### The General

*L*

_{p}

_{p}

### -Dual Mixed Brightness Integrals

### Ping Zhang, Xiaohua Zhang, and Weidong Wang

*Abstract*—Based on general *Lp*-mixed brightness integrals

of convex bodies and general *Lp*-intersection bodies of star

bodies, this paper is going to define the general*Lp*-dual mixed

brightness integrals. After studying their extremum values and
establishing Aleksandrov-Frenchel inequality, cyclic inequality
and the Brunn-Minkowski inequality for the general *Lp*-dual

mixed brightness integrals, we obtain a more general result
than the Brunn-Minkowski inequality for the general*Lp*-dual

mixed brightness integrals.

*Index Terms*—general*Lp*-mixed brightness integrals, general
*Lp*-intersection body, general *Lp*-dual mixed brightness

inte-grals.

I. INTRODUCTION

## L

ET*Kn*denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space

**R**

*n*

_{. For the set of convex bodies containing the origin}in their interiors in

**R**

*n*

_{, we write}

_{K}n*o*.*Son* denotes the set of
star bodies (about the origin) in **R***n*_{. Let} * _{S}n−*1

_{denote the}

unite sphere in**R***n*_{, and let}_{V}_{(}_{K}_{)}_{denote the}_{n}_{-dimensional}
volume of body *K*. For the standard unit ball*B* in**R***n*_{, we}
write*ωn* =*V*(*B*).

If *K* *∈ Kn*_{, then its support function,} _{h}

*K* = *h*(*K, .*) :

**R***n _{→}*

_{(}

_{−∞}_{,}_{∞}_{)}

_{, is defined by (see [2])}

*h*(*K, x*) = max*{x·y*:*y∈K}, x∈***R***n,*

where*x·y* denotes the standard inner product of*x*and*y*.
For a compact set *K* in **R***n*, which is star shaped with
respect to the origin, the radial function,*ρK*(*u*) =*ρ*(*K, u*),
of *K* is defined by (see [2])

*ρK*(*u*) = max*{λ≥*0 :*λu∈K},* *u∈Sn−*1*.* (1)
If *ρK* is positive and continuous, then *K* will be called
a star body (about the origin). Two star bodies *K* and *L*
are said to be dilates (of one another) if *ρK*(*u*)*/ρL*(*u*) is
independent of*u∈Sn−*1.

If*c >*0and*K∈ Sn*

*o*, then*ρ*(*cK,·*) =*cρ*(*K,·*)*.*

Let *GL*(*n*) denote the group of general (nonsingular)
linear transformations, if *ϕ* *∈* *GL*(*n*)*,* from (1), we easily
have

*ρ*(*ϕK, x*) =*ρ*(*K, ϕ−*1*x*)*, x∈***R***n\ {*0*},* (2)

where*ϕ−*1 _{denotes the reverse of transformation}_{ϕ.}

Manuscript received July 5, 2016; revised November 23, 2016. This work was supported by the National Natural Science Foundations of China(Grant No.11371224),and was supported by the Academic Mainstay Foundation of Hubei Province of China (Grant No.B2016030).

P. Zhang is with the Department of Mathematics, China Three Gorges U-niversity, Yichang, 443002, P. R. China e-mail: (zhangping9978@126.com). X. H. Zhang is with the Department of Mathematics, China Three Gorges University, Yichang, 443002, P. R. China e-mail: (zhangxiao-hua07@163.com).

W. D. Wang is with the Department of Mathematics, China Three Gorges University, Yichang, 443002, P. R. China e-mail: (wangwd722@163.com).

The notion of mixed brightness-integrals of convex bodies
was defined by Li(see [8]). After that, Yan and Wang
extended mixed brightness-integrals to the general mixed
brightness-integrals of convex bodies: For *K*1*,· · ·, Kn* *∈*

*Kn*

*o*,*p≥*1 and*τ∈*[*−*1*,*1], the general*Lp*-mixed brightness
integrals, *D*(*pτ*)(*K*1*,· · ·, Kn*), of *K*1*,· · ·, Kn* is defined by

(see [22])

*D*(* _{p}τ*)(

*K*1

*,· · ·, Kn*) = 1

*n*

∫

*Sn−*1

*δ _{p}*(

*τ*)(

*K*1

*, u*)

*· · ·δp*(

*τ*)(

*Kn, u*)

*dS*(

*u*)

*,*where

*δ*(

*pτ*)(

*K, u*) = 1

_{2}

*h*(Π

*τpK, u*) denotes the half general

*Lp*-brightness of

*K∈ Kn*andΠ

_{o}*τ*denotes the general

_{p}K*Lp*-projection body of

*K∈ Kno*. Further, they established some inequalities for the general

*Lp*-mixed brightness integrals(see [22]).

Recently, Wang and Li used the function *φτ* : **R** *→*

[0*,*+*∞*)which is given by

*φτ*(*t*) =*|t| −τ t,* *τ* *∈*[*−*1*,*1] (3)

to define the general*Lp*-intersection body with parameter*τ*
as follows: For*K* *∈ Sn*

*o,* 0 *< p <*1*,* and *τ* *∈* [*−*1*,*1], the
general*Lp*-intersection body,*Iτ*

*pK∈ Son,*of*K*is defined by
(see [20])

*ρ*(*I _{p}τK, u*)

*p*=

*i*(

*τ*)

∫

*K*

*φτ*(*u·x*)*−pdx,* *u∈Sn−*1*,* (4)

where

*i*(*τ*) = (1 +*τ*)

*p*_{(1}_{−}_{τ}_{)}*p*

(1 +*τ*)*p*_{+ (1}*− _{τ}*

_{)}

*p.*

In this paper, based on the general*Lp*-intersection bodies
and the general*Lp*-mixed brightness integrals, we are going
to define the general *Lp*-dual mixed brightness integrals of
star bodies as follows:

For*K*1*,· · ·, Kn* *∈ Son*, 0 *< p <* 1 and *τ* *∈* [*−*1*,*1], the
general*Lp*-dual mixed brightness integrals,*Dτ*

*p*(*K*1*,···, Kn*),

of*K*1*,· · ·, Kn* is defined by

*Dpτ*(*K*1*,· · ·, Kn*)

= 1
*n*

∫

*Sn−*1

*δ _{p}τ*(

*K*1

*, u*)

*· · ·δτp*(

*Kn, u*)

*dS*(

*u*)

*,*(5) where

*δτ*

*p*(*K, u*) =

1
2*ρ*(*I*

*τ*

*pK, u*)denotes the half general*Lp*
-dual brightness of*K∈ Sn*

*o* in direction*u∈Sn−*1*.*
If*τ* = 0*,* we write *Dτ*

*p*(*K*1*,· · ·, Kn*) =*Dp*(*K*1*,· · ·, Kn*)
and*δp*(*K, u*) = 1_{2}*ρ*(*IpK, u*)*,* then

*Dp*(*K*1*,· · ·, Kn*) =

1
*n*

∫

*Sn−*1

*δp*(*K*1*, u*)*· · ·δp*(*Kn, u*)*dS*(*u*)*,*

we call *Dp*(*K*1*,· · ·, Kn*) the *Lp*-dual mixed brightness

integrals of*K*1*,· · ·, Kn* *∈ Son.*

**IAENG International Journal of Applied Mathematics, 47:2, IJAM_47_2_03**

Let*K*1=*···*=*Kn−i* =*K*and*Kn−i*+1=*···*=*Kn*=*L*
(*i*= 0*,*1*,· · ·, n*), we write

*Dτ _{p,i}*(

*K, L*) =

*D*(

_{p}τ*K,· · ·K, L· ··, L*)

*.*

If *i*is any real, *K, L∈ Sn*

*o,*0*< p <*1*, τ* *∈*[*−*1*,*1], then
the general*Lp*-dual mixed brightness integrals, *Dτ*

*p,i*(*K, L*),
of *K* and*L*is defined by

*Dτ _{p,i}*(

*K, L*) = 1

*n*

∫

*Sn−*1

*δ _{p}τ*(

*K, u*)

*n−iδ*(

_{p}τ*L, u*)

*idS*(

*u*)

*.*(6)

Let*τ* = 0in (6), we write*Dτ*

*p,i*(*K, L*) =*Dp,i*(*K, L*)*.*
Let*i*= 0in (6), we write

*D _{p,}τ*

_{0}(

*K, K*) =

*Dτ*(

_{p}*K*) = 1

*n*

∫

*Sn−*1

*δτ _{p}*(

*K, u*)

*ndS*(

*u*)

*,*(7)

which is called the general *Lp*-dual brightness integrals of
*K*.

Let*τ* = 0in (7), we write*Dτ*

*p*(*K*) =*Dp*(*K*); for*τ*=*±*1,
we write *Dτ*

*p*(*K*) =*D±p*(*K*).

In this paper, we will establish the following inequalities
for the general *Lp*-dual mixed brightness integrals.

Initinally, we give the extremal values of the general *Lp*
-dual mixed brightness integrals.

Theorem 1.1. *IfK∈ Sn*

*o,*0*< p <*1*,τ* *∈*[*−*1*,*1]*,* *then*

*Dp*(*K*)*≤Dτp*(*K*)*≤Dp±*(*K*)*,* (8)

*if* *K* *is not origin-symmetric, there is equality in the left*
*inequality if and only if* *τ* = 0 *and equality in the right*
*inequality if and only if* *τ*=*±*1*.*

Furthermore, we establish the following
Fenchel-Aleksandrov type inequality for the general *Lp*-dual mixed
brightness integrals.

Theorem 1.2. *If* *K*1*,· · ·, Kn∈ Son,*0*< p <*1*,τ∈*[*−*1*,*1]

*and* 1*< m≤n,* *then*

*D _{p}τ*(

*K*1

*,· · ·, Kn*)

*m*

*≤*

*m*

∏

*i*=1

*D _{p}τ*(

*K*1

*,· · ·, Kn*+1

_{−}m, Kn_{−}i*,· · ·, Kn*+1) (9)

_{−}i*equality holds if and only ifIτ*

*pKn−m*+1*,···, IpτKnare dilates*

*of each other.*

Let *τ* = 0 in Theorem 1.2, we obtain the following
inequality.

Corollary 1.1. *If* *K*1*,· · ·, Kn* *∈ Son,* 0 *< p <*1 *and* 1 *<*
*m≤n,* *then*

*Dp*(*K*1*,· · ·, Kn*)*m*

*≤*

*m*

∏

*i*=1

*Dp*(*K*1*,· · ·, Kn−m, Kn−i*+1*,· · ·, Kn−i*+1)*,*

*equality holds if and only ifIpKn _{−}m*+1

*,···, IpKnare dilates*

*of each other.*

Additionally, we establish the following cyclic inequality
for the general *Lp*-dual mixed brightness integrals.

Theorem 1.3. *Iet* *K, L∈ Sn*

*o,* 0 *< p <* 1*,* *τ* *∈*[*−*1*,*1]*, if*
*k _{−}i*

*k _{−}j*

*>*1

*,then*

*Dτp,i*(*K, L*)*k−jDτp,k*(*K, L*)*j−i≥Dp,jτ* (*K, L*)*k−i,* (10)

*equality holds if and only if* *Iτ*

*pK* *and* *IpτL* *are dilates. If*
0*<* *k−i*

*k−j* *<*1*, the inequality (10) is reversed.*

Let *τ* = 0 in Theorem 1.3, we obtain the following
inequality.

Corollary 1.2. *IetK, L∈ Sn*

*o,*0*< p <*1*, if*
*k−i*

*k _{−}j*

*>*1

*,*

*then*

*Dp,i*(*K, L*)*k−jDp,k*(*K, L*)*j−i≥Dp,j*(*K, L*)*k−i,* (11)

*equality holds if and only if* *IpK* *and* *IpL* *are dilates. If*

0*<* _{k}k−_{−}_{j}i*<*1*, the inequality (11) is reversed.*

Finally, we obtain the Brunn-Minkowski type inequality
for the general*Lp*-dual mixed brightness integrals as follows:
Theorem 1.4. *IetK, K′, L∈ Son,*0*< p <*1*, τ* *∈*[*−*1*,*1]*,*

*ifi≤n−p, then*

*Dp,iτ* (*K*+˜*n−pK*

*′*

*, L*)*np−i*

*≤D _{p,i}τ* (

*K, L*)

*np−i*+

*Dτ*

*p,i*(*K*

*′*

*, L*)*np−i,* (12)

*equality holds if and only if* *Iτ*

*pK* *and* *IpτK*

*′*

*are dilates. If*

*i≥n−p, the inequality (12) is reversed.*

Actually, we prove a more general result than Theorem 1.4 in Section III.

Our work belongs to a new and rapidly evolving
asymmet-ric*Lp*dual Brunn-Minkowski theory that has its own origin
in the work of Ludwig, Haberl and Schuster (see [3], [4],
[5], [6], [10], [11]). For the further researches of asymmetric
*Lp*Brunn-Minkowski theory, we can refer to papers [1], [7],
[14], [15], [16], [17], [18], [19], [20], [21].

II. PRELIMINARIES

*A. Dual mixed volumes*

In 1975, Lutwak (see [9]) gave the notion of dual mixed
volumes as follows: For *K*1*, K*2*,· · ·, Kn* *∈ Son*, the dual
mixed volume, *V*e(*K*1*, K*2*,· · ·, Kn*), of *K*1*, K*2*,· · ·, Kn* is
defined by

e

*V*(*K*1*,· · ·, Kn*)
= 1

*n*

∫

*Sn−*1

*ρ*(*K*1*, u*)*· · ·ρ*(*Kn, u*)*dS*(*u*)*.* (13)

If *K*1 = *· · ·* = *Kn−i* = *K*, *Kn−i*+1 = *· · ·* =*Kn* = *L*
in (13), we write *V*e*i*(*K, L*) = *V*e(*K, n−i*;*L, i*), where *K*
appears*n−i*times and*L*appears*i*times. Let*i*be any real,
we have

e

*Vi*(*K, L*) = 1
*n*

∫

*Sn−*1

*ρ*(*K, u*)*n−iρ*(*L, u*)*idS*(*u*)*.* (14)

Let*i*= 0 in (14), then

e

*V*0(*K, L*) =*V*(*K*) =

1
*n*

∫

*Sn−*1

*ρ*(*K, u*)*ndS*(*u*)*.* (15)

*B. SomeLp-combinations*

For*K, L* *∈ Son, p >* 0 and *λ, µ* *≥*0(not both zero), the
*Lp*-radial linear combination,*λ◦K*+˜*pµ◦L∈ Son,*of*K* and
*L*is defined by(see [12])

*ρ*(*λ◦K*+˜*pµ◦L,·*)*p* =*λρ*(*K,·*)*p*+*µρ*(*L,·*)*p.* (16)

For*ϕ∈GL*(*n*)*, K, L∈ Son, p >*0and*λ, µ≥*0(not both
zero), from (2) and (16), we have

*ϕ*(*λ◦K*+˜*pµ◦L,·*) =*λ◦ϕK*+˜*pµ◦ϕL.* (17)

For*K, L∈ Sn*

*o,*0*< p <*1,*τ* *∈*[*−*1*,*1], from (4),(16) and
a transformation to polar coordinate, we obtain

*ρ*(*Ipτ*(*K*+˜*n−pL*)*,·*)*p*=*ρ*(*IpτK,·*)
*p*

+*ρ*(*IpτL,·*)
*p*

*,* (18)

**IAENG International Journal of Applied Mathematics, 47:2, IJAM_47_2_03**

i.e.,

*I _{p}τ*(

*K*+˜

*n−pL*) =

*IpτK*+˜

*pIpτL.*(19)

From (17) and (19), we get

*I _{p}τ*(

*ϕ*(

*K*+˜

*n*)) =

_{−}pL*IpτϕK*+˜

*pI*

*τ*

*pϕL.* (20)

III. PROOFS OFTHEOREMS

In this section, firstly we shall prove Theorems 1.1-1.3 ,
then we prove a more general result than Theorem 1.4, i.e.,
a quotient form of the Brunn-Minkowski type inequality for
the general *Lp*-dual mixed brightness integrals.

In order to prove Theorem 1.1, we need the following inequality (see [20]).

Lemma 3.1. *IfK∈ Sn*

*o,* 0*< p <*1*, τ* *∈*[*−*1*,*1]*, then*

*V*(*IpK*)*≤V*(*IpτK*)*≤V*(*Ip±K*)*.* (21)

*If* *K* *is not origin-symmetric, there is equality in the left*
*inequality if and only if* *τ* = 0 *and equality in the right*
*inequality if and only if* *τ*=*±*1*.*

*Proof of Theorem 1.1.* If*K, L∈ Sn*

*o*, from (6), then

*D _{p,i}τ* (

*K, L*) = 1

*n*

∫

*Sn−*1

*δ _{p}τ*(

*K, u*)

*n−iδ*(

_{p}τ*L, u*)

*idS*(

*u*)

= 1
2*n*

1
*n*

∫

*Sn−*1

*ρ*(*I _{p}τK, u*)

*n−iρ*(

*I*)

_{p}τL, u*idS*(

*u*)

= 1
2*nV*e*i*(*I*

*τ*
*pK, I*

*τ*

*pL*)*.* (22)

Let*i*= 0in (22), and from (15), we have

*Dτ _{p,}*

_{0}(

*K, L*) =

*Dτ*(

_{p}*K*) = 1 2

*nV*(

*I*

*τ*
*pK*)*.*

According to (21), we get

1

2*nV*(*IpK*)*≤*
1
2*nV*(*I*

*τ*
*pK*)*≤*

1
2*nV*(*I*

*±*

*pK*)*.*

i.e.,

*Dp*(*K*)*≤Dτ _{p}*(

*K*)

*≤D*(

_{p}±*K*)

*.*(23)

According to (21), we know that if *K* is not
origin-symmetric, there is equality in the left inequality if and only
if *τ* = 0 and equality in the right inequality if and only if
*τ* =*±*1 in (23). And (23) is just the inequality (8).

The proof of Theorem 1.2 requires the following extension
of the H*o*¨lder inequality (see [8] [13]).

Lemma 3.2. *Iff*0*, f*1*,· · ·, fm* *are (strictly) positive *

*contin-uous functions defined onSn−*1 *andλ*1*,· · ·, λmare positive*

*constants the sum of whose reciprocals is unity, then*

∫

*Sn−*1

*f*0(*u*)*f*1(*u*)*· · ·fm*(*u*)*dS*(*u*)

*≤*

*m*

∏

*i*=1

( ∫

*Sn−*1

*f*0(*u*)*fiλi*(*u*)*dS*(*u*)

)1

*λi*

*,* (24)

*with equality if and only if there exist positive constantsα*1*,··*

*·, αm* *such that* *α*1*f*1*λ*1(*u*) = *· · ·* =*αmfmλm*(*u*) *for all* *u∈*
*Sn−*1_{.}

*Proof of Theorem 1.2.* If *K*1*,· · ·, Kn* *∈ Son*,0*< p <* 1,
*τ* *∈*[*−*1*,*1],1*< m≤n,*and let*λi*=*m*(1*≤i≤m*)*,*and

*f*0(*u*) =*δpτ*(*K*1*, u*)*· · ·δpτ*(*Kn−m, u*)*,* (*f*0= 1*if m*=*n*)*,*

*fi*(*u*) =*δτp*(*Kn−i*+1*, u*)*,* (1*≤i≤m*)*.*

According to (24), we have

∫

*Sn−*1

*δτ _{p}*(

*K*1

*, u*)

*· · ·δpτ*(

*Kn−m, u*)

*· · ·δ*

*τ*

*p*(*Kn, u*)*dS*(*u*)

*≤*

*m*

∏

*i*=1

( ∫

*Sn−*1

*δτ _{p}*(

*K*1

*, u*)

*× · · ·*

*×δ _{p}τ*(

*Kn−m, u*)

*δpτ*(

*Kn−i*+1

*, u*)

*mdS*(

*u*)

)1

*m*

*.*

So

( ∫

*Sn−*1

*δτ _{p}*(

*K*1

*, u*)

*· · ·δτp*(

*Kn−m, u*)

*· · ·δ*

*τ*

*p*(*Kn, u*)*dS*(*u*)

)*m*

*≤*

*m*

∏

*i*=1

∫

*Sn−*1

*δ _{p}τ*(

*K*1

*, u*)

*···δpτ*(

*Kn−m, u*)

*δτp*(

*Kn−i*+1

*, u*)

*mdS*(

*u*)

*.*

i.e.,

*Dτ _{p}*(

*K*1

*,· · ·, Kn*)

*m*

*≤*

*m*

∏

*i*=1

*D _{p}τ*(

*K*1

*,· · ·, Kn−m, Kn−i*+1

*,· · ·, Kn−i*+1)

*.*(25)

The equality condition in (25) can be got from the equality
condition in inequality (24) if and only if there exist positive
constants*α*1*,· · ·, αm*such that

*α*1*δpτ*(*Kn−m*+1*, u*)*m*=*α*2*δpτ*(*Kn−m*+2*, u*)*m*

=*· · ·*=*αmδ _{p}τ*(

*Kn, u*)

*m*

for all*u* *∈* *Sn−*1_{.}_{So equality holds in (25) if and only if}

*Iτ*

*pKn−m*+1*,· · ·, IpτKn*are dilates of each other. And (25) is
just the inequality (9).

*Proof of Theorem 1.3.* Let *K, L* *∈ Sn*

*o*, 0 *< p <* 1,
*τ* *∈* [*−*1*,*1], if _{k}k_{−}−_{j}i*>*1*,* according to (6), and the H¨older
inequality, we have

*D _{p,i}τ* (

*K, L*)

*kk−−ji*

_{D}τ*p,k*(*K, L*)

*j−i*
*k−i*

=

(

1
*n*

∫

*Sn−*1

*δ _{p}τ*(

*K, u*)

*n−iδ*(

_{p}τ*L, u*)

*idS*(

*u*)

)*k−j*
*k−i*

*×*

(

1
*n*

∫

*Sn−*1

*δ _{p}τ*(

*K, u*)

*n−kδτ*(

_{p}*L, u*)

*kdS*(

*u*)

)*j−i*
*k−i*

=

(

1
*n*

∫

*Sn−*1

[*δτp*(*K, u*)

(*n _{−}i*)

*k*

_{k}−_{−}j_{i}*δτp*(*L, u*)
*ik _{k}−_{−}j_{i}*

]*kk−−ij _{dS}*

_{(}

_{u}_{)})

*k−j*

*k−i*

*×*

(

1
*n*

∫

*Sn−*1

[*δτ _{p}*(

*K, u*)(

*n−k*)

*jk−−ii*

_{δ}τ*p*(*L, u*)

*k _{k}j−_{−}i_{i}*

_{]}

_{j}k_{−}−_{i}i_{dS}_{(}

_{u}_{)}

)*j−i*
*k−i*

*≥* 1

*n*

∫

*Sn−*1

*δ _{p}τ*(

*K, u*)(

*n−i*)

*kk−−ji*

_{δ}τ*p*(*K, u*)

(*n−k*)_{k}j−_{−}i_{i}

*δ _{p}τ*(

*L, u*)

*kjk−−ii*

_{δ}τ*p*(*L, u*)
*ik−j*

*k−i _{dS}*

_{(}

_{u}_{)}

= 1
*n*

∫

*Sn−*1

*δ _{p}τ*(

*K, u*)

*n−jδτ*(

_{p}*L, u*)

*jdS*(

*u*) =

*D*(

_{p,j}τ*K, L*)

*.*

i.e.,

*Dτ _{p,i}*(

*K, L*)

*kk−−jiDτ*

*p,k*(*K, L*)

*j−i*
*k−i* *≥Dτ*

*p,j*(*K, L*)*.* (26)

The equality condition in (26) can be got from the equality
condition in the H¨older inequalitiy if and only if *IpτK* and

**IAENG International Journal of Applied Mathematics, 47:2, IJAM_47_2_03**

*Iτ*

*pL* are dilates. Similarly, if0 *<*
*k _{−}i*

*k−j* *<*1, we can obtain
the reverse form of (26). And (26) is just the inequality (10).
Now, we give a more general result than Theorem 1.4 as
follows:

Theorem 3.1. *ForK, L, K′* *∈ S _{o}n,ϕ∈GL*(

*n*)

*,*0

*< p <*1

*,*

*τ* *∈*[*−*1*,*1]*,* *if*0*≤n−j≤p≤n−i,* *then*

(

*D _{p,i}τ* (

*ϕ*(

*K*+˜

*n−pK*

*′*

)*, L*)

*Dτ*

*p,j*(*ϕ*(*K*+˜*n _{−}pK′*)

*, L*)

) *p*

*j−i*

*≤*

(_{D}τ

*p,i*(*ϕK, L*)
*Dτ*

*p,j*(*ϕK, L*)

) *p*

*j−i*

+

(_{D}τ*p,i*(*ϕK*

*′*

*, L*)

*Dτ*
*p,j*(*ϕK*

*′ _{, L}*

_{)}

) *p*
*j−i*

*,* (27)

*with equality holds in (27) if and only ifIτ*

*pϕK* *andIpτϕK*

*′*

*are dilates. Ifn−j* *≤*0*< n−i≤p,the inequality (27) is*
*reversed.*

The proof of Theorem 3.1 requires the following Dresher’s inequality(see [23]).

Lemma 3.3. *Let functionsf*1*, f*2*, g*1*, g*2*≥*0*,Eis a bounded*

*measurable subset in***R***n _{. If}*

_{p}_{≥}_{1}

_{≥}_{r}_{≥}_{0}

_{, then}(∫

*E*(*f*1+*f*2)
*p _{dx}*

∫

*E*(*g*1+*g*2)
*r _{dx}*

) 1

*p−r*

*≤*
(∫
*Ef*
*p*
1*dx*
∫
*Eg*
*r*
1*dx*
) 1

*p−r*

+
(∫
*Ef*
*p*
2*dx*
∫
*Eg*
*r*
2*dx*
) 1

*p−r*

*,* (28)

*equality holds if and only if* *f*1
*f*2 =

*g*1

*g*2 *. If*1*≥p >*0*> r, the*
*inequality (28) is reversed.*

*Proof of Theorem 3.1.* For*K, K′, L∈ Sn*

*o*, *ϕ∈GL*(*n*),
0*< p <*1,*τ∈*[*−*1*,*1], if0*≤n−j≤p≤n−i*, according
to (16),(20) and (28), we have

(_{D}τ

*p,i*(*ϕ*(*K*+˜*n−pK*

*′*

)*, L*)

*Dτ*

*p,j*(*ϕ*(*K*+˜*n−pK′*)*, L*)

) *p*
*j−i*

=

( 1

*n*

∫

*Sn−*1*δ*
*τ*

*p*(*ϕ*(*K*+˜*n−pK*

*′*

)*, u*)*n−iδpτ*(*L, u*)*idS*(*u*)

1

*n*

∫

*Sn−*1*δτp*(*ϕ*(*K*+˜*n−pK′*)*, u*)*n−jδpτ*(*L, u*)*jdS*(*u*)

) *p*

*j−i*

=

( 1

*n*

∫

*Sn−*1*ρ*(*I*
*τ*

*p*(*ϕ*(*K*+˜*n _{−}pK*

*′*

))*, u*)*n−i _{ρ}*

_{(}

_{I}τ*pL, u*)*idS*(*u*)

1

*n*

∫

*Sn−*1*ρ*(*Ipτ*(*ϕ*(*K*+˜*n−pK′*))*, u*)*n−jρ*(*IpτL, u*)*jdS*(*u*)

) *p*

*j−i*

*≤*

( 1

*n*

∫

*Sn−*1*ρ*(*I*
*τ*

*pϕK, u*)*n−iρ*(*IpτL, u*)*idS*(*u*)

1

*n*

∫

*Sn−*1*ρ*(*IpτϕK, u*)*n−jρ*(*IpτL, u*)*jdS*(*u*)

) *p*
*j−i*

+

(1

*n*

∫

*Sn−*1*ρ*(*I*
*τ*
*pϕK*

*′*

*, u*)*n−i _{ρ}*

_{(}

_{I}τ*pL, u*)*idS*(*u*)

1

*n*

∫

*Sn−*1*ρ*(*IpτϕK*

*′ _{, u}*

_{)}

*n−j*

_{ρ}_{(}

_{I}τ*pL, u*)*jdS*(*u*)

) *p*

*j−i*

=

(_{D}τ

*p,i*(*ϕK, L*)
*Dτ*

*p,j*(*ϕK, L*)

) *p*

*j−i*

+

(_{D}τ*p,i*(*ϕK*

*′*

*, L*)

*Dτ*
*p,j*(*ϕK*

*′ _{, L}*

_{)}

) *p*
*j−i*

*.*

The equality condition in (27) can be got from the equality
condition in (28) if and only if*Iτ*

*pϕK*and*IpτϕK*

*′*

are dilates.
If*n−j≤*0*< n−i≤p*, similarly, we can prove that the
reverse of the inequality (27) is true.

If*ϕ*is identic, then we get the following inequality.
Theorem 3.2. *For* *K, L, K′* *∈ S _{o}n,* 0

*< p <*1

*, if*0

*≤*

*n−j*

*≤p≤n−i,*

*then*

(_{D}τ

*p,i*(*K*+˜*n−pK*

*′*

*, L*)

*Dτ*

*p,j*(*K*+˜*n _{−}pK′, L*)

) *p*

*j−i*

*≤*

(_{D}τ*p,i*(*K, L*)
*Dτ*

*p,j*(*K, L*)

) *p*

*j−i*

+

(_{D}τ*p,i*(*K*

*′*

*, L*)

*Dτ*
*p,j*(*K*

*′ _{, L}*

_{)}

) *p*
*j−i*

*,* (29)

*with equality holds in (29) if and only ifIτ*

*pK* *andIpτK*

*′*
*are*
*dilates. If* *n−j* *≤* 0 *< n−i* *≤* *p,* *the inequality (29) is*
*reversed.*

*Proof of Theorem 1.4.* Let *j* =*n* in the inequality (29)
and notice that*Dτ*

*p,n*(*M, L*) =*Dτp*(*L*)by (6), for*i≤n−p*
and any*L∈ Sn*

*o*, we get

*Dτ _{p,i}*(

*K*+˜

*n−pK*

*′*

*, L*)*np−i*

*≤D _{p,i}τ* (

*K, L*)

*np−i*

_{+}

_{D}τ*p,i*(*K*

*′*

*, L*)*np−i _{,}*

_{(30)}

which is just the inequality (12). From the equality condition
of (29), we see that equality holds in (30) if and only if*IpτK*
and*Iτ*

*pK*

*′*

are dilates.

Similarly, let*j* =*n*in the reverse of the inequality (29),
and for *i* *≥* *n−p*, we can obtain that the reverse of the
inequality (30) is true.

ACKNOWLEDGMENT

The author is indebted to the editors and the anonymous referees for many valuable suggestions and comments.

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