Zhang et al. Journal of Inequalities and Applications (2015) 2015:340 DOI 10.1186/s13660-015-0871-5

RESEARCH

Open Access

On dual mixed quermassintegral quotient functions Ping Zhang, Weidong Wang and Xiaohua Zhang* *

Correspondence: [email protected] Department of Mathematics, China Three Gorges University, Yichang, 443002, P.R. China

Abstract We introduce the notion of dual mixed quermassintegral quotient functions and establish the Brunn-Minkowski inequalities for them in this paper. MSC: 52A20; 52A40 Keywords: dual quermassintegral; dual mixed quermassintegral quotient function; Brunn-Minkowski inequality

1 Introduction and main results The setting for this paper is Euclidean n-space Rn . Let Son denote the set of star bodies containing the origin in their interiors in Rn . Let Sn– denote the unit sphere in Rn , and let V (K) denote the n-dimensional volume of body K . For the standard unit ball B in Rn , we use ωn = V (B) to denote its volume. In , Lutwak (see []) gave the notion of dual mixed volumes as follows: For (K , K , . . . , Kn ), of K , K , . . . , Kn is defined K , K , . . . , Kn ∈ Son , the dual mixed volume, V by    ρ(K , u) · · · ρ(Kn , u) dS(u). (.) V (K , . . . , Kn ) = n Sn– i (K, L) = V (K, n – Taking K = · · · = Kn–i = K , Kn–i+ = · · · = Kn = L in (.), we write V i; L, i), where K appears n – i times and L appears i times. Then   i (K, L) = V ρ(K, u)n–i ρ(L, u)i dS(u). (.) n Sn– Let L = B in (.) and notice ρ(B, ·) = , and allow i is any real, then the dual quermassintegrals can be defined as follows: For K ∈ Son and i is any real, the dual quermassintegrals,  i (K), of K are given by (see []) W    i (K) = ρ(K, u)n–i dS(u). (.) W n Sn– Associated with dual quermassintegrals, Zhao (see []) defined the dual quermassintegral quotient functions of a star body K by QW  i,j (K) =

 i (K) W  j (K) W

(i, j ∈ R).

(.)

© 2015 Zhang et al. 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.

Zhang et al. Journal of Inequalities and Applications (2015) 2015:340

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Further, in [] the Brunn-Minkowski type inequalities for the dual quermassintegral quotient functions of star bodies were established as follows. Theorem A If K, L ∈ Son and reals i, j satisfy i ≤ n –  ≤ j ≤ n, then  j–i

 j–i

QW 

˜ L) i,j (K +

 j–i

≤ QW 

i,j (K)

+ QW 

i,j (L)

.

Here +˜ is the radial Minkowski sum. Theorem B If K, L ∈ Son and reals i, j satisfy i ≤  ≤ j ≤ n, then n– j–i

QW 

i,j

n– j–i

≤ QW  (K +˘ L)

i,j

n– j–i

+ QW  (K)

i,j (L)

.

Here +˘ is the radial Blaschke sum. Theorem C If K, L ∈ Son and reals i, j satisfy i ≤ – ≤ j ≤ n, then n+ j–i

n+ j–i

QW 

ˆ L) i,j (K +

V (K +L) ˆ

≤

n+ j–i

QW 

i,j (K)

V (K)

+

QW 

i,j (L)

V (L)

.

Here +ˆ is the harmonic Blaschke sum. Motivated by the work of Zhao, we give the following definition of dual mixed quermassintegral quotient function.  i (K, L) = Let K = · · · = Kn–i– = K , Kn–i = · · · = Kn– = B, Kn = L in (.), then we write W  V (K, n – i – ; B, i; L, ), where K appears n – i –  times, B appears i times and L appears  time. Here, we allow i to be any real and define as follows: For K, L ∈ Son and i any real,  i (K, L), of K and L are given by the dual mixed quermassintegrals, W  i (K, L) = W

 n

 ρ(K, u)n–i– ρ(L, u) dS(u).

(.)

Sn–

 i (K, K) = W  i (K). According to (.), we define Obviously, from (.) and (.), we have W the following. Definition . Let K, L ∈ Son and i, j ∈ R, the dual mixed quermassintegral quotient function, QW  i,j (K,L) , of K and L can be defined by QW  i,j (K,L) =

 i (K, L) W .  j (K, L) W

(.)

Obviously, if L = K , then (.) is just (.). The aim of this paper is to establish the following Brunn-Minkowski type inequalities for dual mixed quermassintegral quotient functions of star bodies. Theorem . For K, K  , L ∈ Son , if i ≤ n –  ≤ j < n – , then  j–i

QW 

i,j

 j–i

(K +˜ K  ,L)

≤ QW 

i,j

 j–i

+ QW  (K,L)

 i,j (K ,L)

;

(.)

Zhang et al. Journal of Inequalities and Applications (2015) 2015:340

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if n –  ≤ i < n –  < j, then  j–i

QW 

i,j

 j–i

(K +˜ K  ,L)

 j–i

≥ QW 

i,j

+ QW  (K,L)

 i,j (K ,L)

.

(.)

In each case, equality holds if and only if K and K  are dilates. Here +˜ is the radial Minkowski sum. Theorem . For K, K  , L ∈ Son , if i ≤  ≤ j < n – , then n– j–i

QW 

i,j

n– j–i

(K +˘ K  ,L)

≤ QW 

i,j

n– j–i

+ QW  (K,L)

 i,j (K ,L)

;

(.)

.

(.)

if  ≤ i < n –  < j, then n– j–i

n– j–i

QW 

˘ K  ,L) i,j (K +

≥ QW 

i,j (K,L)

n– j–i

+ QW 

 i,j (K ,L)

In each case, equality holds if and only if K and K  are dilates. Here +˘ is the radial Blaschke sum. Theorem . For K, K  , L ∈ Son , if i ≤ – ≤ j < n – , then n+ j–i

n+ j–i

QW 

ˆ K  ,L) i,j (K +

V (K

+ˆ K  )

≤

QW 

i,j (K,L)

V (K)

n+ j–i

+

QW 

 i,j (K ,L)

V (K  )

;

(.)

.

(.)

if – ≤ i < n –  < j, then n+ j–i

QW 

i,j

n+ j–i

(K +ˆ K  ,L)

V (K +ˆ K  )

≥

QW 

i,j (K,L)

V (K)

n+ j–i

+

QW 

 i,j (K ,L)

V (K  )

In each case, equality holds if and only if K and K  are dilates. Here +ˆ is the harmonic Blaschke sum.

2 Preliminaries For a compact set K in Rn which is star shaped with respect to the origin, we define the radial function ρK (u) = ρ(K, u) of K by ρ(K, u) = max{λ ≥  : λu ∈ K},

u ∈ Sn– .

If ρK 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 ρK (u)/ρL (u) is independent of u ∈ Sn– . For K , K ∈ Son , and λ , λ ≥  (not both ), the radial function of the radial Minkowski linear combination λ K +˜ λ K is given by Zhang (see []): ρ(λ K +˜ λ K , u) = λ ρ(K , u) + λ ρ(K , u).

(.)

Zhang et al. Journal of Inequalities and Applications (2015) 2015:340

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For K , K ∈ Son , and λ , λ ≥  (not both ), the radial Blaschke linear combination λ · K +ˇ λ · K is a star body whose radial function is given by Lutwak (see []): ρ(λ · K +ˇ λ · K , u)n– = λ ρ(K , u)n– + λ ρ(K , u)n– .

(.)

For K , K ∈ Son , and λ , λ ≥  (not both ), the harmonic Blaschke linear combination λ ◦ K +ˆ λ ◦ K is a star body whose radial function is given by Lutwak (see []): ρ(K , u)n+ ρ(K , u)n+ ρ(λ ◦ K +ˆ λ ◦ K , u)n+ = λ + λ . V (λ ◦ K +ˆ λ ◦ K ) V (K ) V (K )

(.)

3 Proofs of theorems According to a generalization of the Dresher inequality (see []), we get the reverse Dresher inequality. Lemma . (Dresher’s inequality) Let functions f , f , g , g ≥ , E is a bounded measurable subset in Rn . If p ≥  ≥ r ≥ , then        p  p–r   p  p–r (f + f )p dx p–r f dx f dx E E r E r ≤ + , r E (g + g ) dx E g dx E g dx

(.)

equality holds if and only if f /f = g /g . Lemma . (Reverse Dresher’s inequality) Let functions f , f , g , g ≥ , E is a bounded measurable subset in Rn . If  ≥ p >  > r, then        p  p–r   p  p–r (f + f )p dx p–r f dx f dx E E E  ≥  r +  r , r E (g + g ) dx E g dx E g dx

(.)

equality holds if and only if f /f = g /g . Proof of Lemma . If f , f , g , g ≥ , and  ≥ p >  > r, according to the Minkowski inequality,  p

 p

(f + f ) dx

 p f dx

≥

E

E

 r

 r

(g + g ) dx

 p

 ≥

E

E

gr

 + E

 r

 +

dx

p f dx

E

gr dx

 p ,

 r .

For  ≥ p >  > r, we have 

 (f + f ) dx ≥ p

E

E

p f dx



 (g + g ) dx ≤ r

E

 p

E

gr

 + E

 r dx

p f dx

 p p

 + E

gr dx

,

(.)

.

(.)

 r r

Zhang et al. Journal of Inequalities and Applications (2015) 2015:340

According to the Hölder inequality,

p–r p

Page 5 of 9

> , and (.), (.),

 p          p (f + f )p dx p–r (( E f dx) p + ( E f dx) p )p p–r E  ≥     r (( E gr dx) r + ( E gr dx) r )r E (g + g ) dx p  p  p   [( E f dx) p + ( E f dx) p ] p–r =   r   [( E gr dx) r + ( E gr dx) r ] p–r 

  p–r  p–r p

p

= E

× E

 p f dx

E

gr

–   p–r

  p–r  p  p–r p p–r

p

E



≥

 +

f dx

f dx



–(p–r) r

+

dx

E

   p–r

E

gr

–  p–r

–  –(p–r)  –r  p–r r p–r

 +

dx

gr dx

E

    p  p–r   p  p–r f dx f dx E =  r + E r . E g dx E g dx

p f dx

   p–r

E

gr dx

–  p–r

According to the equality condition of the Minkowski inequality and the Hölder inequality, equality holds in (.) if and only if f /f = g /g .  Proof of Theorem . From (.), for K, K  , L ∈ Son ,

  n–p– K +˜ K  , L = W n = =

 n  n



p

Sn–



Sn–



Sn–

ρK +˜ K  (u)ρL (u) dS(u)

p ρK (u) + ρK  (u) ρL (u) dS(u)  

p p p ρK (u)ρL (u) + ρK  (u)ρL (u) dS(u)

(.)

and

  n–r– K +˜ K  , L = W n





Sn–

 

r ρK (u)ρLr (u) + ρK  (u)ρLr (u) dS(u).

From (.), (.), and (.), for p ≥  ≥ r > , we have         p–r p p p  Wn–p– (K +˜ K  , L) p–r Sn– (ρK (u)ρL (u) + ρK (u)ρL (u)) dS(u) =     n–r– (K +˜ K  , L) W (ρ (u)ρ r (u) + ρ  (u)ρ r (u))r dS(u)

Sn–

K

 p

Sn– (ρK (u)ρL

≤



 r

Sn– (ρK (u)ρL

 +

K

L



(u))p dS(u)

L

  p–r

(u))r dS(u)  p



(u)ρL (u))p dS(u)

Sn– (ρK

(u)ρLr (u))r dS(u)

Sn– (ρK







  p–r  p n– ρK (u)ρL (u) dS(u) S =  r Sn– ρK (u)ρL (u) dS(u)

  p–r

(.)

Zhang et al. Journal of Inequalities and Applications (2015) 2015:340

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  p–r  p Sn– ρK  (u)ρL (u) dS(u)  + r Sn– ρK  (u)ρL (u) dS(u)

      Wn–p– (K, L) p–r Wn–p– (K  , L) p–r = + .  n–r– (K, L)  n–r– (K  , L) W W

(.)

According to the equality condition of inequality (.), we see that equality holds in (.) if and only if K and L, K  , and L are dilates, respectively. So K and K  are dilates. Let i = n – p – , j = n – r – , then p ≥  ≥ r >  and i ≤ n –  ≤ j < n –  are equivalent. This and (.) yield inequality (.) and its equality condition. Similarly, if  ≥ p >  > r, according to (.), (.), and (.), we have          Wn–p– (K, L) p–r Wn–p– (K  , L) p–r Wn–p– (K +˜ K  , L) p–r ≥ + ,  n–r– (K +˜ K  , L)  n–r– (K, L)  n–r– (K  , L) W W W

(.)

and equality holds if and only if K and K  are dilates. Let i = n – p – , j = n – r – , then (.) gives inequality (.) and its equality condition.  Proof of Theorem . From (.), for K, K  , L ∈ Son , we have

  n–p– K +˘ K  , L = W n = =

 n  n



p

Sn–



Sn–



Sn–

ρK +˘ K  (u)ρL (u) dS(u) p n–

n– ρK (u) + ρKn– ρL (u) dS(u)  (u) n– n– n–

p p p ρK (u)ρL (u) + ρKn– (u) n– dS(u)  (u)ρL

(.)

and

  n–r– K +˘ K  , L = W n =

 n

 ρKr +˘ K  (u)ρL (u) dS(u)

Sn–





Sn–

n–

n–

r ρKn– (u)ρL r (u) + ρKn– (u)  (u)ρL

r

n–

(.)

dS(u).

According to (.), (.), and (.), for p ≥ n –  ≥ r > , n– p–r

QW 

˘ K  ,L) n–p–,n–r– (K +

  n– Wn–p– (K +˘ K  , L) p–r =  n–r– (K +˘ K  , L) W 

n–

= 

n–



n–

≤ 

n–

p n– Sn– (ρK (u)ρL

n– r Sn– (ρK (u)ρL

p n– Sn– (ρK (u)ρL

p

(u)) n– dS(u)

n–

r

 n– p–r

r

(u)) n– dS(u)



n–

+ 

p

r (u) + ρKn– (u)) n– dS(u)  (u)ρL

n– r Sn– (ρK (u)ρL Sn–

n– p

(u) + ρKn– (u)) n– dS(u)  (u)ρL

p

p (ρKn– (u)) n– dS(u)  (u)ρL n–

n– r Sn– (ρK  (u)ρL

r

(u)) n– dS(u)

 n– p–r

 n– p–r

Zhang et al. Journal of Inequalities and Applications (2015) 2015:340

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 n–  n–   p p p–r p–r Sn– ρK (u)ρL (u) dS(u) Sn– ρK  (u)ρL (u) dS(u)   = + r r Sn– ρK (u)ρL (u) dS(u) Sn– ρK  (u)ρL (u) dS(u)  n–    n–  Wn–p– (K, L) p–r Wn–p– (K  , L) p–r = +  n–r– (K, L)  n–r– (K  , L) W W n– p–r

n– p–r

= QW 

n–p–,n–r– (K,L)

+ QW 

 n–p–,n–r– (K ,L)

.

Then n– p–r

n– p–r

n– p–r

≤ QW 

QW 

˘  ,L) n–p–,n–r– (K +K

n–p–,n–r– (K,L)

+ QW 

 n–p–,n–r– (K ,L)

.

(.)

According to the equality condition of inequality (.), we see that equality holds in (.) if and only if K and K  are dilates. Let i = n – p –  and j = n – r – , then p ≥ n –  ≥ r >  and i ≤  ≤ j < n –  are equivalent. This and (.) yield inequality (.) and its equality condition. Similarly, if n –  ≥ p >  > r, according to (.), (.), and (.), we have          Wn–p– (K +˜ K  , L) p–r Wn–p– (K, L) p–r Wn–p– (K  , L) p–r ≥ + ,  n–r– (K +˜ K  , L)  n–r– (K, L)  n–r– (K  , L) W W W

(.)

and equality holds if and only if K and K  are dilates. Let i = n – p –  and j = n – r – , then (.) gives inequality (.) and its equality condition.  Proof of Theorem . From (.), for K, K  , L ∈ Son , p  n–p– (K +ˆ K  , L)   ρK +ˆ K  (u)ρL (u) W = dS(u) V (K +K ˆ  )p/(n+) n Sn– V (K +ˆ K  )p/(n+) p  n+   n+ ρK +K  ˆ  (u) ρL (u) dS(u) = n Sn– V (K +ˆ K  ) p  n+   n+  ρK (u) ρKn+  (u) = + ρL (u) dS(u) n Sn– V (K) V (K  )

 = n



 Sn–

n+ p

n+

p (u) ρKn+ (u)ρL (u) ρKn+  (u)ρL +  V (K) V (K )

p  n+

(.)

dS(u)

and n+ n+  n+  r r  n–r– (K +ˆ K  , L)   (u) n+ ρK (u)ρL r (u) ρKn+ W  (u)ρL = dS(u). + V (K +K ˆ  )r/(n+) n Sn– V (K) V (K  )

(.)

According to (.), (.), and (.), for p ≥ n +  > r > , n+ p–r

QW 

ˆ K  ,L) n–p–,n–r– (K +

 n+  Wn–p– (K +ˆ K  , L) p–r =  n–r– (K +ˆ K  , L) W

= V K +ˆ K







n+

p ρ n+ (u) n+ p K ρ (u)) n+ V (K  ) L

dS(u)

n+ (u) n+ ρ n+ (u) n+ r ρK r r K n+ Sn– ( V (K) ρL (u) + V (K  ) ρL (u))

dS(u)

n+ (u) ρK p Sn– ( V (K) ρL



(u) +

 n+ p–r

Zhang et al. Journal of Inequalities and Applications (2015) 2015:340



≤ V K +ˆ K



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+ V K +ˆ K

n+

Sn– (V (K)

p – n+ ρK (u)ρL

Sn– (V (K)

– ρ n+ (u)ρ r K L







p

(u)) n+ dS(u)

n+



r

(u)) n+ dS(u) n+

Sn– (V (K

p  – n+ ) ρK  (u)ρL

Sn– (V (K

 )– ρ n+ (u)ρ r L K



 n+ p–r

p

(u)) n+ dS(u)

n+

 n+ p–r

r

(u)) n+ dS(u)

p   n+

n– (V (K)– ) n+ ρKp (u)ρL (u) dS(u) p–r = V K +ˆ K  S r – n+ ρ r (u)ρ (u) dS(u) L K Sn– (V (K) ) p   n+ p

Sn– (V (K  )– ) n+ ρK  (u)ρL (u) dS(u) p–r  + V K +ˆ K  r  – n+ ρ r (u)ρ (u) dS(u) L K Sn– (V (K ) )  n+   p V (K +ˆ K  ) Sn– ρK (u)ρL (u) dS(u) p–r  = r V (K) Sn– ρK (u)ρL (u) dS(u)  n+  p V (K +ˆ K  ) Sn– ρK  (u)ρL (u) dS(u) p–r  + r V (K  ) Sn– ρK  (u)ρL (u) dS(u)

=

  n+ p–r V (K +ˆ K  ) W n–p– (K, L)  n–r– (K, L) V (K) W  n+   p–r V (K +ˆ K  ) W n–p– (K , L) +    V (K ) Wn–r– (K , L) n+



= V K +ˆ K

 p–r  n–p–,n–r– (K,L)

QW  V (K)

n+ p–r

+

QW 

 n–p–,n–r– (K ,L)

V (K  )

 ,

i.e., n+ p–r

n+ p–r

QW 

n–p–,n–r–

V (K

(K +K ˆ  ,L)

+ˆ K  )

≤

n+ p–r

QW 

n–p–,n–r– (K,L)

V (K)

+

QW 

 n–p–,n–r– (K ,L)

V (K  )

.

(.)

According to the equality condition of inequality (.), we see that equality holds in (.) if and only if K and K  are dilates. Let i = n–p– and j = n–r –, then p ≥ n+ ≥ r >  and i ≤ – ≤ j < n– are equivalent. This and (.) yield inequality (.) and its equality condition. If n +  ≥ p >  > r, according to (.), (.), and (.), we have n+ p–r

QW 

n–p–,n–r–

n+ p–r

(K +K ˆ  ,L)

V (K +ˆ K  )

≥

n+ p–r

QW 

n–p–,n–r– (K,L)

V (K)

+

QW 

 n–p–,n–r– (K ,L)

V (K  )

,

(.)

with equality if and only if K and K  are dilates. Let i = n – p –  and j = n – r – , then (.) gives inequality (.) and its equality condition.  Competing interests The authors declare that they have no competing interests. Authors’ contributions The main idea of this paper was proposed by the second author. All authors contributed equally to the writing of the paper. All authors read and approved the final manuscript.

Zhang et al. Journal of Inequalities and Applications (2015) 2015:340

Acknowledgements This work was supported by the Natural Science Foundation of China (Grant Nos. 11371224 and 11102101). Received: 25 March 2015 Accepted: 16 October 2015 References 1. Lutwak, E: Dual mixed volumes. Pac. J. Math. 58, 531-538 (1975) 2. Zhao, CJ: On volume quotient functions. Indag. Math. 24, 57-67 (2013) 3. Zhang, GY: Centered bodies and dual mixed volumes. Trans. Am. Math. Soc. 345, 777-801 (1994) 4. Lutwak, E: Intersection bodies and dual mixed volumes. Adv. Math. 71, 232-261 (1988) 5. Lutwak, E: Centroid bodies and dual mixed volumes. Proc. Lond. Math. Soc. 60, 365-391 (1990) 6. Peˇcari´c, JE, Beeasck, PR: On Jessen’s inequality for convex functions. J. Math. Anal. Appl. 118, 125-144 (1986)

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