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volume 4, issue 4, article 68, 2003.

Received 7 November, 2002;

accepted 20 March, 2003.

Communicated by:B. Opi´c

Abstract Contents

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Journal of Inequalities in Pure and Applied Mathematics

WEIGHTED GEOMETRIC MEAN INEQUALITIES OVER CONES INRN

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

c

2000Victoria University ISSN (electronic): 1443-5756 118-02

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Weighted Geometric Mean Inequalities Over Cones inRN

Babita Gupta, Pankaj Jain, Lars-Erik Persson and

Anna Wedestig

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Abstract

Let0< p≤q <∞.LetAbe a measurable subset of the unit sphere inRN,let

E =

x∈RN:x=sσ,0≤s <∞, σ∈A be a cone inRN and letSxbe the part ofEwith ’radius’≤ |x|.A characterization of the weightsuandvonEis given such that the inequality

Z

E

exp

1

|Sx| Z

Sx

lnf(y)dy q

v(x)dx 1q

≤C Z

E

fp(x)u(x)dx 1p

holds for allf ≥0and some positive and finite constantC.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 constantCis discussed.

2000 Mathematics Subject Classification:26D15, 26D07.

Key words: Inequalities, Multidimensional inequalities, Geometric mean inequalities, Hardy type inequalities, Cones inRN, Sharp constant.

We thank Professor Alexandra ˇCižmešija for some valuable advice and the referee for pointing out an inaccuracy in our original manuscript (see Remark4.3) 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

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Weighted Geometric Mean Inequalities Over Cones inRN

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 (1.1)

Z 0

exp

εx−ε

Z x 0

yε−1lnf(y)dy

xadx≤ea+1ε Z

0

xaf(x)dx, where a, ε are real numbers with ε > 0, f is a positive function defined on (0,∞)and the constantea+1ε is the best possible. This inequality, in fact, is a generalization of what sometimes is referred to as Knopp’s inequality1 , which is obtained by takingε= 1anda= 0in (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. ˇCižmešija, J. Peˇcari´c and I. Peri´c [1, Th. 9, formula (23)] proved anN−dimensional analogue of (1.1) by replacing the in- terval(0,∞)byRN and the means are considered over the balls inRN centered at the origin. Their inequality reads:

(1.2) Z

RN

exp

ε|Bx|−ε Z

Bx

|By|ε−1lnf(y)dy

|Bx|adx

≤ea+1ε Z

RN

f(x)|Bx|adx, where a ∈ R, ε > 0, f is a positive function onRN, Bx is a ball in RN with radius|x|,x∈RN,centered at the origin and|Bx|is its volume.

1See 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.

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In this paper we prove a more general result, namely we characterize the weightsuandvonRN such that for0< p≤q <∞

Z

RN

exp

1

|Bx| Z

Bx

lnf(y)dy q

v(x)dx 1q

≤C Z

RN

fp(x)u(x)dx 1p

holds for some finite positive constantC (See Corollary3.2). In the case when v(x) =|Sx|aandu(x) = |Sx|b we obtain a genuine generalization of (1.2) (see Proposition3.3and Remark3.4).

In this paper we also generalize the results in another direction, namely when the geometric averages over spheres inRN 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 byRN+ or even more general cones inRN.

The paper is organized in the following way. In Section2we 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 Theorem4.1, Corollary4.2and Proposition4.3).

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Babita Gupta, Pankaj Jain, Lars-Erik Persson and

Anna Wedestig

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2. Preliminaries

LetΣN−1 be the unit sphere inRN, that is,ΣN−1 ={x∈RN :|x|= 1}, where

|x|denotes the Euclidean norm of the vectorx ∈ RN. LetAbe a measurable subset ofΣN−1, and letE ⊆RN be a spherical cone, i.e.,

E =

x∈RN :x=sσ,0≤s <∞, σ ∈A . LetSx,x∈RN denote the part ofEwith ‘radius’≤ |x|,i.e.,

Sx=

y∈RN :y=sσ,0≤s≤ |x|, σ∈A .

For 0 < p < ∞ and a non-negative measurable function w onE, by Lpw :=

Lpw(E) we denote the weighted Lebesgue space with the weight function w, consisting of all measurable functionsf onE such that

kfkLp

w = Z

E

|f(x)|pw(x)dx 1p

<∞, and make use of the abbreviationsLpandkfkLp whenw(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 ofSx. Using polar coordinates we obtain (dσdenotes the usual surface measure onΣN−1)

|Sx|= Z |x|

0

Z

A

sN−1dσds= |x|N N |A|.

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Moreover, we say that uis a weight function if it is a positive and measurable function onS.Throughout the paper, for anyp > 1we denotep0 = p−1p .

For later purposes but also of independent interest we now state and prove our announced Hardy inequality.

Theorem 2.1. LetE be a cone inRN andSx, Abe defined as above. Suppose that 1 < p ≤ q < ∞ and that u, v are weight functions on E. Then, the inequality

(2.1)

Z

E

Z

Sx

f(y)dy q

v(x)dx 1q

≤C Z

E

fp(x)u(x)dx 1p

holds for allf ≥0if and only if (2.2) D:= sup

t>0

Z

tS

u1−p0(x)dx 1p

× Z

tS

v(x) Z

Sx

u1−p0(y)dy q

dx 1q

<∞.

Moreover, the best constantC in (2.1) can be estimated as follows:

D≤C ≤p0D.

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 re- garded 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 dimen- sional 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 inequal- ity (2.1) is equivalent to that the inequality

(2.3) Z

E

Z

E\Sx

g(y)dy p0

u1−p0(x)dx

!p10

≤C Z

E

gq0(x)v1−q0(x)dx q10

holds for all g ≥ 0and with the same best constantC.First assume that (2.2) holds. Using polar coordinates and putting

(2.4) eg(t) =

Z

A

g(tσ)tN−1dσ, t∈(0,∞) and

(2.5) eu(t) = Z

A

u1−p0(tτ)tN−11−p

, t∈(0,∞) we have

Z

E

Z

E\Sx

g(y)dy p0

u1−p0(x)dx

= Z

0

Z

A

Z t

Z

A

g(sσ)sN−1dσds p0

u1−p0(tτ)tN−1dτ dt

= Z

0

Z

t eg(s)ds p0

eu1−p0(t)dt.

Thus, using this, changing the order of integration and finally using Hölder’s

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inequality, we get I :=

Z

E

Z

E\Sx

g(y)dy p0

u1−p0(x)dx (2.6)

= Z

0

Z

t eg(s)ds p0

eu1−p0(t)dt

= Z

0

Z z

−d dt

Z t

eg(s)ds p0

dt

!

eu1−p0(z)dz

=p0 Z

0

Z z

Z

t eg(s)ds p0−1

eg(t)dt

!

ue1−p0(z)dz

=p0 Z

0

Z

t eg(s)ds p0−1

eg(t) Z t

0 ue1−p0(z)dz

dt

=p0 Z

0

Z

A

Z t

eg(s)ds

p0−1Z t 0

eu1−p0(s)ds

g(tτ)tN−1dτ dt

≤p0 Z

0

Z

A

gq0(tτ)v1−q0(tτ)tN−1dτ dt q10

× Z

0

Z

A

Z t

eg(s)ds

(p0−1)q

× Z t

0

ue1−p0(s)ds q

v(tτ)tN−1dτ dt 1q

=p0 Z

E

gq0(x)v1−q0(x)dx q10

J1q,

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where

J = Z

0

Z

t ge(s)ds

(p0−1)qZ t

0 eu1−p0(s)ds q

ev(t)dt with

(2.7) ev(t) =

Z

A

v(tτ)tN−1dτ.

Using Fubini’s theorem, (2.2), (2.5) and (2.7), we get J =

Z 0

Z t

d dz −

Z

z eg(s)ds

(p0−1)q! dz

Z t

0 eu1−p0(s)ds q

ev(t)dt

= Z

0

"

d dz −

Z

z eg(s)ds

(p0−1)q!#

Z z 0

Z t

0 ue1−p0(s)ds q

ev(t)dtdz

= Z

0

"

d dz −

Z z

eg(s)ds

(p0−1)q!#

× Z z

0

Z

A

Z t 0

Z

A

u1−p0(sσ)sN−1dσds q

v(tτ)tN−1dτ dt

dz

= Z

0

"

d dz −

Z

z eg(s)ds

(p0−1)q!#

× Z

zS

Z

Sx

u1−p0(y)dy q

v(x)dx

dz

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≤Dq Z

0

"

d dz −

Z z

eg(s)ds

(p0−1)q!#

Z

zS

u1−p0(x)dx qp

dz

=Dq Z

0

"

d dz −

Z

z eg(s)ds

(p0−1)q!#

Z z

0 eu1−p0(t)dt qp

dz.

Thus, using Minkowski’s integral inequality, (2.4) and (2.5) we have

J ≤Dq

 Z

0

Z t

"

d dz −

Z z

eg(s)ds

(p0−1)q!#

dz

!pq

eu1−p0(t)dt

q p

=Dq Z

0

Z

t eg(s)ds p0

ue1−p0(t)dt

!qp

=Dq Z

E

Z

E\Sx

g(y)dy p0

u1−p0(x)dx

!qp . Assume first that in (2.6)I <∞.Then

Z

E

Z

E\Sx

g(y)dy p0

u1−p0(x)dx

!p10

≤p0D Z

E

gq0(x)v1−q0(x)dx q10

i.e., (2.3) holds for allg ≥0and also the constantC in (2.3) satisfiesC ≤p0D.

For the caseI =∞replaceg(y)by an approximating sequencegn(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 allf ≥ 0.In this inequality, taking for any fixedt >0the functionfttSu1−p0, we find that

C ≥ Z

E

Z

Sx

ft(y)dy q

v(x)dx 1q Z

E

ftp(x)u(x)dx p1

≥ Z

tS

Z

Sx

u1−p0(y)dy q

v(x)dx 1q Z

tS

u1−p0(x)dx 1p

. By taking the supremum we find that (2.2) holds and, moreover,D ≤ C. The proof is complete.

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3. Geometric Mean Inequalities

Here we prove our main geometric mean inequality by making a limit procedure in Theorem2.1.

Theorem 3.1. Let 0 < p≤ q < ∞and suppose that all other assumptions of Theorem2.1are satisfied. Then the inequality

(3.1) Z

E

exp

1

|Sx| Z

Sx

lnf(y)dy q

v(x)dx 1q

≤C Z

E

fp(x)u(x)dx 1p

holds for allf >0if and only if D1 := sup

t>0

|tS|1p Z

tS

w(x)dx 1q

<∞, where

(3.2) w(t) :=v(x)

exp 1

|Sx| Z

Sx

ln 1 u(y)dy

qp

<∞.

Moreover, the best constantC satisfiesD1 ≤C ≤e1pD1. Proof. It is easy to see that (3.1) is equivalent to

Z

E

exp

1

|Sx| Z

Sx

lnf(y)dy q

w(x)dx 1q

≤C Z

E

fp(x)dx 1p

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withw(x)defined by (3.2). Letv(x) = w(x)|Sx|−q andu(x) =1 in Theorem 2.1 and choose anαsuch that0 < α < p ≤ q < ∞.Then1 < pααq < ∞.

Now, replacing f, p, q and v(x) by fα,αp,αq in Theorem 2.1, we find that the inequality

(3.3)

Z

E

1

|Sx| Z

Sx

fα(y)dy αq

w(x)dx

!1q

≤Cα

Z

E

fp(x)dx 1p

holds for all functionsf >0if and only ifD1holds. Moreover, it is easy to see that (c.f. [20])

(3.4) D1 ≤Cα

p p−α

1α D1.

By lettingα→0+in (3.3) and (3.4) we find that

p p−α

α1

→e1p and 1

|Sx| Z

Sx

fα(y)dy α1

→exp 1

|Sx| Z

Sx

lnf(y)dy

,

i.e. the scale of power means converge to the geometric mean, and the proof follows.

Remark 3.1. Our proof above shows that (3.1) in Theorem3.1may 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 Theorem2.1is very natural from this point of view.

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As a special case, if we takeE = RN andSx =Bxthe ball centered at the origin and with radius|x|,and|Bx|its volume, then we immediately obtain the following corollary to Theorem3.1that averages functions over balls inRN: Corollary 3.2. Let0< p≤ q <∞andu, v be weight functions inRN. Then the inequality

Z

RN

exp

1

|Bx| Z

Bx

lnf(y)dy q

v(x)dx 1q

≤C Z

RN

fp(x)u(x)dx 1p

holds for allf >0if and only if

D2 := sup

z∈RN\{0}

|Bz|1p Z

Bz

v(x)

exp 1

|Bx| Z

Bx

ln 1 u(y)dy

qp dx

!1q

<∞.

Moreover, the best constantC satisfiesD2 ≤C ≤e1pD2.

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 = 1and with a completely different proof.

Remark 3.3. SettingE = RN+ =

(x1, . . . , xN)∈RN, x1 ≥0, . . . , xN ≥0 in Theorem 3.1 we obtain that Corollary3.2 holds also forRN+ instead ofRN andBx∩RN+ instead ofBx.

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+,andE, Sxbe defined as in Theorem2.1. Then

(3.5) Z

E

exp

ε|Sx|−ε Z

Sx

|Sy|ε−1lnf(y)dy q

|Sx|adx 1q

≤C Z

E

fp(x)|Sx|bdx

p1

holds for all positive functionsf for some finite constantC if and only if

(3.6) a+ 1

q = b+ 1 p and the least constantCin (3.5) satisfies

p q

1q

ε1p1qeb+1εp1p ≤C ≤ p

q 1q

ε1p1qeb+1εp . Proof. By writing (3.5) in polar coordinates we find that

Z 0

Z

A

exp εNε tN ε|A|ε

× Z t

0

Z

A

|A|

N ε−1

sN ε−1lnf(sσ)dσds

#q

tN a+N−1 |A|

N a

dτ dt

!1q

≤ Z

0

Z

A

fp(tτ) |A|

N b

tN b+N−1dτ dt

!1p .

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Exchanging variables, s = r1ε and t = z1ε we find that this inequality can be rewritten as

Z 0

Z

A

exp

N

|A|zN Z z

0

Z

A

lnf r1εσ

rN−1dσdr q

× |A|

N a

zN(a+1ε −1)zN−11 εdτ dz

1q

≤C Z

0

Z

A

fp

z1ετ|A|

N b

zN(b+1ε −1)zN−11 εdτ dz

!1p , that is,

(3.7) Z

E

exp

1

|Sx| Z

Sx

lnf1(y)dy q

|Sx|a+1ε −1dx 1q

≤C |A|

N

(b+1p a+1q )(1−1ε) ε1q1p

Z

E

f1p(x)|Sx|b+1ε −1dx 1p

, wheref1(rσ) = f(r1εσ).This means that (3.5) is equivalent to (3.7) i.e., (3.1) holds with the weights v(x) = |Sx|a+1ε −1 andu(x) = |Sx|b+1ε −1. We note that for these weights we find after a direct calculation that the constant D1 from Theorem3.1is

D1 = sup

t>0

|tS|a+1εq b+1εp e1p(b+1ε −1) a+1

εqp b+1ε −11q

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so we conclude that (3.6) must hold and then D1 =e1p(b+1ε −1)

p q

1q . Thus, the proof follows from Theorem3.1.

Remark 3.4. Settingp=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, thena=b.In this situation (3.5) holds with the constantC =e(b+1)/p.Indeed, this constant is sharp. In order to show this forδ >0,we consider the function

fδ(x) =





eb+1εp |S|−(b+1)|x|Np(b+1−εδ), x∈S, eb+1εp |S|−(b+1)|x|Np(b+1+εδ), x∈E\S.

By using this function in (3.5), we find that 1≤ RHS

LHS ≤eδp →1 as δ →0

and consequently the constant is sharp. Note that the sharpness of the constant forp = q,in Proposition3.3has been proved in the more general setting than that in [1].

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Weighted Geometric Mean Inequalities Over Cones inRN

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4. The Companion Inequalities

We present the following result which is a companion of Theorem3.1:

Theorem 4.1. Let 0 < p ≤ q < ∞, ε > 0, and suppose that all other hypotheses of Theorem3.1are satisfied. Then the inequality

(4.1) Z

E

exp

ε|Sx|ε

Z

E\Sx

|Sy|−ε−1lnf(y)dy q

v(x)dx 1q

≤C Z

E

fp(x)u(x)dx 1p

holds for allf >0if and only if

D3 := sup

t>0

|tS|p1 Z

tS

v(x)

exp 1

|Sx| Z

Sx

ln 1 u(y)dy

qp dx

!1q

<∞, where

u(y) :=u(s1εσ)1

εs−N(1+1ε), v(y) :=v(s1εσ)1

εs−N(1+1ε). Moreover, the constantCsatisfiesD3 ≤C ≤e1pD3.

Proof. Note that forx∈RN

|Sx|= Z |x|

0

Z

A

tN−1dτ dt = |x|N N |A|.

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Weighted Geometric Mean Inequalities Over Cones inRN

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Now, using polar coordinates, (4.1) can be written as Z

0

Z

A

expε|A|εtN ε N

× Z

t

Z

A

|A|

N

−ε−1

s−N ε−1lnf(sσ)dσds

!q

v(tτ)tN−1dτ dt

!1q

≤C Z

0

Z

A

fp(tτ)u(tτ)tN−1dτ dt 1p

. Using the exchange of variabless=r−1/εandt=z−1/εwe obtain

Z 0

Z

A

exp

N

|A|zN Z

A

Z z 0

lnf(r1εσ)rN−1dσdr q

×v(z1ετ)z−N(1+1ε)1

εzN−1dτ dz 1q

≤C Z

0

Z

A

fp(z1ετ)u(z1ετ)z−N(1+1ε)1

εzN−1dτ dz 1p

and putf(tτ) =f(t1ετ). (4.1) can be equivalently rewritten as Z

E

exp

1

|Sx| Z

Sx

lnf(y)dy q

v(x)dx 1q

≤C Z

E

fp(x)u(x)dx 1p

. Now, the result is obtained by using Theorem3.1.

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Analogously to Corollary3.2, we can immediately obtain a special case of Theorem4.1that averages functions over balls inRN centered at origin.

Corollary 4.2. Let 0 < p ≤ q < ∞, ε > 0, and u, v be weight functions in RN. Then the inequality

(4.2) Z

RN

exp

ε|Bx|ε

Z

RN\Bx

|By|−ε−1lnf(y)dy q

v(x)dx 1q

≤C Z

RN

fp(x)u(x)dx 1p

holds for allf >0if and only if

Be:= sup

z∈RN

|Bz|1p Z

Bz

v0(x)

exp 1

|Bx| Z

Bx

ln 1 u0(y)dy

qp dx

!1q

<∞, where

u0(x) := u(t1ετ)1

εt−N(1+1ε), v0(x) :=v(t1ετ)1

εt−N(1+1ε). Moreover, the best constantC satisfiesBe ≤C ≤e1pB.e

Remark 4.1. Note that by choosingE as in Remark 3.3we see that Corollary 4.2in fact holds also whenRN is replaced byRN+ or more general cones inRN. The corresponding result to Proposition3.3reads 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 Theorem2.1. Then the inequality

(4.3) Z

E

expε|Sx|ε Z

E\Sx

|Sy|−ε−1lnf(y)dy q

|Sx|adx 1q

≤C Z

E

fp(x)|Sx|bdx 1p

holds for allf >0and some finite positive constantC if and only if a+ 1

q = b+ 1 p and the least constantCin (4.3) satisfies

p q

1q

ε1p1qe(b+1εp+1p) ≤C ≤ p

q 1q

ε1p1qeb+1εp .

Remark 4.2 (Sharp Constant). Analogously to Proposition 3.3, in the above proposition we also find that if we takep=q,thena=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, forδ >0,the function

fδ(x) =





eb+1εp |S|−(b+1)|x|Np(b+1−εδ), x∈S eb+1p |S|−(b+1)|x|Np(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 inRN (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 inRN.

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References

[1] A. ˇCIŽMEŠIJA, J. PE ˇCARI ´CAND I. PERI ´C, Mixed means and inequal- ities of Hardy and Levin-Cochran-Lee type for multidimensional balls, Proc. Amer. Math. Soc., 128(9) (2000), 2543–2552.

[2] J.A. COCHRAN AND C.S. LEE, Inequalities related to Hardy’s and Heinig’s, Math. Proc. Cambridge Phil. Soc., 96 (1984), 1–7.

[3] P. DRÁBEK, H.P. HEINIGANDA. KUFNER, Higher dimensional Hardy inequality, Int. Ser. Num. Math., 123 (1997), 3–16.

[4] G.H. HARDY, Notes on some points in the integral calculus, LXIV (1925), 150–156.

[5] H.P. HEINIG, Weighted inequalities in Fourier analysis, Nonlinear Analy- sis, Function Spaces and Applications, Vol. 4, Teubner-Texte Math., band 119, Teubner, Leipzig, (1990), 42–85.

[6] H.P. HEINIG, R. KERMAN AND M. KRBEC, Weighted exponential in- equalities, Georgian Math. J., (2001), 69–86.

[7] P. JAINANDA.P. SINGH, A characterization for the boundedness of geo- metric mean operator, Applied Math. Letters (Washington), 13(8) (2000), 63–67.

[8] P. JAIN, L.E. PERSSON ANDA. WEDESTIG, From Hardy to Carleman and general mean-type inequalities, Function Spaces and Applications, CRC Press (New York)/Narosa Publishing House (New Delhi)/Alpha Sci- ence (Pangbourne) (2000), 117–130 .

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[9] P. JAIN, L.E. PERSSON AND A. WEDESTIG, Carleman-Knopp type inequalities via Hardy inequalities, Math. Ineq. Appl., 4(3) (2001), 343–

355.

[10] A.M. JARRAH AND A.P. SINGH, A limiting case of Hardy’s inequality, Indian J. Math., 43(1) (2001), 21–36.

[11] S. KAIJSER, L.E. PERSSON AND A. ÖBERG, On Carleman’s and Knopp’s inequalities, J. Approx. Theory, to appear 2002.

[12] K. KNOPP, Über Reihen mit positiven Gliedern, J. London Math. Soc., 3 (1928), 205–211.

[13] A. KUFNER AND L.E. PERSSON, Weighted Inequalities of Hardy Type, World Scientific, New Jersey/London/Singapore/Hong Kong, 2003.

[14] E.R. LOVE, Inequalities related to those of Hardy and of Cochran and Lee, Math. Proc. Camb. Phil. Soc., 99 (1986), 395–408.

[15] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´CANDA.M. FINK, Inequalities Involv- ing Functions and their Integrals and Derivatives , Kluwer Academic Pub- lishers, 1991.

[16] M. NASSYROVA, Weighted inequalities involving Hardy-type and limit- ing geometric mean operators, PhD Thesis, Department of Mathematics, Luleå University of Technology, 2002.

[17] M. NASSYROVA, L.E. PERSSON AND V.D. STEPANOV, On weighted inequalities with geometric mean operator by the Hardy-type integral

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transform, J. Inequal. Pure Appl. Math., 3(4) (2002), Art. 48. [ONLINE:

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[18] B. OPI ´C AND P. GURKA, Weighted inequalities for geometric means, Proc. Amer. Math. Soc., 3 (1994), 771–779.

[19] V.D. STEPANOV, Weighted norm inequalities of Hardy type for a class of integral operators, J. London Math. Soc., 50(2) (1994), 105–120.

[20] L.E. PERSSON AND V.D. STEPANOV, Weighted integral inequalities with the geometric mean operator, J. Inequal. & Appl., 7(5) (2002), 727–

746 (an abbreviated version can also be found in Russian Akad. Sci. Dokl.

Math., 63 (2001), 201–202).

[21] L. PICK AND B. OPI ´C, On the geometric mean operator, J. Math. Anal.

Appl, 183(3) (1994), 652–662.

[22] G. SINNAMON, One-dimensional Hardy-type inequalities in many di- mensions, Proc. Royal Soc. Edinburgh, 128A (1998), 833–848.

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