© Wuhan University 2022

0 Introduction and Main Results

The setting for this paper is $n$-dimensional Eucli-dean spaces ${\mathbb{R}}^n$. Let ${\mathcal{K}}_{}^n$ denote the set of convex bodies (compact, convex subsets with non-empty interiors) in Euclidean space ${\mathbb{R}}^n$. ${\mathcal{K}}_o^n$ denotes the set of convex bodies containing the origin in their interiors. $V(K)$ denotes the n-dimensional volume of a body $K$, $B$ the standard unit ball, and $V(B)={\omega }_n$. Let $S^{n-\mathrm{1}}$ denote the unit sphere in ${\mathbb{R}}^n$.

Blaschke^[1] considered the classical width-integral of convex bodies first and Hadwiger^[2] studied it further. In 1975 Lutwak^[3] introduced the i-th width-integral of convex bodies. In 1977, Lutwak^[4] generalized the i-th width-integral to the mixed width-integral of convex bodies. In 2016, Feng^[5] gave the definitions of mixed width-integral and the general i-th width-integral of convex bodies. In 2017, Zhou^[6] defined the general L_p-mixed width-integral of convex bodies. For the more results of the mixed width-integral of convex bodies, we refer the interested reader to Refs. [7-14].

In this paper, we first establish the Brunn-Minkowski type inequality for the i-th general L_p-mixed width-integral of convex bodies.

Theorem 1   Let $M,N,K,L$$\in$${\mathcal{K}}_o^n$, $\tau \in [-\mathrm{1,1}]$, $p>\mathrm{0}$,$M$ and $N$ have similar general L_p-width,for $K\subset M$, $L\subset N$, $n-p<i<n$ or $M\subset K$, $N\subset L$, $i>n$ , we have

$\begin{array}{l}{\left[B_{p,i}^{(\tau )}(M{+}_{p}N)-B_{p,i}^{(\tau )}(K{+}_{p}L)\right]}^{\frac{p}{n-i}}\\ \le {\left[B_{p,i}^{(\tau )}(M)-B_{p,i}^{(\tau )}(K)\right]}^{\frac{p}{n-i}}+{\left[B_{p,i}^{(\tau )}(N)-B_{p,i}^{(\tau )}(L)\right]}^{\frac{p}{n-i}}\end{array}$(1)

for $K\subset M,L\subset N$, $\mathrm{0}\le i<n-p$, we have

${\left[B_{p,i}^{(\tau )}(M{+}_{p}N)-B_{p,i}^{(\tau )}(K{+}_{p}L)\right]}^{\frac{p}{n-i}}\ge {\left[B_{p,i}^{(\tau )}(M)-B_{p,i}^{(\tau )}(K)\right]}^{\frac{p}{n-i}}+{\left[B_{p,i}^{(\tau )}(N)-B_{p,i}^{(\tau )}(L)\right]}^{\frac{p}{n-i}}$(2)

the equality holds in (1) or (2) if and only if $K$and $L$ have similar general L_p-width, and $(B_{p,i}^{(\tau )}(M),B_{p,i}^{(\tau )}(K))=c(B_{p,i}^{(\tau )}(N),B_{p,i}^{(\tau )}(L))$ , where $c$ is a constant.

We also establish two cyclic inequalities for the differences of i-th general L_p-mixed width-integral of convex bodies.

Theorem 2   Let $M$, $K$$\in {\mathcal{K}}_o^n$ , $K\subset M$, $\tau \in [-\mathrm{1,1}]$, $p>\mathrm{0}$, $M$ has constant general L_p-width, for $\mathrm{0}\le i<j<k\le n$, $i,j,k\in \mathbb{R}$, we get

${\left[B_{p,j}^{(\tau )}(M)-B_{p,j}^{(\tau )}(K)\right]}^{k-i}\ge {\left[B_{p,i}^{(\tau )}(M)-B_{p,i}^{(\tau )}(K)\right]}^{k-j}{\left[B_{p,k}^{(\tau )}(M)-B_{p,k}^{(\tau )}(K)\right]}^{j-i}$(3)

with equality if and only if $K$ has constant general L_p-width.

Theorem 3   Let $M,N,K,L$$\in {\mathcal{K}}_o^n$, $K\subset M$, $L\subset N$, $\tau \in [-\mathrm{1,1}]$, $p>\mathrm{0}$, $M$ and $N$ have similar general L_p-width, for $\mathrm{0}\le i<j<k\le n$, $i,j,k\in \mathbb{R}$, we can obtain

${\left[B_{p,j}^{(\tau )}(M,N)-B_{p,j}^{(\tau )}(K,L)\right]}^{k-i}\ge {\left[B_{p,i}^{(\tau )}(M,N)-B_{p,i}^{(\tau )}(K,L)\right]}^{k-j}{\left[B_{p,k}^{(\tau )}(M,N)-B_{p,k}^{(\tau )}(K,L)\right]}^{j-i}$(4)

with equality if and only if $(b_p^{(\tau )}(M,u),b_p^{(\tau )}(K,u))=c(b_p^{(\tau )}(N,u),b_p^{(\tau )}(L,u))$ , where $c$ is a constant, and $K$ and $L$ have similar general L_p -width.

Meanwhile, we establish a cyclic Brunn-Minkowski inequality for the i-th general L_p-mixed width-integral of convex bodies.

Theorem 4   Let $K$,$L$$\in {\mathcal{K}}_o^n$, $p>\mathrm{0}$, $\tau \in [-\mathrm{1,1}]$, if $j<n-p$ and $i<j<k$, we can obtain

$B_{p,j}^{(\tau )}(K{+}_p{L)}^{\frac{p}{n-j}}\le B_{p,i}^{(\tau )}{(K)}^{\frac{p(k-j)}{(k-i)(n-j)}}{B}_{p,k}^{(\tau )}{(K)}^{\frac{p(j-i)}{(k-i)(n-j)}}+B_{p,i}^{(\tau )}{(L)}^{\frac{p(k-j)}{(k-i)(n-j)}}{B}_{p,k}^{(\tau )}{(L)}^{\frac{p(j-i)}{(k-i)(n-j)}}$(5)

with equality if and only if $K$ and $L$ both have constant general L_p-width. If $n-p<j<n$ and $j\le i<k$, or $j>n$ and $i\le j<k$, the inequality is reversed.

1 Preliminaries

1.1 Support Function and Firey L_p-Combination

If $K\in {\mathcal{K}}_{}^n$, the support function, $h_K=h(K,\cdot ):{\mathbb{R}}^n\to (-\mathrm{\infty },\mathrm{\infty })$, is defined by^[15,16]

$h(K,x)=\mathrm{m}\mathrm{a}\mathrm{x}\{x\cdot y:y\in K\},x\in {\mathbb{R}}^n$

where $x\cdot y$ denotes the standard inner product of $x$ and $y$.

For $K,L\in {\mathcal{K}}_o^n$, $p\ge \mathrm{1}$ and $\lambda$, $\mu \ge \mathrm{0}$ (not both zero), the Firey L_p-combination$\lambda \cdot K{+}_p\mu \cdot L\in K_o^n$ of $K$and $L$ is defined by ^[17]

$h(\lambda \cdot K{+}_p{\mu \cdot L,\cdot )}^p=\lambda h{(K,\cdot )}^p+\mu h{(L,\cdot )}^p$(6)

where the operation "${+}_p$" is called Firey addition and $\lambda \cdot K$ denotes the Firey scalar multiplication.

1.2 General L_p-Mixed Width-Integral of Order i

For$\tau \in [-\mathrm{1,1}]$ and $p>\mathrm{0}$, the general L_p-mixed width-integral$B_p^{(\tau )}(K_{\mathrm{1}},\cdots ,K_n)$ of $K_{\mathrm{1}},\cdots ,K_n\in {\mathcal{K}}_o^n$ is defined by ^[6]

$B_p^{(\tau )}(K_{\mathrm{1}},\cdots ,K_n)=\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{b}_p^{(\tau )}(K_{\mathrm{1}},u)\cdots b_p^{(\tau )}(K_n,u)\mathrm{d}S(u)$(7)

where $b_p^{(\tau )}(K,u)={\left(f_{\mathrm{1}}(\tau )h^p(K,u)+f_{\mathrm{2}}(\tau )h^p(K,-u)\right)}^{\frac{\mathrm{1}}{p}}$ for any $u\in S^{n-\mathrm{1}}$, and $f_{\mathrm{1}}(\tau )$, $f_{\mathrm{2}}(\tau )$ are chosen as follows:

$f_{\mathrm{1}}(\tau )=\frac{{(\mathrm{1}+\tau )}^{\mathrm{2}p}}{{(\mathrm{1}+\tau )}^{\mathrm{2}p}+{(\mathrm{1}-\tau )}^{\mathrm{2}p}},f_{\mathrm{2}}(\tau )=\frac{{(\mathrm{1}-\tau )}^{\mathrm{2}p}}{{(\mathrm{1}+\tau )}^{\mathrm{2}p}+{(\mathrm{1}-\tau )}^{\mathrm{2}p}}$

$f_{\mathrm{1}}(\tau )$ and $f_{\mathrm{2}}(\tau )$ satisfy

$f_{\mathrm{1}}(\tau )+f_{\mathrm{2}}(\tau )=\mathrm{1};f_{\mathrm{1}}(-\tau )=f_{\mathrm{2}}(\tau );f_{\mathrm{2}}(-\tau )=f_{\mathrm{1}}(\tau ).$

$K$ and $L$ are said to have similar general L_p-width if there exists a constant $\lambda >\mathrm{0}$ such that $b_p^{(\tau )}(K,u)=\lambda b_p^{(\tau )}(L,u)$ for all $u\in S^{n-\mathrm{1}}$. If $b_p^{(\tau )}(K,u)=b_p^{(\tau )}(L,u)$ for all $u\in S^{n-\mathrm{1}}$, then we call $K$ and $L$ have the same general L_p-width. If $b_p^{(\tau )}(K,u)$ is a constant for all $u\in S^{n-\mathrm{1}}$, we call $K$ has the constant general L_p-width.

Taking$K_{\mathrm{1}}=\cdots =K_{n-i}=K$ and $K_{n-i+\mathrm{1}}=\cdots =K_n=L$ in (7), the general L_p-mixed width-integral $B_{p,i}^{(\tau )}(K,L)$ of $K,L\in {\mathcal{K}}_o^n$ is given by

$B_{p,i}^{(\tau )}(K,L)=\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{b}_p^{(\tau )}{(K,u)}^{n-i}{b}_p^{(\tau )}{(L,u)}^i\mathrm{d}S(u)$(8)

Further, let $L=B$ in (8), since $b_p^{(\tau )}(B,u)=\mathrm{1}$, and write $B_{p,i}^{(\tau )}(K)$ for $B_{p,i}^{(\tau )}(K,B)$, we get

$B_{p,i}^{(\tau )}(K)=\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{b}_p^{(\tau )}{(K,u)}^{n-i}\mathrm{d}S(u)$(9)

where $B_{p,i}^{(\tau )}(K)$ is called the i-th general L_p-mixed width-integral of $K\in {\mathcal{K}}_o^n$. If $i=\mathrm{0}$, we write $B_{p,\mathrm{0}}^{(\tau )}(K)=B_p^{(\tau )}(K)$, $B_p^{(\tau )}(K)$ is called the general L_p-width-integral of $K\in {\mathcal{K}}_o^n$.

2 Proofs of Theorems

In this section, we give the proofs of the Theorems 1-4. The proof of Theorem 1 requires the following lemmas.

Lemma 1^[6]If $K,L\in {\mathcal{K}}_o^n$, $\tau \in [-\mathrm{1,1}]$ and $p>\mathrm{0}$, for $i\le n-p\le j\le n$ and $i\ne j$, then

${\left(\frac{B_{p,i}^{(\tau )}(K{+}_{p}L)}{B_{p,j}^{(\tau )}(K{+}_{p}L)}\right)}^{\frac{p}{j-i}}\le {\left(\frac{B_{p,i}^{(\tau )}(K)}{B_{p,j}^{(\tau )}(K)}\right)}^{\frac{p}{j-i}}+{\left(\frac{B_{p,i}^{(\tau )}(L)}{B_{p,j}^{(\tau )}(L)}\right)}^{\frac{p}{j-i}}$(10)

for $j\ge n\ge i\ge n-p$ and $i\ne j$, then

${\left(\frac{B_{p,i}^{(\tau )}(K{+}_{p}L)}{B_{p,j}^{(\tau )}(K{+}_{p}L)}\right)}^{\frac{p}{j-i}}\ge {\left(\frac{B_{p,i}^{(\tau )}(K)}{B_{p,j}^{(\tau )}(K)}\right)}^{\frac{p}{j-i}}+{\left(\frac{B_{p,i}^{(\tau )}(L)}{B_{p,j}^{(\tau )}(L)}\right)}^{\frac{p}{j-i}}$(11)

with equality in every inequality if and only if $K$ and $L$ have similar general L_p-width.

Lemma 2^[18,19]Let $x=(x_{\mathrm{1}},\cdots ,x_n)$ and $y=(y_{\mathrm{1}},\cdots ,y_n)$ be two series of non-negative real numbers, and ${x_{\mathrm{1}}}^p\ge {\displaystyle \sum _{i=\mathrm{2}}^n}{x}_i^p$, ${y_{\mathrm{1}}}^p\ge {\displaystyle \sum _{i=\mathrm{2}}^n}{y}_i^p$, for $p>\mathrm{1}$, then

${\left({x_{\mathrm{1}}}^p-{\displaystyle \sum _{i=\mathrm{2}}^n}{x}_i^p\right)}^{\frac{\mathrm{1}}{p}}+{\left({y_{\mathrm{1}}}^p-{\displaystyle \sum _{i=\mathrm{2}}^n}{y}_i^p\right)}^{\frac{\mathrm{1}}{p}}\le {\left((x_{\mathrm{1}}+y_{\mathrm{1}}{)}^p-{\displaystyle \sum _{i=\mathrm{2}}^n}(x_i+y_i{)}^p\right)}^{\frac{\mathrm{1}}{p}}$(12)

for $p<\mathrm{0}$ or $\mathrm{0}<p<\mathrm{1}$,

${\left({x_{\mathrm{1}}}^p-{\displaystyle \sum _{i=\mathrm{2}}^n}{x}_i^p\right)}^{\frac{\mathrm{1}}{p}}+{\left({y_{\mathrm{1}}}^p-{\displaystyle \sum _{i=\mathrm{2}}^n}{y}_i^p\right)}^{\frac{\mathrm{1}}{p}}\ge {\left((x_{\mathrm{1}}+y_{\mathrm{1}}{)}^p-{\displaystyle \sum _{i=\mathrm{2}}^n}(x_i+y_i{)}^p\right)}^{\frac{\mathrm{1}}{p}}$(13)

with equality in every inequality if and only if $x$ and $y$ are proportional.

Proof of Theorem 1   Let $j=n$ in Lemma 1, since $B_{p,n}^{(\tau )}(K{+}_{p}L)={\omega }_n$ is a constant, for $i\le n-p$, we immediately obtain

$B_{p,i}^{(\tau )}(K{+}_p{L)}^{\frac{p}{n-i}}\le B_{p,i}^{(\tau )}{(K)}^{\frac{p}{n-i}}+B_{p,L}^{(\tau )}{(L)}^{\frac{p}{n-i}}$(14)

and for $n-p<i<n$,

$B_{p,i}^{(\tau )}(K{+}_p{L)}^{\frac{p}{n-i}}\ge B_{p,i}^{(\tau )}{(K)}^{\frac{p}{n-i}}+B_{p,L}^{(\tau )}{(L)}^{\frac{p}{n-i}}$(15)

the equality holds if and only if $K$ and $L$have similar general L_p-width, which is just Corollary 3.

Since $M$,$N$,$K$,$L$$\in {\mathcal{K}}_o^n$, $M$ and $N$have similar general L_p-width, for $n-p<i<n$, we have

$B_{p,i}^{(\tau )}(K{+}_p{L)}^{\frac{p}{n-i}}\ge B_{p,i}^{(\tau )}{(K)}^{\frac{p}{n-i}}+B_{p,i}^{(\tau )}{(L)}^{\frac{p}{n-i}}$(16)

$B_{p,i}^{(\tau )}(M{+}_p{N)}^{\frac{p}{n-i}}=B_{p,i}^{(\tau )}{(M)}^{\frac{p}{n-i}}+B_{p,i}^{(\tau )}{(N)}^{\frac{p}{n-i}}$(17)

By the definition of the i-th general L_p-mixed width-integral of convex bodies, we know that $B_{p,i}^{(\tau )}(K)\le B_{p,i}^{(\tau )}(M)$ and $B_{p,i}^{(\tau )}(L)\le B_{p,i}^{(\tau )}(N)$ if$K\subset M$, $L\subset N$ for $n-p<i<n$; $B_{p,i}^{(\tau )}(M)>B_{p,i}^{(\tau )}(K)$ and $B_{p,i}^{(\tau )}(N)>B_{p,i}^{(\tau )}(L)$ if $M\subset K$, $N\subset L$ for $i>n$. According to (13), combining (16) with (17), we can obtain

${\left[B_{p,i}^{(\tau )}(M{+}_{p}N)-B_{p,i}^{(\tau )}(K{+}_{p}L)\right]}^{\frac{p}{n-i}}\le {\left[{\left(B_{p,i}^{(\tau )}{(M)}^{\frac{p}{n-i}}+B_{p,i}^{(\tau )}{(N)}^{\frac{p}{n-i}}\right)}^{\frac{n-i}{p}}-{\left(B_{p,i}^{(\tau )}{(K)}^{\frac{p}{n-i}}+B_{p,i}^{(\tau )}{(L)}^{\frac{p}{n-i}}\right)}^{\frac{n-i}{p}}\right]}^{\frac{p}{n-i}}$

$\le {\left[B_{p,i}^{(\tau )}(M)-B_{p,i}^{(\tau )}(K)\right]}^{\frac{p}{n-i}}+{\left[B_{p,i}^{(\tau )}(N)-B_{p,i}^{(\tau )}(L)\right]}^{\frac{p}{n-i}}$(18)

the equality holds if and only if $(B_{p,i}^{(\tau )}(M),B_{p,i}^{(\tau )}(K))$ is proportional to $(B_{p,i}^{(\tau )}(N),B_{p,i}^{(\tau )}(L))$, and $K$ and $L$have similar general L_p-width. The proof of inequality (2) is similar. This proves the theorem.

The proof of Theorem 2-3 requires the following lemma.

Lemma 3^[20]Suppose that $f_i$ , $g_i$ ($i$= 1,2) are non-negative continuous functions on $S^{n-\mathrm{1}}$ such that${\int }_{S^{n-\mathrm{1}}}{f}_{\mathrm{1}}^p(\xi )\mathrm{d}\xi \ge {\int }_{S^{n-\mathrm{1}}}{f}_{\mathrm{2}}^p(\xi )\mathrm{d}\xi$ and ${\int }_{S^{n-\mathrm{1}}}{g}_{\mathrm{1}}^q(\xi )\mathrm{d}\xi \ge {\int }_{S^{n-\mathrm{1}}}{g}_{\mathrm{2}}^q(\xi )\mathrm{d}\xi$ for $p>\mathrm{1}$, $\frac{\mathrm{1}}{p}+\frac{\mathrm{1}}{q}=\mathrm{1}$ , and for all $\xi \in S^{n-\mathrm{1}}$, $f_{\mathrm{1}}^p(\xi )=\lambda g_{\mathrm{1}}^q(\xi )$ where $\lambda$ is a constant, then

${\left({\int }_{S^{n-\mathrm{1}}}\left(f_{\mathrm{1}}^p(\xi )-f_{\mathrm{2}}^p(\xi )\right)\mathrm{d}\xi \right)}^{\frac{\mathrm{1}}{p}}{\left({\int }_{S^{n-\mathrm{1}}}\left(g_{\mathrm{1}}^q(\xi )-g_{\mathrm{2}}^q(\xi )\right)\mathrm{d}\xi \right)}^{\frac{\mathrm{1}}{q}}\le {\int }_{S^{n-\mathrm{1}}}\left(f_{\mathrm{1}}(\xi )g_{\mathrm{1}}(\xi )-f_{\mathrm{2}}(\xi )g_{\mathrm{2}}(\xi )\right)\mathrm{d}\xi$(19)

with equality if and only if $f_{\mathrm{2}}^p(\xi )=\lambda g_{\mathrm{2}}^q(\xi )$ for any $\xi \in S^{n-\mathrm{1}}$.

Proof of Theorem 2   Suppose that $\mathrm{0}\le i<j<k\le n$, let

$f_{\mathrm{1}}={\left(b_p^{(\tau )}{(M,u)}^{n-k}\right)}^{\frac{\mathrm{1}}{\lambda }},f_{\mathrm{2}}={\left(b_p^{(\tau )}{(K,u)}^{n-k}\right)}^{\frac{\mathrm{1}}{\lambda }},g_{\mathrm{1}}={\left(b_p^{(\tau )}{(M,u)}^{n-i}\right)}^{\frac{\mathrm{1}}{\mu }},g_{\mathrm{2}}={\left(b_p^{(\tau )}{(K,u)}^{n-i}\right)}^{\frac{\mathrm{1}}{\mu }},\lambda =(k-i)/(j-i),\mu =(k-i)/(k-j)$

in Lemma 3, since $K\subset M$, $M$ has constant general L_p-width, $f_{\mathrm{1}}^{\lambda }$/$g_{\mathrm{1}}^{\mu }$=$b_p^{(\tau )}{(M,u))}^{i-k}$ is a constant, according to Lemma 3, we can get the inequality (3).

By Lemma 3, the equality in (3) holds if and only if $b_p^{(\tau )}{(K,u))}^{i-k}=b_p^{(\tau )}{(M,u))}^{i-k}$ for all $u\in S^{n-\mathrm{1}}$, that means $K$ has constant general L_p-width. This proves the theorem.

Proof of Theorem 3   We can prove Theorem 3 by Lemma 3 as well. Suppose that $\mathrm{0}\le i<j<k\le n,$$\lambda =(k-i)/(j-i)$ and $\mu =(k-i)/(k-j)$, let

$f_{\mathrm{1}}={\left(b_p^{(\tau )}{(M,u)}^{n-k}{b}_p^{(\tau )}{(N,u)}^k\right)}^{\frac{\mathrm{1}}{\lambda }},f_{\mathrm{2}}={\left(b_p^{(\tau )}{(K,u)}^{n-k}{b}_p^{(\tau )}{(L,u)}^k\right)}^{\frac{\mathrm{1}}{\lambda }},g_{\mathrm{1}}={\left(b_p^{(\tau )}{(M,u)}^{n-i}{b}_p^{(\tau )}{(N,u)}^i\right)}^{\frac{\mathrm{1}}{\mu }},g_{\mathrm{2}}={\left(b_p^{(\tau )}{(K,u)}^{n-i}{b}_p^{(\tau )}{(L,u)}^i\right)}^{\frac{\mathrm{1}}{\mu }}$

in Lemma 3, since $K\subset M$, $L\subset N$, $M$ and $N$ have similar general L_p-width, $f_{\mathrm{1}}^{\lambda }/g_{\mathrm{1}}^{\mu }=(b_p^{(\tau )}(N,u)/b_p^{(\tau )}{(M,u))}^{k-i}$ is a constant. We get the consequence.

By Lemma 3, the equality in (4) holds if and only if $(b_p^{(\tau )}(L,u)/b_p^{(\tau )}{(K,u))}^{k-i}$$=$$(b_p^{(\tau )}(N,u)/b_p^{(\tau )}{(M,u))}^{k-i}$ for all $u\in S^{n-\mathrm{1}}$, that means $K$ and $L$have similar general L_p-width. This proves the theorem.

Taking $i=\mathrm{0}$, $j=\mathrm{1}$, $k=n$ in Theorem 3, we obtain

Corollary 1   Let $M$,$N$,$K$,$L$$\in {\mathcal{K}}_o^n$, $\tau \in [-\mathrm{1,1}]$, $p>\mathrm{0}$, $M$ and $N$ have similar general L_p-width, for $K\subset M$, $L\subset N$, one gets

${\left[B_{p,\mathrm{1}}^{(\tau )}(M,N)-B_{p,\mathrm{1}}^{(\tau )}(K,L)\right]}^n\ge {\left[B_p^{(\tau )}(M)-B_p^{(\tau )}(K)\right]}^{n-\mathrm{1}}\left[B_p^{(\tau )}(N)-B_p^{(\tau )}(L)\right]$(20)

with equality if and only if $(b_p^{(\tau )}(M,u),b_p^{(\tau )}(K,u))=c(b_p^{(\tau )}(N,u),b_p^{(\tau )}(L,u))$ , where $c$ is a constant.

Proof of Theorem 4   If $j<n-p$, combined (6) with (9), according to the Minkowski's inequality^[21], it follows that

$\begin{array}{l}{B}_{p,j}^{(\tau )}(K{+}_{p}L)=\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{b}_p^{(\tau )}(K{+}_p{L,u)}^{n-j}\mathrm{d}S(u)=\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{\left(f_{\mathrm{1}}(\tau )h^p(K{+}_{p}L,u)+f_{\mathrm{2}}(\tau )h^p(K{+}_{p}L,-u)\right)}^{\frac{n-j}{p}}\mathrm{d}S(u)\\ =\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{\left(f_{\mathrm{1}}(\tau )h^p(K,u)+f_{\mathrm{2}}(\tau )h^p(K,-u)+f_{\mathrm{1}}(\tau )h^p(L,u)+f_{\mathrm{2}}(\tau )h^p(L,-u)\right)}^{\frac{n-j}{p}}\mathrm{d}S(u)\\ =\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{\left(b_p^{(\tau )}{(K,u)}^p+b_p^{(\tau )}{(L,u)}^p\right)}^{\frac{n-j}{p}}\mathrm{d}S(u)\\ =\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}(b_p^{(\tau )}{(K,u)}^{\frac{p(k-j)(n-i)}{(k-i)(n-j)}}{b}_p^{(\tau )}{(K,u)}^{\frac{p(j-i)(n-k)}{(k-i)(n-j)}}{+b_p^{(\tau )}{(L,u)}^{\frac{p(k-j)(n-i)}{(k-i)(n-j)}}{b}_p^{(\tau )}{(L,u)}^{\frac{p(j-i)(n-k)}{(k-i)(n-j)}})}^{\frac{n-j}{p}}\mathrm{d}S(u)\\ \le [{\left(\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{b}_p^{(\tau )}{(K,u)}^{\frac{(k-j)(n-i)}{k-i}}{b}_p^{(\tau )}{(K,u)}^{\frac{(j-i)(n-k)}{k-i}}\mathrm{d}S(u)\right)}^{\frac{p}{n-j}}{+{\left(\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{b}_p^{(\tau )}{(L,u)}^{\frac{(k-j)(n-i)}{k-i}}{b}_p^{(\tau )}{(L,u)}^{\frac{(j-i)(n-k)}{k-i}}\mathrm{d}S(u)\right)}^{\frac{p}{n-j}}]}^{\frac{n-j}{p}}\end{array}$(21)

Since $i<j<k$ means $\frac{k-i}{k-j}>\mathrm{1}$, using Hölder's inequality ^[22] we have

$\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{b}_p^{(\tau )}{(K,u)}^{\frac{(k-j)(n-i)}{k-i}}{b}_p^{(\tau )}{(K,u)}^{\frac{(j-i)(n-k)}{k-i}}\mathrm{d}S(u)$

$\le {\left(\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{\left(b_p^{(\tau )}{(K,u)}^{\frac{(k-j)(n-i)}{k-i}}\right)}^{\frac{k-i}{k-j}}\mathrm{d}S(u)\right)}^{\frac{k-j}{k-i}}\times {\left(\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{\left(b_p^{(\tau )}{(K,u)}^{\frac{(j-i)(n-k)}{k-i}}\right)}^{\frac{k-i}{j-i}}\mathrm{d}S(u)\right)}^{\frac{j-i}{k-i}}=B_{p,i}^{(\tau )}{(K)}^{\frac{k-j}{k-i}}{B}_{p,k}^{(\tau )}{(K)}^{\frac{j-i}{k-i}}$(22)

Hence, we can get the following inequality

${\left[\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{b}_p^{(\tau )}{(K,u)}^{\frac{(k-j)(n-i)}{k-i}}{b}_p^{(\tau )}{(K,u)}^{\frac{(j-i)(n-k)}{k-i}}\mathrm{d}S(u)\right]}^{\frac{p}{n-j}}\le B_{p,i}^{(\tau )}{(K)}^{\frac{p(k-j)}{(n-j)(k-i)}}{B}_{p,k}^{(\tau )}{(K)}^{\frac{p(j-i)}{(n-j)(k-i)}}$(23)