Open Access

On dual mixed quermassintegral quotient functions

Journal of Inequalities and Applications20152015:340

https://doi.org/10.1186/s13660-015-0871-5

Received: 25 March 2015

Accepted: 16 October 2015

Published: 26 October 2015

Abstract

We introduce the notion of dual mixed quermassintegral quotient functions and establish the Brunn-Minkowski inequalities for them in this paper.

Keywords

dual quermassintegraldual mixed quermassintegral quotient functionBrunn-Minkowski inequality

MSC

52A2052A40

1 Introduction and main results

The setting for this paper is Euclidean n-space \(\mathbb{R}^{n}\). Let \(\mathcal{S}^{n}_{o}\) denote the set of star bodies containing the origin in their interiors in \(\mathbb{R}^{n}\). Let \(S^{n-1}\) denote the unit sphere in \(\mathbb{R}^{n}\), and let \(V(K)\) denote the n-dimensional volume of body K. For the standard unit ball B in \(\mathbb{R}^{n}\), we use \(\omega_{n}=V(B)\) to denote its volume.

In 1975, Lutwak (see [1]) gave the notion of dual mixed volumes as follows: For \(K_{1}, K_{2}, \ldots, K_{n}\in\mathcal{S}^{n}_{o}\), the dual mixed volume, \(\widetilde{V}(K_{1}, K_{2}, \ldots, K_{n})\), of \(K_{1}, K_{2}, \ldots, K_{n}\) is defined by
$$ \widetilde{V}(K_{1},\ldots,K_{n})=\frac{1}{n}\int _{S^{n-1}}\rho (K_{1},u)\cdots\rho(K_{n},u) \,dS(u). $$
(1.1)
Taking \(K_{1}=\cdots=K_{n-i}=K\), \(K_{n-i+1}=\cdots=K_{n}=L\) in (1.1), we write \(\widetilde{V}_{i}(K, L)=\widetilde{V}(K, n-i; L, i )\), where K appears \(n-i\) times and L appears i times. Then
$$ \widetilde{V}_{i}(K, L)=\frac{1}{n}\int_{S^{n-1}} \rho(K,u)^{n-i}\rho (L,u)^{i}\, dS(u). $$
(1.2)
Let \(L=B\) in (1.2) and notice \(\rho(B, \cdot)=1\), and allow i is any real, then the dual quermassintegrals can be defined as follows: For \(K\in\mathcal{S}^{n}_{o}\) and i is any real, the dual quermassintegrals, \(\widetilde{W}_{i}(K)\), of K are given by (see [1])
$$ \widetilde{W}_{i}(K)=\frac{1}{n}\int_{S^{n-1}} \rho(K,u)^{n-i}\,dS(u). $$
(1.3)
Associated with dual quermassintegrals, Zhao (see [2]) defined the dual quermassintegral quotient functions of a star body K by
$$ Q_{\widetilde{W}_{i,j}(K)}=\frac{\widetilde{W}_{i}(K)}{\widetilde {W}_{j}(K)}\quad (i,j\in\mathbb{R}). $$
(1.4)

Further, in [2] the Brunn-Minkowski type inequalities for the dual quermassintegral quotient functions of star bodies were established as follows.

Theorem A

If \(K, L\in\mathcal{S}^{n}_{o}\) and reals i, j satisfy \(i\leq n-1\leq j\leq n\), then
$$Q^{\frac{1}{j-i}}_{\widetilde{W}_{i,j}(K\, \,\tilde{+}\,\, L)} \leq Q^{\frac {1}{j-i}}_{\widetilde{W}_{i,j}(K)}+Q^{\frac{1}{j-i}}_{\widetilde{W}_{i,j}(L)}. $$
Here \(\tilde{+}\) is the radial Minkowski sum.

Theorem B

If \(K, L\in\mathcal{S}^{n}_{o}\) and reals i, j satisfy \(i\leq1\leq j\leq n\), then
$$Q^{\frac{n-1}{j-i}}_{\widetilde{W}_{i,j}(K\,\breve{+}\,L)} \leq Q^{\frac {n-1}{j-i}}_{\widetilde{W}_{i,j}(K)}+Q^{\frac{n-1}{j-i}}_{\widetilde {W}_{i,j}(L)}. $$
Here \(\breve{+}\) is the radial Blaschke sum.

Theorem C

If \(K, L\in\mathcal{S}^{n}_{o}\) and reals i, j satisfy \(i\leq-1\leq j\leq n\), then
$$\frac{Q^{\frac{n+1}{j-i}}_{\widetilde{W}_{i,j}(K\,\hat{+}\,L)}}{V(K\hat {+}L)} \leq\frac{Q^{\frac{n+1}{j-i}}_{\widetilde {W}_{i,j}(K)}}{V(K)}+\frac{Q^{\frac{n+1}{j-i}}_{\widetilde{W}_{i,j}(L)}}{V(L)}. $$
Here \(\hat{+}\) is the harmonic Blaschke sum.

Motivated by the work of Zhao, we give the following definition of dual mixed quermassintegral quotient function.

Let \(K_{1}=\cdots=K_{n-i-1}=K\), \(K_{n-i}=\cdots=K_{n-1}=B\), \(K_{n}=L\) in (1.1), then we write \(\widetilde{W}_{i}(K, L)=\widetilde{V}(K, n-i-1; B, i; L, 1)\), where K appears \(n-i-1\) times, B appears i times and L appears 1 time. Here, we allow i to be any real and define as follows: For \(K, L\in\mathcal{S}^{n}_{o}\) and i any real, the dual mixed quermassintegrals, \(\widetilde{W}_{i}(K, L)\), of K and L are given by
$$ \widetilde{W}_{i}(K, L)=\frac{1}{n}\int_{S^{n-1}} \rho(K,u)^{n-i-1}\rho (L,u)\,dS(u). $$
(1.5)

Obviously, from (1.3) and (1.5), we have \(\widetilde{W}_{i}(K, K)=\widetilde{W}_{i}(K)\). According to (1.5), we define the following.

Definition 1.1

Let \(K, L\in\mathcal{S}^{n}_{o}\) and \(i,j\in\mathbb{R}\), the dual mixed quermassintegral quotient function, \(Q_{\widetilde{W}_{i,j}(K,L)}\), of K and L can be defined by
$$ Q_{\widetilde{W}_{i,j}(K,L)}=\frac{\widetilde{W}_{i}(K,L)}{\widetilde {W}_{j}(K, L)}. $$
(1.6)

Obviously, if \(L=K\), then (1.6) is just (1.4).

The aim of this paper is to establish the following Brunn-Minkowski type inequalities for dual mixed quermassintegral quotient functions of star bodies.

Theorem 1.1

For \(K, K', L\in\mathcal {S}^{n}_{o}\), if \(i\leq n-2\leq j< n-1\), then
$$ Q^{\frac{1}{j-i}}_{\widetilde{W}_{i,j}(K\,\tilde{+}\,K',L)} \leq Q^{\frac{1}{j-i}}_{\widetilde{W}_{i,j}(K,L)}+Q^{\frac {1}{j-i}}_{\widetilde{W}_{i,j}(K',L)}; $$
(1.7)
if \(n-2\leq i< n-1<j\), then
$$ Q^{\frac{1}{j-i}}_{\widetilde{W}_{i,j}(K\,\tilde{+}\,K',L)} \geq Q^{\frac{1}{j-i}}_{\widetilde{W}_{i,j}(K,L)}+Q^{\frac {1}{j-i}}_{\widetilde{W}_{i,j}(K',L)}. $$
(1.8)
In each case, equality holds if and only if K and \(K'\) are dilates. Here \(\tilde{+}\) is the radial Minkowski sum.

Theorem 1.2

For \(K, K', L\in\mathcal {S}^{n}_{o}\), if \(i\leq0\leq j< n-1\), then
$$ Q^{\frac{n-1}{j-i}}_{\widetilde{W}_{i,j}(K\,\breve{+}\,K',L)} \leq Q^{\frac{n-1}{j-i}}_{\widetilde{W}_{i,j}(K,L)}+Q^{\frac {n-1}{j-i}}_{\widetilde{W}_{i,j}(K',L)}; $$
(1.9)
if \(0\leq i< n-1< j\), then
$$ Q^{\frac{n-1}{j-i}}_{\widetilde{W}_{i,j}(K\,\breve{+}\,K',L)} \geq Q^{\frac{n-1}{j-i}}_{\widetilde{W}_{i,j}(K,L)}+Q^{\frac {n-1}{j-i}}_{\widetilde{W}_{i,j}(K',L)}. $$
(1.10)
In each case, equality holds if and only if K and \(K'\) are dilates. Here \(\breve{+}\) is the radial Blaschke sum.

Theorem 1.3

For \(K, K', L\in\mathcal {S}^{n}_{o}\), if \(i\leq-2\leq j< n-1\), then
$$ \frac{Q^{\frac{n+1}{j-i}}_{\widetilde{W}_{i,j}(K\,\hat{+}\,K',L)} }{V(K\,\hat{+}\,K')}\leq\frac{Q^{\frac{n+1}{j-i}}_{\widetilde {W}_{i,j}(K,L)}}{V(K)} +\frac{Q^{\frac{n+1}{j-i}}_{\widetilde{W}_{i,j}(K',L)}}{V(K')}; $$
(1.11)
if \(-2\leq i< n-1< j\), then
$$ \frac{Q^{\frac{n+1}{j-i}}_{\widetilde{W}_{i,j}(K\,\hat{+}\,K',L)} }{V(K\,\hat{+}\,K')}\geq\frac{Q^{\frac{n+1}{j-i}}_{\widetilde {W}_{i,j}(K,L)}}{V(K)} +\frac{Q^{\frac{n+1}{j-i}}_{\widetilde{W}_{i,j}(K',L)}}{V(K')}. $$
(1.12)
In each case, equality holds if and only if K and \(K'\) are dilates. Here \(\hat{+}\) is the harmonic Blaschke sum.

2 Preliminaries

For a compact set K in \(\mathbb{R}^{n}\) which is star shaped with respect to the origin, we define the radial function \(\rho_{K}(u)=\rho(K,u)\) of K by
$$\rho(K,u)=\max\{\lambda\geq0: \lambda u\in K\}, \quad u\in S^{n-1}. $$
If \(\rho_{K}\) is positive and continuous, 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 \(\rho_{K}(u)/\rho_{L}(u)\) is independent of \(u\in S^{n-1}\).
For \(K_{1}, K_{2}\in\mathcal{S}^{n}_{o}\), and \(\lambda_{1},\lambda_{2}\geq 0\) (not both 0), the radial function of the radial Minkowski linear combination \(\lambda_{1}K_{1}\,\tilde{+}\,\lambda_{2}K_{2}\) is given by Zhang (see [3]):
$$ \rho(\lambda_{1}K_{1}\,\tilde{+}\,\lambda_{2}K_{2},u)= \lambda_{1}\rho (K_{1},u)+\lambda_{2} \rho(K_{2},u). $$
(2.1)
For \(K_{1}, K_{2}\in\mathcal{S}^{n}_{o}\), and \(\lambda_{1},\lambda_{2}\geq 0\) (not both 0), the radial Blaschke linear combination \(\lambda _{1}\cdot K_{1} \,\check{+}\,\lambda_{2}\cdot K_{2}\) is a star body whose radial function is given by Lutwak (see [4]):
$$ \rho(\lambda_{1}\cdot K_{1}\,\check{+}\, \lambda_{2}\cdot K_{2},u)^{n-1}= \lambda_{1}\rho(K_{1},u)^{n-1}+\lambda_{2} \rho (K_{2},u)^{n-1}. $$
(2.2)
For \(K_{1}, K_{2}\in\mathcal{S}^{n}_{o}\), and \(\lambda_{1},\lambda_{2}\geq 0\) (not both 0), the harmonic Blaschke linear combination \(\lambda _{1} \circ K_{1} \,\hat{+}\, \lambda_{2}\circ K_{2}\) is a star body whose radial function is given by Lutwak (see [5]):
$$ \frac{\rho(\lambda_{1}\circ K_{1} \,\hat{+}\, \lambda_{2}\circ K_{2},u)^{n+1}}{V(\lambda_{1}\circ K_{1} \,\hat{+}\, \lambda_{2}\circ K_{2})}=\lambda_{1}\frac{\rho(K_{1},u)^{n+1}}{V( K_{1})}+\lambda _{2}\frac{\rho(K_{2},u)^{n+1}}{V( K_{2})}. $$
(2.3)

3 Proofs of theorems

According to a generalization of the Dresher inequality (see [6]), we get the reverse Dresher inequality.

Lemma 3.1

(Dresher’s inequality)

Let functions \(f_{1}, f_{2}, g_{1}, g_{2} \geq0\), E is a bounded measurable subset in \(\mathbb{R}^{n}\). If \(p \geq1 \geq r \geq0\), then
$$ \biggl(\frac{\int_{E}(f_{1}+f_{2})^{p}\,dx}{\int_{E}(g_{1}+g_{2})^{r}\,dx} \biggr)^{\frac{1}{p-r}} \leq \biggl(\frac{\int_{E}f_{1}^{p}\,dx}{\int_{E}g_{1}^{r}\,dx} \biggr)^{\frac{1}{p-r}} + \biggl(\frac{\int_{E}f_{2}^{p}\,dx}{\int_{E}g_{2}^{r}\,dx} \biggr)^{\frac{1}{p-r}}, $$
(3.1)
equality holds if and only if \(f_{1}/ f_{2}= g_{1}/g_{2} \).

Lemma 3.2

(Reverse Dresher’s inequality)

Let functions \(f_{1}, f_{2}, g_{1}, g_{2} \geq0\), E is a bounded measurable subset in \(\mathbb{R}^{n}\). If \(1\geq p>0>r\), then
$$ \biggl(\frac{\int_{E}(f_{1}+f_{2})^{p}\,dx}{\int_{E}(g_{1}+g_{2})^{r}\,dx} \biggr)^{\frac{1}{p-r}} \geq \biggl(\frac{\int_{E}f_{1}^{p}\,dx}{\int_{E}g_{1}^{r}\,dx} \biggr)^{\frac{1}{p-r}} + \biggl(\frac{\int_{E}f_{2}^{p}\,dx}{\int_{E}g_{2}^{r}\,dx} \biggr)^{\frac{1}{p-r}}, $$
(3.2)
equality holds if and only if \(f_{1}/ f_{2}= g_{1}/g_{2} \).

Proof of Lemma 3.2

If \(f_{1}, f_{2}, g_{1}, g_{2} \geq0\), and \(1\geq p>0>r\), according to the Minkowski inequality,
$$\begin{aligned}& \biggl(\int_{E}(f_{1}+f_{2})^{p} \,dx \biggr)^{\frac{1}{p}}\geq \biggl(\int_{E}f_{1}^{p} \,dx \biggr)^{\frac{1}{p}}+ \biggl(\int_{E} f_{2}^{p}\,dx \biggr)^{\frac{1}{p}}, \\& \biggl(\int_{E}(g_{1}+g_{2})^{r} \,dx \biggr)^{\frac{1}{r}}\geq \biggl(\int_{E}g_{1}^{r} \,dx \biggr)^{\frac{1}{r}}+ \biggl(\int_{E} g_{2}^{r}\,dx \biggr)^{\frac{1}{r}}. \end{aligned}$$
For \(1\geq p>0>r\), we have
$$\begin{aligned}& \int_{E}(f_{1}+f_{2})^{p} \,dx \geq \biggl(\biggl(\int_{E}f_{1}^{p} \,dx\biggr)^{\frac{1}{p}}+\biggl(\int_{E} f_{2}^{p}\,dx\biggr)^{\frac{1}{p}} \biggr)^{p}, \end{aligned}$$
(3.3)
$$\begin{aligned}& \int_{E}(g_{1}+g_{2})^{r} \,dx \leq \biggl(\biggl(\int_{E}g_{1}^{r} \,dx\biggr)^{\frac{1}{r}}+\biggl(\int_{E} g_{2}^{r}\,dx\biggr)^{\frac{1}{r}} \biggr)^{r}. \end{aligned}$$
(3.4)
According to the Hölder inequality, \(\frac{p-r}{p}>1\), and (3.3), (3.4),
$$\begin{aligned} \biggl(\frac{\int_{E}(f_{1}+f_{2})^{p}\,dx}{\int_{E}(g_{1}+g_{2})^{r}\,dx} \biggr)^{\frac{1}{p-r}} \geq& \biggl[\frac{((\int_{E}f_{1}^{p}\,dx)^{\frac{1}{p}}+(\int_{E} f_{2}^{p}\,dx)^{\frac{1}{p}})^{p}}{((\int_{E}g_{1}^{r}\,dx)^{\frac{1}{r}}+(\int_{E} g_{2}^{r}\,dx)^{\frac{1}{r}})^{r}} \biggr]^{\frac{1}{p-r}} \\ =&\frac{ [(\int_{E}f_{1}^{p}\,dx)^{\frac{1}{p}}+(\int_{E} f_{2}^{p}\,dx)^{\frac {1}{p}} ]^{\frac{p}{p-r}}}{ [(\int_{E}g_{1}^{r}\,dx)^{\frac {1}{r}}+(\int_{E} g_{2}^{r}\,dx)^{\frac{1}{r}} ]^{\frac{r}{p-r}}} \\ =& \biggl[ \biggl(\biggl(\int_{E}f_{1}^{p} \,dx\biggr)^{\frac{1}{p-r}} \biggr)^{\frac {p-r}{p}}+ \biggl(\biggl(\int _{E} f_{2}^{p}\,dx\biggr)^{\frac{1}{p-r}} \biggr)^{\frac {p-r}{p}} \biggr]^{\frac{p}{p-r}} \\ &{}\times \biggl[ \biggl(\biggl(\int_{E}g_{1}^{r} \,dx\biggr)^{\frac{-1}{p-r}} \biggr)^{\frac {-(p-r)}{r}}+ \biggl(\biggl(\int _{E} g_{2}^{r}\,dx\biggr)^{\frac{-1}{p-r}} \biggr)^{\frac {-(p-r)}{r}} \biggr]^{\frac{-r}{p-r}} \\ \geq& \biggl(\int_{E}f_{1}^{p}\,dx \biggr)^{\frac{1}{p-r}} \biggl(\int_{E}g_{1}^{r} \,dx \biggr)^{\frac{-1}{p-r}}+ \biggl(\int_{E}f_{2}^{p} \,dx \biggr)^{\frac {1}{p-r}} \biggl(\int_{E}g_{2}^{r} \,dx \biggr)^{\frac{-1}{p-r}} \\ =& \biggl(\frac{\int_{E}f_{1}^{p}\,dx}{\int_{E}g_{1}^{r}\,dx} \biggr)^{\frac {1}{p-r}}+ \biggl(\frac{\int_{E}f_{2}^{p}\,dx}{\int_{E}g_{2}^{r}\,dx} \biggr)^{\frac{1}{p-r}}. \end{aligned}$$
According to the equality condition of the Minkowski inequality and the Hölder inequality, equality holds in (3.2) if and only if \(f_{1}/ f_{2}= g_{1}/g_{2} \). □

Proof of Theorem 1.1

From (2.1), for \(K, K', L\in\mathcal {S}^{n}_{o}\),
$$\begin{aligned} \widetilde{W}_{n-p-1}\bigl(K\,\tilde{+}\,K',L\bigr) =& \frac{1}{n}\int_{S^{n-1}}\rho_{K\,\tilde{+}\,K'}^{p}(u) \rho_{L}(u)\,dS(u) \\ =&\frac{1}{n}\int_{S^{n-1}}\bigl(\rho_{K}(u)+ \rho_{K'}(u)\bigr)^{p}\rho_{L}(u)\,dS(u) \\ =&\frac{1}{n}\int_{S^{n-1}}\bigl(\rho_{K}(u) \rho^{\frac {1}{p}}_{L}(u)+\rho_{K'}(u)\rho^{\frac{1}{p}} _{L}(u)\bigr)^{p}\,dS(u) \end{aligned}$$
(3.5)
and
$$ \widetilde{W}_{n-r-1}\bigl(K\,\tilde{+}\,K',L\bigr)= \frac{1}{n}\int_{S^{n-1}}\bigl(\rho_{K}(u) \rho^{\frac{1}{r}} _{L}(u)+\rho_{K'}(u)\rho ^{\frac{1}{r}} _{L}(u)\bigr)^{r}\,dS(u). $$
(3.6)
From (3.1), (3.5), and (3.6), for \(p\geq1\geq r>0\), we have
$$\begin{aligned} \biggl(\frac{\widetilde{W}_{n-p-1}(K\, \tilde {+}\, K',L)}{\widetilde{W}_{n-r-1}(K\,\tilde{+}\,K',L)} \biggr) ^{\frac{1}{p-r}} =& \biggl(\frac{\int_{S^{n-1}}(\rho_{K}(u)\rho^{\frac {1}{p}}_{L}(u)+\rho_{K'}(u)\rho^{\frac{1}{p}} _{L}(u))^{p}\,dS(u)}{\int_{S^{n-1}}(\rho_{K}(u)\rho^{\frac{1}{r}} _{L}(u)+\rho_{K'}(u)\rho^{\frac{1}{r}} _{L}(u))^{r}\,dS(u)} \biggr)^{\frac{1}{p-r}} \\ \leq& \biggl(\frac{\int_{S^{n-1}}(\rho_{K}(u)\rho^{\frac{1}{p}} _{L}(u))^{p}\,dS(u)}{\int_{S^{n-1}}(\rho_{K}(u)\rho^{\frac{1}{r}} _{L}(u))^{r}\,dS(u)} \biggr)^{\frac{1}{p-r}} \\ &{}+ \biggl( \frac{\int_{S^{n-1}}(\rho_{K'}(u)\rho^{\frac{1}{p}} _{L}(u))^{p}\,dS(u)}{\int_{S^{n-1}}(\rho_{K'}(u)\rho^{\frac{1}{r}} _{L}(u))^{r}\,dS(u)} \biggr)^{\frac{1}{p-r}} \\ =& \biggl(\frac{\int_{S^{n-1}}\rho_{K}^{p}(u)\rho_{L}(u)\,dS(u)}{\int_{S^{n-1}}\rho_{K}^{r}(u)\rho_{L}(u)\,dS(u)} \biggr)^{\frac{1}{p-r}} \\ &{}+ \biggl(\frac{\int_{S^{n-1}}\rho_{K'}^{p}(u)\rho_{L}(u)\,dS(u)}{\int_{S^{n-1}}\rho_{K'}^{r}(u)\rho_{L}(u)\,dS(u)} \biggr)^{\frac{1}{p-r}} \\ =& \biggl(\frac{\widetilde{W}_{n-p-1}(K,L)}{\widetilde {W}_{n-r-1}(K,L)} \biggr)^{\frac{1}{p-r}}+ \biggl(\frac{\widetilde{W}_{n-p-1}(K',L)}{\widetilde {W}_{n-r-1}(K',L)} \biggr)^{\frac{1}{p-r}}. \end{aligned}$$
(3.7)

According to the equality condition of inequality (3.1), we see that equality holds in (3.7) if and only if K and L, \(K'\), and L are dilates, respectively. So K and \(K'\) are dilates.

Let \(i=n-p-1\), \(j=n-r-1\), then \(p\geq1\geq r>0\) and \(i\leq n-2\leq j< n-1\) are equivalent. This and (3.7) yield inequality (1.7) and its equality condition.

Similarly, if \(1\geq p>0>r\), according to (3.2), (3.5), and (3.6), we have
$$ \biggl(\frac{\widetilde{W}_{n-p-1}(K\,\tilde{+}\,K',L)}{\widetilde {W}_{n-r-1}(K\,\tilde{+}\,K',L)} \biggr) ^{\frac{1}{p-r}}\geq \biggl(\frac{\widetilde{W}_{n-p-1}(K,L)}{\widetilde{W}_{n-r-1}(K,L)} \biggr)^{\frac{1}{p-r}}+ \biggl(\frac{\widetilde{W}_{n-p-1}(K',L)}{\widetilde {W}_{n-r-1}(K',L)} \biggr)^{\frac{1}{p-r}}, $$
(3.8)
and equality holds if and only if K and \(K'\) are dilates.

Let \(i=n-p-1\), \(j=n-r-1\), then (3.8) gives inequality (1.8) and its equality condition. □

Proof of Theorem 1.2

From (2.2), for \(K, K', L \in\mathcal {S}^{n}_{o}\), we have
$$\begin{aligned} \widetilde{W}_{n-p-1}\bigl(K\,\breve{+}\,K',L\bigr) =& \frac{1}{n}\int_{S^{n-1}}\rho _{K\,\breve{+}\,K'}^{p}(u) \rho_{L}(u)\,dS(u) \\ =&\frac{1}{n}\int_{S^{n-1}}\bigl(\rho_{K}^{n-1}(u)+ \rho _{K'}^{n-1}(u)\bigr)^{\frac{p}{n-1}}\rho_{L}(u) \,dS(u) \\ =&\frac{1}{n}\int_{S^{n-1}} \bigl(\rho_{K}^{n-1}(u) \rho_{L}^{\frac {n-1}{p}}(u)+\rho_{K'}^{n-1}(u) \rho_{L}^{\frac{n-1}{p}}(u) \bigr)^{\frac{p}{n-1}}\,dS(u) \end{aligned}$$
(3.9)
and
$$\begin{aligned} \widetilde{W}_{n-r-1}\bigl(K\,\breve{+}\,K',L\bigr) =& \frac{1}{n}\int_{S^{n-1}}\rho _{K\,\breve{+}\,K'}^{r}(u) \rho_{L}(u)\,dS(u) \\ =&\frac{1}{n}\int_{S^{n-1}} \bigl(\rho_{K}^{n-1}(u) \rho_{L}^{\frac {n-1}{r}}(u)+\rho_{K'}^{n-1}(u) \rho_{L}^{\frac{n-1}{r}}(u) \bigr)^{\frac{r}{n-1}}\,dS(u). \end{aligned}$$
(3.10)
According to (3.1), (3.9), and (3.10), for \(p\geq n-1\geq r>0\),
$$\begin{aligned} Q^{\frac{n-1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K\,\breve{+}\,K',L)} =& \biggl[\frac{\widetilde{W}_{n-p-1}(K\,\breve{+}\,K',L)}{\widetilde {W}_{n-r-1}(K\,\breve{+}\,K',L)} \biggr]^{\frac{n-1}{p-r}} \\ =& \biggl[\frac{\int_{S^{n-1}} (\rho_{K}^{n-1}(u)\rho_{L}^{\frac {n-1}{p}}(u)+\rho_{K'}^{n-1}(u)\rho_{L}^{\frac{n-1}{p}}(u) )^{\frac{p}{n-1}}\,dS(u)}{\int_{S^{n-1}} (\rho_{K}^{n-1}(u)\rho _{L}^{\frac{n-1}{r}}(u)+\rho_{K'}^{n-1}(u)\rho_{L}^{\frac {n-1}{r}}(u) )^{\frac{r}{n-1}}\,dS(u)} \biggr]^{\frac{n-1}{p-r}} \\ \leq& \biggl[\frac{\int_{S^{n-1}} (\rho_{K}^{n-1}(u)\rho_{L}^{\frac {n-1}{p}}(u) )^{\frac{p}{n-1}}\,dS(u)}{\int_{S^{n-1}} (\rho _{K}^{n-1}(u)\rho_{L}^{\frac{n-1}{r}}(u) )^{\frac {r}{n-1}}\,dS(u)} \biggr]^{\frac{n-1}{p-r}} \\ &{}+ \biggl[\frac{\int_{S^{n-1}} (\rho_{K'}^{n-1}(u)\rho_{L}^{\frac {n-1}{p}}(u) )^{\frac{p}{n-1}}\,dS(u)}{\int_{S^{n-1}} (\rho _{K'}^{n-1}(u)\rho_{L}^{\frac{n-1}{r}}(u) )^{\frac {r}{n-1}}\,dS(u)} \biggr]^{\frac{n-1}{p-r}} \\ =& \biggl[\frac{\int_{S^{n-1}}\rho_{K}^{p}(u)\rho _{L}(u)\,dS(u)}{\int_{S^{n-1}}\rho_{K}^{r}(u)\rho_{L}(u)\,dS(u)} \biggr]^{\frac{n-1}{p-r}}+ \biggl[\frac{\int_{S^{n-1}}\rho_{K'}^{p}(u)\rho_{L}(u)\,dS(u)}{\int_{S^{n-1}}\rho_{K'}^{r}(u)\rho_{L}(u)\,dS(u)} \biggr]^{\frac{n-1}{p-r}} \\ =& \biggl[\frac{\widetilde{W}_{n-p-1}(K,L)}{\widetilde {W}_{n-r-1}(K,L)} \biggr]^{\frac{n-1}{p-r}} + \biggl[\frac{\widetilde{W}_{n-p-1}(K',L)}{ \widetilde{W}_{n-r-1}(K',L)} \biggr]^{\frac{n-1}{p-r}} \\ =&Q^{\frac{n-1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K,L)}+Q^{\frac {n-1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K',L)}. \end{aligned}$$
Then
$$ Q^{\frac{n-1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K\breve {+}K',L)}\leq Q^{\frac{n-1}{p-r}}_{\widetilde {W}_{n-p-1,n-r-1}(K,L)}+Q^{\frac{n-1}{p-r}}_{\widetilde {W}_{n-p-1,n-r-1}(K',L)}. $$
(3.11)

According to the equality condition of inequality (3.1), we see that equality holds in (3.11) if and only if K and \(K'\) are dilates.

Let \(i=n-p-1\) and \(j=n-r-1\), then \(p\geq n-1\geq r>0\) and \(i\leq0\leq j< n-1\) are equivalent. This and (3.11) yield inequality (1.9) and its equality condition.

Similarly, if \(n-1\geq p>0>r\), according to (3.2), (3.9), and (3.10), we have
$$ \biggl(\frac{\widetilde{W}_{n-p-1}(K\,\tilde{+}\,K',L)}{\widetilde {W}_{n-r-1}(K\,\tilde{+}\,K',L)} \biggr) ^{\frac{1}{p-r}}\geq \biggl(\frac{\widetilde{W}_{n-p-1}(K,L)}{\widetilde{W}_{n-r-1}(K,L)} \biggr)^{\frac{1}{p-r}}+ \biggl(\frac{\widetilde{W}_{n-p-1}(K',L)}{\widetilde {W}_{n-r-1}(K',L)} \biggr)^{\frac{1}{p-r}}, $$
(3.12)
and equality holds if and only if K and \(K'\) are dilates.

Let \(i=n-p-1\) and \(j=n-r-1\), then (3.12) gives inequality (1.10) and its equality condition. □

Proof of Theorem 1.3

From (2.3), for \(K, K', L \in\mathcal {S}^{n}_{o}\),
$$\begin{aligned} \frac{\widetilde{W}_{n-p-1}(K\,\hat{+}\,K', L)}{V(K\hat {+}K')^{p/(n+1)}} =&\frac{1}{n}\int_{S^{n-1}} \frac{\rho_{K\,\hat{+}\,K'}^{p}(u)\rho _{L}(u)}{V(K\,\hat{+}\,K')^{p/(n+1)}}\,dS(u) \\ =&\frac{1}{n}\int_{S^{n-1}} \biggl(\frac{\rho_{K\hat {+}K'}^{n+1}(u)}{V(K\,\hat{+}\,K')} \biggr)^{\frac{p}{n+1}}\rho _{L}(u)\,dS(u) \\ =&\frac{1}{n}\int_{S^{n-1}} \biggl(\frac{\rho_{K}^{n+1}(u)}{V(K)}+ \frac {\rho_{K'}^{n+1}(u)}{V(K')} \biggr)^{\frac{p}{n+1}}\rho_{L}(u)\,dS(u) \\ =&\frac{1}{n}\int_{S^{n-1}} \biggl(\frac{\rho_{K}^{n+1}(u)\rho _{L}^{\frac{n+1}{p}}(u)}{V(K)}+ \frac{\rho_{K'}^{n+1}(u)\rho_{L}^{\frac{n+1}{p}}(u)}{V(K')} \biggr)^{\frac{p}{n+1}}\,dS(u) \end{aligned}$$
(3.13)
and
$$ \frac{\widetilde{W}_{n-r-1}(K\,\hat{+}\,K', L)}{V(K\hat {+}K')^{r/(n+1)}} =\frac{1}{n}\int_{S^{n-1}} \biggl(\frac{\rho_{K}^{n+1}(u)\rho _{L}^{\frac{n+1}{r}}(u)}{V(K)}+ \frac{\rho_{K'}^{n+1}(u)\rho_{L}^{\frac{n+1}{r}}(u)}{V(K')} \biggr)^{\frac{r}{n+1}}\,dS(u). $$
(3.14)
According to (3.1), (3.13), and (3.14), for \(p\geq n+1>r>0\),
$$\begin{aligned} Q^{\frac{n+1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K\,\hat{+}\,K',L)} =& \biggl(\frac{\widetilde{W}_{n-p-1}(K\,\hat{+}\,K',L)}{ \widetilde{W}_{n-r-1}(K\,\hat{+}\,K',L)} \biggr)^{\frac{n+1}{p-r}} \\ =&V\bigl(K\,\hat{+}\,K'\bigr) \biggl[\frac{\int_{S^{n-1}} (\frac{\rho_{K}^{n+1}(u)}{V(K)}\rho_{L}^{\frac {n+1}{p}}(u)+\frac{\rho_{K'}^{n+1}(u)}{V(K')}\rho_{L}^{\frac {n+1}{p}}(u) )^{\frac{p}{n+1}}\,dS(u)}{\int_{S^{n-1}} (\frac{\rho _{K}^{n+1}(u)}{V(K)}\rho_{L}^{\frac{n+1}{r}}(u)+\frac {\rho_{K'}^{n+1}(u)}{V(K')}\rho_{L}^{\frac{n+1}{r}}(u) )^{\frac{r}{n+1}}\,dS(u)} \biggr]^{\frac{n+1}{p-r}} \\ \leq& V\bigl(K\,\hat{+}\,K'\bigr) \biggl[\frac{\int_{S^{n-1}} (V(K)^{-1}\rho_{K}^{n+1}(u)\rho_{L}^{\frac{n+1}{p}}(u) )^{\frac {p}{n+1}}\,dS(u)}{\int_{S^{n-1}} (V(K)^{-1}\rho_{K}^{n+1}(u)\rho _{L}^{\frac{n+1}{r}}(u) )^{\frac{r}{n+1}}\,dS(u)} \biggr]^{\frac{n+1}{p-r}} \\ &{}+V\bigl(K\,\hat{+}\,K'\bigr) \biggl[\frac{\int_{S^{n-1}} (V(K')^{-1}\rho_{K'}^{n+1}(u)\rho_{L}^{\frac{n+1}{p}}(u) )^{\frac{p}{n+1}}\,dS(u)}{\int_{S^{n-1}} (V(K')^{-1}\rho _{K'}^{n+1}(u)\rho_{L}^{\frac{n+1}{r}}(u) )^{\frac {r}{n+1}}\,dS(u)} \biggr]^{\frac{n+1}{p-r}} \\ =&V\bigl(K\,\hat{+}\,K'\bigr) \biggl[\frac{\int_{S^{n-1}}(V(K)^{-1})^{\frac {p}{n+1}}\rho_{K}^{p}(u)\rho_{L}(u)\,dS(u)}{\int_{S^{n-1}}(V(K)^{-1})^{\frac{r}{n+1}}\rho_{K}^{r}(u)\rho _{L}(u)\,dS(u)} \biggr]^{\frac{n+1}{p-r}} \\ &{}+V\bigl(K\,\hat{+}\,K'\bigr) \biggl[\frac{\int_{S^{n-1}}(V(K')^{-1})^{\frac{p}{n+1}}\rho_{K'}^{p}(u)\rho _{L}(u)\,dS(u)}{\int_{S^{n-1}}(V(K')^{-1})^{\frac{r}{n+1}}\rho _{K'}^{r}(u)\rho_{L}(u)\,dS(u)} \biggr]^{\frac{n+1}{p-r}} \\ =&\frac{V(K\,\hat{+}\,K')}{V(K)} \biggl(\frac{\int_{S^{n-1}}\rho _{K}^{p}(u)\rho_{L}(u)\,dS(u)}{\int_{S^{n-1}}\rho_{K}^{r}(u)\rho _{L}(u)\,dS(u)} \biggr)^{\frac{n+1}{p-r}} \\ &{}+\frac{V(K\,\hat{+}\,K')}{V(K')} \biggl(\frac{\int_{S^{n-1}}\rho _{K'}^{p}(u)\rho_{L}(u)\,dS(u)}{\int_{S^{n-1}}\rho_{K'}^{r}(u)\rho _{L}(u)\,dS(u)} \biggr)^{\frac{n+1}{p-r}} \\ =&\frac{V(K\,\hat{+}\,K')}{V(K)} \biggl(\frac{\widetilde {W}_{n-p-1}(K,L)}{\widetilde{W}_{n-r-1}(K,L)} \biggr) ^{\frac{n+1}{p-r}} \\ &{}+\frac{V(K\,\hat{+}\,K')}{V(K')} \biggl(\frac{\widetilde {W}_{n-p-1}(K',L)}{\widetilde{W}_{n-r-1}(K',L)} \biggr) ^{\frac{n+1}{p-r}} \\ =&V\bigl(K\,\hat{+}\,K'\bigr) \biggl(\frac{Q^{\frac {n+1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K,L)}}{ V(K)}+ \frac{Q^{\frac{n+1}{p-r}}_{\widetilde {W}_{n-p-1,n-r-1}(K',L)}}{V(K')} \biggr), \end{aligned}$$
i.e.,
$$ \frac{Q^{\frac{n+1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K\hat {+}K',L)} }{V(K\,\hat{+}\,K')}\leq\frac{Q^{\frac {n+1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K,L)}}{V(K)} +\frac{Q^{\frac{n+1}{p-r}}_{\widetilde {W}_{n-p-1,n-r-1}(K',L)}}{V(K')}. $$
(3.15)

According to the equality condition of inequality (3.1), we see that equality holds in (3.15) if and only if K and \(K'\) are dilates.

Let \(i=n-p-1\) and \(j=n-r-1\), then \(p\geq n+1\geq r>0\) and \(i\leq-2\leq j< n-1\) are equivalent. This and (3.15) yield inequality (1.11) and its equality condition.

If \(n+1\geq p>0>r\), according to (3.2), (3.13), and (3.14), we have
$$ \frac{Q^{\frac{n+1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K\hat {+}K',L)} }{V(K\,\hat{+}\,K')}\geq\frac{Q^{\frac {n+1}{p-r}}_{\widetilde{W}_{n-p-1,n-r-1}(K,L)}}{V(K)} +\frac{Q^{\frac{n+1}{p-r}}_{\widetilde {W}_{n-p-1,n-r-1}(K',L)}}{V(K')}, $$
(3.16)
with equality if and only if K and \(K'\) are dilates.

Let \(i=n-p-1\) and \(j=n-r-1\), then (3.16) gives inequality (1.12) and its equality condition. □

Declarations

Acknowledgements

This work was supported by the Natural Science Foundation of China (Grant Nos. 11371224 and 11102101).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, China Three Gorges University

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