Advances in Pure Mathematics
Vol.06 No.12(2016), Article ID:72072,9 pages
10.4236/apm.2016.612067
Inequalities for Dual Orlicz Mixed Quermassintegrals
Lijuan Liu
School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan, China
Copyright © 2016 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: October 14, 2016; Accepted: November 14, 2016; Published: November 17, 2016
ABSTRACT
In this paper, we establish the dual Orlicz-Minkowski inequality and the dual Orlicz- Brunn-Minkowski inequality for dual Orlicz mixed quermassintegrals.
Keywords:
Star Body, Orlicz Radial Sum, Dual Orlicz Mixed Volume
1. Introduction
Recently, Convex Geometry Analysis has made great achievement in Orlicz space (see [1] - [14] ). Zhu, Zhou and Xu [12] defined the Orlicz radial sum and dual Orlicz mixed volumes. Let be the set of convex and strictly decreasing functions
such that
,
and
.
Let K and L be two star bodies about the origin in and
; the Orlicz radial sum
was defined by [13]
(1.1)
The case of the Orlicz radial sum is the
harmonic radial sum, which was defined by Lutwak (see [15] ).
Let denote the right derivative of a real-valued function
. For
, there is
because
is convex and strictly decreasing. The dual Orlicz mixed volume
is defined by
(1.2)
In this paper, we will define the dual Orlicz mixed quermassintegral by
(1.3)
The main purpose of this paper is to establish the dual Orlicz-Minkowski inequality and the dual Orlicz-Brunn-Minkowski inequality for dual Orlicz mixed quermassintegrals.
Theorem 1.1 Let K and L be two star bodies about the origin in and
. If
, then
(1.4)
with equality if and only if K and L are dilates of each other.
Theorem 1.2 Let K and L be two star bodies about the origin in and
. If
, then
(1.5)
with equality if and only if K and L are dilates of each other.
This paper is organized as follows: In Section 2 we introduce above interrelated notations and their background materials. Section 3 contains the proofs of our main results.
2. Notation and Background Material
The radial function of a compact star-shaped about the origin
is defined, for
, by
(2.1)
If is positive and continuous, then K is called a star body about the origin. The set of star bodies about the origin in
is denoted by
. Obviously, for
,
(2.2)
If is independent of
, then we say star bodies K and L are dilates of
each other.
If and
are nonnegative real numbers, then the volume of
is a homogeneous polynomial of degree n in
given by
where the sum is taken over all n-tuples of positive integers not exceeding m. The coefficient
depends only on the bodies
, and is uniquely determined by the above identity, it is called the dual mixed volume of
. More explicitly, the dual mixed volume
has the following integral representation [16] :
(2.3)
where S is the Lebesgue measure on
The coefficients are nonnegative, symmetric and monotone (with respect to set inclusion). They are also multilinear with respect to the radial sum and
. Let
and
, then the dual mixed volume
is usually written as
. If L = B, then
is the dual quermassintegral
. For
, the dual mixed quermassinte-
gral denotes the dual mixed volume
. For
,
then.
The dual mixed quermassintegral has the following integral representation:
(2.4)
where S is the Lebesgue measure on.
By using the Minkowski’s integral inequality, we can obtain the dual Minkowski inequality for dual mixed quermassintegrals: If, and
, then
(2.5)
equality holds if and only if K and L are dilates of each other.
Suppose that m is a probability measure on a space X and is a m- intergrable function, where I is a possibly infinite interval. Jessen’s inequality states that if
is a convex function, then
(2.6)
If is strictly convex, equality holds if and only if
is a constant for m-almost all
(see [17] ).
3. Main Results
Let and
. For
, the dual Orlicz mixed quermassintegral
is defined by
(3.1)
For, then
. The case
of the dual Orlicz mixed quermassintegral
is the dual Orlicz mixed volume
, which was defined by Zhu, Zhou and Xu [12] .
Corollary 3.1 The dual Orlicz mixed quermassintegral is monotone with respect to set inclusion.
Proof. Let and
. By (3.1), (2.2) and the fact that
is strictly decreasing on
, we have
Lemma 3.1 [12] Let and
. If
, then
if and only if
Lemma 3.2 [12] Let and
. Then
(3.2)
uniformly for all.
Theorem 3.1 Let and
. For
, then
Proof. Suppose, and
. Note that
as
(see [12] ). By Lemma 3.2, it follows that
uniformly on.
Hence
We complete the proof of Theorem 3.1. ,
From (3.1) and Theorem 3.1, we have
(3.3)
For, since
, then
is a probabil-
ity measure on.
Proof of Theorem 1.1
By (3.1), (2.6), (2.5) and the fact that is decreasing on
, we obtain
This gives the desired inequality. Since is strictly decreasing, from the equality condition of the dual Minkowski inequality (2.5), we have that K and L are dilates of each other.
Conversely, when, by (3.1), we have
,
The following uniqueness is a direct consequence of the dual Orlicz-Minkowski inequality (1.4).
Corollary 3.2 Suppose, and
such that
. For
, if
(3.4)
or
(3.5)
then.
Proof. Suppose (3.4) holds. If we take K for M, then from (3.1), we obtain
Hence, from the dual Orlicz-Minkowski inequality (1.4), we have
with equality if and only if K and L are dilates of each other. Since is strictly decreasing on
, we have
with equality if and only if K and L are dilates of each other. If we take L for M, we similarly have. Hence,
and from the equality condition we can conclude that K and L are dilates of each other. However, since they have the same volume they must be equal.
Next, suppose (3.5) holds. If we take K for M, then from (3.1), we obtain
Then, from the dual Orlicz-Minkowski inequality (1.4), we have
with equality if and only if K and L are dilates of each other. Since is strictly decreasing on
, we have
with equality if and only if K and L are dilates of each other. If we take L for M, we similarly have. Hence,
and from the equality condition we can conclude that K and L are dilates of each other. However, since they have the same volume they must be equal.
From the dual Orlicz-Minkowski inequality, we will prove the following dual Orlicz-Brunn-Minkowski inequality which is more general than Theorem 1.2.
Theorem 3.2 Let,
and
. If
, then
with equality if and only if K and L are dilates of each other.
Proof. Let. From (2.3), Lemma 3.1 and (1.4), it follows that
By the equality condition of the dual Orlicz-Minkowski inequality (1.4), equality in (3.6) holds if and only if K and L are dilates of each other.
Indeed, we also can prove the dual Orilcz-Minkowski inequality by the dual Orilcz- Brunn-Minkowski inequality.
Proof. For, let
. Note that
as
. By the dual Orlicz-Brunn-Minkowski inequality, the following function
is non-positive. Obviously,. Thus
(3.7)
On the other hand, we have
(3.8)
Let and note that
as
. Consequently,
(3.9)
By (3.3), we have
(3.10)
From (3.8), (3.9), and (3,10), it follows that
(3.11)
Combing (3.7) and (3.11), we have
(3.12)
Therefore, the equality in (3.12) holds if and only if, this implies that K and L are dilates of each other.
Remark 3.1 The case of Theorem 1.1 and Theorem 1.2 were established by Zhu, Zhou and Xu [12] . The dual forms of Theorem 1.1 and Theorem 1.2 were established by Xiong and Zou [11] .
Cite this paper
Liu, L.J. (2016) Inequalities for Dual Orlicz Mixed Quermassintegrals. Advances in Pure Mathematics, 6, 894-902. http://dx.doi.org/10.4236/apm.2016.612067
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