt
A
p0
∞
0
ge(s)ds
u
e1−p (t)dt.
Thus, using this, changing the order of integration and finally using Hölder’s
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inequality, we get
Z Z
(2.6) I :=
E
Z
p0
E\Sx
∞
Z
∞
∞
Z
=
0
Z
t
0
Z
∞
0
=p
0
Z
∞
z
∞
=p
Z
≤p
p0 −1
ge(s)ds
t
∞
Z Z
0
0
t
∞
Z
0
0
p0
0
ge(s)ds
u
e1−p (t)dt
Z ∞
p0 !
d
0
ge(s)ds
dt u
e1−p (z)dz
−
dt
t
z
!
p0 −1
Z ∞ Z ∞
0
ge(s)ds
ge(t)dt u
e1−p (z)dz
=
= p0
0
u1−p (x)dx
g(y)dy
A
Z
∞
ge(s)ds
q0
1−q 0
0
N −1
(tτ )t
∞
Z Z
∞
×
0
Z
×
0
= p0
E
t
1−p0
u
e
1−p0
u
e
A
t
t
(s)ds g(tτ )tN −1 dτ dt
10
q
dτ dt
0
u
e1−p (s)ds
v(tτ )tN −1 dτ dt
10
q
1
q0
1−q 0
g (x)v
(x)dx
Jq,
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(p0 −1)q
ge(s)ds
q
Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
Anna Wedestig
(z)dz dt
A
Z
Z
0
t
p0 −1 Z
g (tτ )v
0
ge(t)
∞
t
Z
Z
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Inequalities Over Cones in RN
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where
Z
∞
Z
J=
0
(p0 −1)q Z
∞
ge(s)ds
t
0
t
1−p0
u
e
q
(s)ds
ve(t)dt
with
Z
ve(t) =
(2.7)
v(tτ )tN −1 dτ.
A
Using Fubini’s theorem, (2.2), (2.5) and (2.7), we get
Z ∞
(p0 −1)q ! Z t
q
Z ∞Z ∞
d
1−p0
dz
u
e
(s)ds ve(t)dt
J=
−
ge(s)ds
dz
z
0
0
t
Z ∞
(p0 −1)q !# Z z Z t
q
Z ∞"
d
1−p0
=
−
ge(s)ds
u
e
(s)ds ve(t)dtdz
dz
0
z
0
0
Z ∞
(p0 −1)q !#
Z ∞"
d
−
ge(s)ds
=
dz
z
0
Z z Z Z t Z
q
1−p0
N −1
N −1
×
u
(sσ)s
dσds v(tτ )t
dτ dt dz
0
A
0
A
Z ∞
(p0 −1)q !#
Z ∞"
d
=
−
ge(s)ds
dz
0
z
Z Z
q
1−p0
×
u
(y)dy v(x)dx dz
zS
Weighted Geometric Mean
Inequalities Over Cones in RN
Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
Anna Wedestig
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Sx
J. Ineq. Pure and Appl. Math. 4(4) Art. 68, 2003
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≤ Dq
∞
Z
"
0
= Dq
∞
Z
"
0
−
∞
Z
d
dz
−
z
pq
(x)dx dz
1−p0
pq
(t)dt dz.
u
ge(s)ds
z
1−p0
(p0 −1)q !# Z
∞
Z
d
dz
zS
(p0 −1)q !# Z
ge(s)ds
z
u
e
0
Thus, using Minkowski’s integral inequality, (2.4) and (2.5) we have
 pq

Z ∞
(p0 −1)q !# ! pq
Z ∞ Z ∞"
d
0
u
e1−p (t)dt
J ≤ Dq 
−
ge(s)ds
dz
dz
z
0
t
! pq
0
Z Z
∞
0
=D
q
p
∞
= Dq
t
0
u
e1−p (t)dt
ge(s)ds
g(y)dy
E
! pq
p0
Z Z
u
1−p0
(x)dx
p
g(y)dy
E
E\Sx
.
p
u
1−p0
(x)dx
0
Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
Anna Wedestig
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E\Sx
Assume first that in (2.6) I < ∞. Then
! 10
0
Z Z
Weighted Geometric Mean
Inequalities Over Cones in RN
Z
≤pD
q0
g (x)v
1−q 0
10
q
(x)dx
E
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0
i.e., (2.3) holds for all g ≥ 0 and also the constant C in (2.3) satisfies C ≤ p D.
For the case I = ∞ replace g(y) by an approximating sequence gn (y) ≤ g(y)
(such that the corresponding In < ∞) and use the Monotone Convergence
Theorem to obtain the result.
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Conversely, suppose that (2.1) holds for all f ≥ 0. In this inequality, taking
0
for any fixed t > 0 the function ft = χtS u1−p , we find that
q
Z Z
C≥
ft (y)dy
E
Sx
E
Z Z
≥
u
tS
1q Z
− p1
p
v(x)dx
ft (x)u(x)dx
Sx
1−p0
1q Z
q
(y)dy
v(x)dx
u
1−p0
− p1
(x)dx
.
tS
By taking the supremum we find that (2.2) holds and, moreover, D ≤ C. The
proof is complete.
Weighted Geometric Mean
Inequalities Over Cones in RN
Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
Anna Wedestig
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3.
Geometric Mean Inequalities
Here we prove our main geometric mean inequality by making a limit procedure
in Theorem 2.1.
Theorem 3.1. Let 0 < p ≤ q < ∞ and suppose that all other assumptions of
Theorem 2.1 are satisfied. Then the inequality
Z exp
(3.1)
E
1
|Sx |
q
Z
ln f (y)dy
1q
v(x)dx
Sx
p1
p
f (x)u(x)dx
Z
≤C
E
Weighted Geometric Mean
Inequalities Over Cones in RN
Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
Anna Wedestig
holds for all f > 0 if and only if
− p1
1q
w(x)dx
< ∞,
Z
D1 := sup |tS|
t>0
Contents
tS
where
(3.2)
w(t) := v(x) exp
1
|Sx |
Z
ln
Sx
Title Page
1
dy
u(y)
pq
< ∞.
1
p
Moreover, the best constant C satisfies D1 ≤ C ≤ e D1 .
Proof. It is easy to see that (3.1) is equivalent to
q
1q
Z
p1
Z Z
1
p
ln f (y)dy
w(x)dx
exp
≤C
f (x)dx
|Sx | Sx
E
E
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with w(x) defined by (3.2). Let v(x) = w(x) |Sx |−q and u(x) =1 in Theorem
2.1 and choose an α such that 0 < α < p ≤ q < ∞. Then 1 < αp ≤ αq < ∞.
Now, replacing f, p, q and v(x) by f α , αp , αq in Theorem 2.1, we find that the
inequality
Z (3.3)
E
1
|Sx |
Z
! 1q
αq
α
f (y)dy
w(x)dx
Z
≤ Cα
Sx
p1
f (x)dx
p
E
holds for all functions f > 0 if and only if D1 holds. Moreover, it is easy to see
that (c.f. [20])
D1 ≤ Cα ≤
(3.4)
p
p−α
By letting α → 0 in (3.3) and (3.4) we find that
1
|Sx |
Z
α
f (y)dy
Sx
α1
→ exp
Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
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α1
+
1
|Sx |
D1 .
p
p−α
Z
Weighted Geometric Mean
Inequalities Over Cones in RN
α1
1
→ e p and
ln f (y)dy ,
Sx
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i.e. the scale of power means converge to the geometric mean, and the proof
follows.
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Remark 3.1. Our proof above shows that (3.1) in Theorem 3.1 may be regarded
as a natural limiting case of Hardy’s inequality (2.1) as it is in the classical
one-dimensional situation. This fact indicates that our formulation of Hardy’s
inequality in Theorem 2.1 is very natural from this point of view.
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As a special case, if we take E = RN and Sx = Bx the ball centered at the
origin and with radius |x| , and |Bx | its volume, then we immediately obtain the
following corollary to Theorem 3.1 that averages functions over balls in RN :
Corollary 3.2. Let 0 < p ≤ q < ∞ and u, v be weight functions in RN . Then
the inequality
Z
exp
RN
1
|Bx |
q
Z
ln f (y)dy
1q
Z
v(x)dx
≤C
p1
f p (x)u(x)dx
RN
Bx
Weighted Geometric Mean
Inequalities Over Cones in RN
holds for all f > 0 if and only if
D2 :=
sup
z∈RN \{0}
− p1
Z
|Bz |
v(x) exp
Bz
1
|Bx |
Z
1
ln
dy
u(y)
Bx
pq
Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
Anna Wedestig
! 1q
dx
< ∞.
Title Page
1
p
Moreover, the best constant C satisfies D2 ≤ C ≤ e D2 .
Contents
Remark 3.2. Corollary 3.2 extends a result of P. Drábek, H.P. Heinig and A.
Kufner [3, Theorem 4.1], who obtained it for the case p = q = 1 and with a
completely different proof.
N
Remark 3.3. Setting E = RN
+ = (x1 , . . . , xN ) ∈ R , x1 ≥ 0, . . . , xN ≥ 0
N
in Theorem 3.1 we obtain that Corollary 3.2 holds also for RN
+ instead of R
and Bx ∩ RN
+ instead of Bx .
JJ
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We shall now consider the special weights discussed in our introduction and
in [1].
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Proposition 3.3. Let 0 < p ≤ q < ∞, a, b ∈ R, ε ∈ R+ , and E, Sx be defined
as in Theorem 2.1. Then
1q
Z q
Z
−ε
ε−1
a
(3.5)
exp ε |Sx |
|Sy | ln f (y)dy
|Sx | dx
E
Sx
Z
≤C
p1
f p (x) |Sx |b dx
E
holds for all positive functions f for some finite constant C if and only if
(3.6)
a+1
b+1
=
q
p
and the least constant C in (3.5) satisfies
1q
1q
1
1 b+1
1
1
1 b+1
p
p
−
−
ε p q e εp p ≤ C ≤
ε p − q e εp .
q
q
Proof. By writing (3.5) in polar coordinates we find that
Z ∞ Z εN ε
exp N ε ε
t |A|
0
A
#q
! 1q
a
Z t Z ε−1
|A|
|A|
×
sN ε−1 ln f (sσ)dσds tN a+N −1
dτ dt
N
N
0
A
! p1
b
Z ∞Z
|A|
≤
f p (tτ )
tN b+N −1 dτ dt
.
N
0
A
Weighted Geometric Mean
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Lars-Erik Persson and
Anna Wedestig
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1
1
Exchanging variables, s = r ε and t = z ε we find that this inequality can be
rewritten as
Z ∞ Z q
Z zZ
1 N
N
−1
exp
ln f r ε σ r
dσdr
|A| z N 0 A
0
A
1q
a
1
|A|
N ( a+1
−1
N
−1
)z
ε
z
dτ dz
×
N
ε
! p1
Z ∞Z
1 |A| b
b+1
1
N
−1
,
≤C
fp z ε τ
z ( ε ) z N −1 dτ dz
ε
N
0
A
Weighted Geometric Mean
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Babita Gupta, Pankaj Jain,
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Anna Wedestig
that is,
Z (3.7)
E
q
1q
Z
a+1
1
−1
exp
ln f1 (y)dy
|Sx | ε
dx
|Sx | Sx
( b+1
Z
p1
− a+1
1− 1ε )
p
q )(
b+1
1
|A|
−1
p
− p1
ε
q
≤C
ε
f1 (x) |Sx |
dx ,
N
E
1
where f1 (rσ) = f (r ε σ). This means that (3.5) is equivalent to (3.7) i.e., (3.1)
a+1
b+1
holds with the weights v(x) = |Sx | ε −1 and u(x) = |Sx | ε −1 . We note that
for these weights we find after a direct calculation that the constant D1 from
Theorem 3.1 is
a+1
b+1
1 b+1
|tS| εq − εp e p ( ε −1)
D1 = sup 1q
t>0
q b+1
a+1
−
−
1
ε
p
ε
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so we conclude that (3.6) must hold and then
b+1
D1 = e ( ε −1)
1
p
1q
p
.
q
Thus, the proof follows from Theorem 3.1.
Remark 3.4. Setting p = q = 1, a = b, we have that (3.5) implies the estimate
(1.2).
Remark 3.5 (Sharp Constant). In the above proposition, if we take p = q,
then a = b. In this situation (3.5) holds with the constant C = e(b+1)/p . Indeed,
this constant is sharp. In order to show this for δ > 0, we consider the function

N
−(b+1)
− b+1

|x|− p (b+1−εδ) , x ∈ S,
 e εp |S|
fδ (x) =

N
 − b+1
e εp |S|−(b+1) |x|− p (b+1+εδ) , x ∈ E\S.
By using this function in (3.5), we find that
1≤
δ
RHS
≤ ep → 1
LHS
as
δ→0
and consequently the constant is sharp. Note that the sharpness of the constant
for p = q, in Proposition 3.3 has been proved in the more general setting than
that in [1].
Weighted Geometric Mean
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Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
Anna Wedestig
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4.
The Companion Inequalities
We present the following result which is a companion of Theorem 3.1:
Theorem 4.1. Let 0 < p ≤ q < ∞, ε > 0, and suppose that all other
hypotheses of Theorem 3.1 are satisfied. Then the inequality
Z ε
Z
−ε−1
exp ε |Sx |
(4.1)
E
|Sy |
q
ln f (y)dy
1q
v(x)dx
E\Sx
p1
p
f (x)u(x)dx
Z
≤C
Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
Anna Wedestig
E
holds for all f > 0 if and only if
− p1
D3 := sup |tS|
t>0
Z
v∗ (x) exp
tS
1
|Sx |
Z
1
ln
dy
u∗ (y)
Sx
pq
! 1q
dx
where
1
1
1
u∗ (y) := u(s− ε σ) s−N (1+ ε ) ,
ε
1
1
1
v∗ (y) := v(s− ε σ) s−N (1+ ε ) .
ε
1
p
Moreover, the constant C satisfies D3 ≤ C ≤ e D3 .
Proof. Note that for x ∈ RN
Z
|Sx | =
0
Weighted Geometric Mean
Inequalities Over Cones in RN
< ∞,
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|x|
Z
N −1
t
A
|x|N
dτ dt =
|A|.
N
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Now, using polar coordinates, (4.1) can be written as
Z ∞ Z ε |A|ε tN ε
exp
N
0
A
!q
! 1q
Z ∞ Z −ε−1
|A|
s−N ε−1 ln f (sσ)dσds v(tτ )tN −1 dτ dt
×
N
t
A
Z ∞ Z
p1
≤C
f p (tτ )u(tτ )tN −1 dτ dt .
0
A
Using the exchange of variables s = r−1/ε and t = z −1/ε we obtain
Z ∞ Z q
Z Z z
N
− 1ε
N −1
exp
ln f (r σ)r
dσdr
|A| z N A 0
0
A
1q
− 1ε
−N (1+ 1ε ) 1 N −1
×v(z τ )z
z
dτ dz
ε
Z ∞ Z
p1
p − 1ε
− 1ε
−N (1+ 1ε ) 1 N −1
z
dτ dz
≤C
f (z τ )u(z τ )z
ε
0
A
1
and put f∗ (tτ ) = f (t− ε τ ). (4.1) can be equivalently rewritten as
Z exp
E
1
|Sx |
q
Z
ln f∗ (y)dy
1q
p1
Z
p
v∗ (x)dx
≤C
f∗ (x)u∗ (x)dx .
Sx
Weighted Geometric Mean
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Now, the result is obtained by using Theorem 3.1.
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Analogously to Corollary 3.2, we can immediately obtain a special case of
Theorem 4.1 that averages functions over balls in RN centered at origin.
Corollary 4.2. Let 0 < p ≤ q < ∞, ε > 0, and u, v be weight functions in
RN . Then the inequality
Z
ε
Z
exp ε |Bx |
(4.2)
RN
q
−ε−1
|By |
ln f (y)dy
1q
v(x)dx
RN \Bx
p1
f p (x)u(x)dx
Z
≤C
RN
Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
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holds for all f > 0 if and only if
Z
− p1
e := sup |Bz |
B
v0 (x) exp
z∈RN
Bz
1
|Bx |
Z
1
ln (y)dy
u0
Bx
Weighted Geometric Mean
Inequalities Over Cones in RN
pq
! 1q
dx
< ∞,
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where
− 1ε
u0 (x) := u(t
1
1
τ ) t−N (1+ ε ) ,
ε
− 1ε
v0 (x) := v(t
1
1
τ ) t−N (1+ ε ) .
ε
e ≤ C ≤ e p1 B.
e
Moreover, the best constant C satisfies B
Remark 4.1. Note that by choosing E as in Remark 3.3 we see that Corollary
N
4.2 in fact holds also when RN is replaced by RN
+ or more general cones in R .
The corresponding result to Proposition 3.3 reads as follows and the proof is
analogous.
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Proposition 4.3. Let 0 < p ≤ q < ∞, ε > 0, and a, b ∈ R, and E, Sx be
defined as in Theorem 2.1. Then the inequality
Z ε
Z
exp ε |Sx |
(4.3)
E
−ε−1
|Sy |
q
ln f (y)dy
1q
|Sx | dx
a
E\Sx
Z
≤C
p1
f (x) |Sx | dx
p
b
E
holds for all f > 0 and some finite positive constant C if and only if
a+1
b+1
=
q
p
Weighted Geometric Mean
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Babita Gupta, Pankaj Jain,
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and the least constant C in (4.3) satisfies
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− 1q
− 1q
b+1
1
1
1
p
p
− 1q −( b+1
+ p1 )
p
εp
ε
e
≤C≤
ε p − q e− εp .
q
q
Remark 4.2 (Sharp Constant). Analogously to Proposition 3.3, in the above
proposition we also find that if we take p = q, then a = b. In this situation (4.3)
holds with the constant C = e−(b+1)/εp and the constant is sharp. This can be
shown by considering, for δ > 0, the function
 b+1
N
−(b+1)

|x|− p (b+1−εδ) , x ∈ S
 e εp |S|
fδ (x) =

N
 b+1
e p |S|−(b+1) |x|− p (b+1+εδ) , x ∈ E\S.
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Remark 4.3. It is tempting to think that the results in this paper hold also in
general star-shaped regions in RN (c.f. [22]) but this is not true in general as
was pointed out to us by the referee. See also [22] and note that the results there
also hold at least for cones in RN .
Weighted Geometric Mean
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Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
Anna Wedestig
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References
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Weighted Geometric Mean
Inequalities Over Cones in RN
[4] G.H. HARDY, Notes on some points in the integral calculus, LXIV
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Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
Anna Wedestig
[5] H.P. HEINIG, Weighted inequalities in Fourier analysis, Nonlinear Analysis, Function Spaces and Applications, Vol. 4, Teubner-Texte Math., band
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[6] H.P. HEINIG, R. KERMAN AND M. KRBEC, Weighted exponential inequalities, Georgian Math. J., (2001), 69–86.
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[9] P. JAIN, L.E. PERSSON AND A. WEDESTIG, Carleman-Knopp type
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[15] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequalities Involving Functions and their Integrals and Derivatives , Kluwer Academic Publishers, 1991.
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[17] M. NASSYROVA, L.E. PERSSON AND V.D. STEPANOV, On weighted
inequalities with geometric mean operator by the Hardy-type integral
Weighted Geometric Mean
Inequalities Over Cones in RN
Babita Gupta, Pankaj Jain,
Lars-Erik Persson and
Anna Wedestig
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transform, J. Inequal. Pure Appl. Math., 3(4) (2002), Art. 48. [ONLINE:
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Weighted Geometric Mean
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