A Prékopa-Leindler type inequality of the L p Brunn-Minkowski inequality
aa r X i v : . [ m a t h . M G ] J u l A PR ´EKOPA-LEINDLER TYPE INEQUALITY OF THE L p BRUNN-MINKOWSKI INEQUALITY
YUCHI WU
Abstract.
In this paper, we prove a Pr´ekopa-Leindler type inequality of the L p Brunn-Minkowski inequality. It extends an inequality proved by Das Gupta [8] and Klartag [16],and thus recovers the Pr´ekopa-Leindler inequality. In addition, we prove a functional L p Minkowski inequality. Introduction
One of the cornerstones of the Brunn-Minkowski theory is the celebrated Brunn-Minkowskiinequality (see, e.g., the books by Gardner [12], Gruber [14] and Schneider [26] for references).It has had far reaching consequences for subjects quite distant from geometric convexity.For this, see the wonderful survey by Gardner [11]. By the middle of the last century, theBrunn-Minkowski inequality had been successfully extended to nonconvex sets.
Theorem 1.1.
Let A and B be nonempty bounded measurable sets in n -dimensional Euclideanspace R n such that (1 − λ ) A + λB is also measurable. Then V n ((1 − λ ) A + λB ) n ≥ (1 − λ ) V n ( A ) n + λV n ( B ) n . (1.1)Here, V n denotes the n -dimensional Lebesgue measure and A + B = { x + y : x ∈ A, y ∈ B } is the Minkowski sum of A and B .Denote by R f the integral of a function f on its domain with respect to the Lebesguemeasure. The following Pr´ekopa-Leindler inequality [22] is a functional type of the Brunn-Minkowski inequality (1.1). Theorem 1.2.
Let < λ < and let f, g and h be nonnegative integrable functions on R n satisfying h ((1 − λ ) x + λy ) ≥ f ( x ) − λ g ( y ) λ for all x, y ∈ R n . Then Z h ≥ (cid:18)Z f (cid:19) − λ (cid:18)Z g (cid:19) λ . Mathematics Subject Classification.
Key words and phrases. L p Brunn-Minkowski inequality, M -addition, s-concave.The author is supported by Project funded by China Postdoctoral Science Foundation 2019TQ0097, NSFC11671249 and a research grant from Shanghai Key Laboratory of PMMP 18dz2271000. The Pr´ekopa-Leindler inequality can quickly imply the Brunn-Minkowksi inequality (1.1),see section 7 in [11] for details. This connection helps trigger a fruitful development of func-tional analogues of several geometric parameters into the class of log-concave functions cur-rently undergoing (see [1, 2, 4, 5, 9, 17, 21, 18]).The following Borell-Brascamp-Lieb inequality [6, 7] generalizes the Pr´ekopa-Leindler in-equality, which is just the case α = 0. Theorem 1.3.
Let < λ < , let − /n ≤ α ≤ ∞ , and let f, g, and h be nonnegative integrablefunctions on R n satisfying h ((1 − λ ) x + λy ) ≥ M α ( f ( x ) , g ( y ) , λ ) (1.2) for all x, y ∈ R n . Then Z h ≥ M α/ ( nα +1) (cid:18)Z f, Z g, λ (cid:19) (1.3)Here, for a, b ≥ , if 0 < λ < α = 0 , we define M α ( a, b, λ ) = ((1 − λ ) a α + λb α ) /α if ab = 0 and M α ( a, b, λ ) = 0 if ab = 0; we also define M ( a, b, λ ) = a − λ b λ M −∞ ( a, b, λ ) = min { a, b } , and M ∞ ( a, b, λ ) = max { a, b } . Firey [10] generalizes Minkowski addition to L p addition for convex bodies (compact, convexsubsets with nonempty interiors) containing the origin and proves the L p Brunn-Minkowskiinequality, see also Lutwak [19].If
K, L are two convex bodies containing the origin, then the L p sum K + p L of K and L isdefined by h K + p L ( x ) p = h K ( x ) p + h L ( x ) p (1.4)for all x ∈ R n , where h A ( x ) = max { x · y : y ∈ A } is the support function of A . Here, wedenote by “ · ” the standard scalar product.Lutwak, Yang and Zhang [20] provided the explicit pointwise formula of L p addition : K + p L = (cid:8) (1 − λ ) /q x + λ /q y : x ∈ K, y ∈ L, ≤ λ ≤ (cid:9) (1.5)where q is the H¨older conjugate of p, i.e. p + q = 1 . When p = 1 , q = ∞ , and 1 /q is defined as0 . It is worth to point out that the pointwise L p addition (1.5) is suited for nonconvex sets.Lutwak, Yang and Zhang [20] established the following L p Brunn-Minkowski inequality forcompact sets:
Theorem 1.4.
Suppose p ≥ . If K and L are nonempty compact sets in R n , then V ( K + p L ) pn ≥ V ( K ) pn + V ( L ) pn . (1.6) PR´EKOPA-LEINDLER TYPE INEQUALITY OF THE L p BRUNN-MINKOWSKI INEQUALITY 3
We consider the following problem: is there a Pr´ekopa-Leindler type inequality that cansimply imply (1.6)?For any nonnegative function f on R n , we denote supp f by the support of f , i.e., the closureof { x ∈ R n : f ( x ) > } . In this paper, we will prove the following theorem: Theorem 1.5.
Let p ≥ , let s, µ, ω > , and let f, g, h : R n → [0 , ∞ ) be integrable withnonepmty supports. If for all x ∈ supp f, y ∈ supp g and λ ∈ [0 , ,h ((1 − λ ) q µ p x + λ q ω p y ) s ≥ (1 − λ ) q µ p f ( x ) s + λ q ω p g ( y ) s , (1.7) where q is the H¨older conjugate of p .Then, (cid:18)Z h (cid:19) pn + s ≥ µ (cid:18)Z f (cid:19) pn + s + ω (cid:18)Z g (cid:19) pn + s . (1.8)Let µ = ω = 1 and f = χ K , g = χ L , h = χ K + p L , where χ E denotes the characteristicfunction of E . It follows from (1.5) that (1.7) holds. Thus, we obtain (1.6) by letting s → + in (1.8). This shows that Theorem 1.5 can be viewed as a functional generalization of Theorem1.4.Theorem 1.5 for p = 1 is proved by Das Gupta [8] and Klartag [16], which recovers thePr´ekopa-Leindler inequality, see Corollary 2.2 in [16] for details.Let p = 1, s = α and µ + ω = 1 in Theorem 1.5. When f, g are positive in their supports andhave positive integrals, Theorem 1.5 and Theorem 1.3 coincide with each other for α ∈ (0 , ∞ ).This paper was completed and sumbitted to a Journal on February 18, 2020. We recentlyfound that [25] was published on Arxiv on April 20, 2020. Proposition 3.1 of [25] is the sameas our Theorem 1.5.This paper is organized as follows: In section 2, some basic facts and definitions for quickreference are provided. In section 3, some useful lemmas are given. In Section 4, we proveThreorem 1.5 and a functional L p Minkowski inequality.2.
Preliminaries
In this section, we collect some terminologies and notations. We recommend the books ofGardner [12], Gruber [14] and Schneider [26] as excellent references on convex geometry.For a nonempty set M ⊂ R , the M -addition of two sets K, L ⊂ R n is defined as K ⊕ M L = { ax + by : ( a, b ) ∈ M, x ∈ K, y ∈ L } . If M = { (1 , } , then M -addition is the classical Minkowski addition. For p ≥ q , if M = { ( a, b ) : a q + b q = 1 , a ≥ , b ≥ } , then M -addition is the explicitpointwise formula of L p addition (1.5). Y. WU
For 0 < s < ∞ , we say that f : R n → [0 , ∞ ) is s -concave if supp( f ) is nonempty, compactand convex, and f s is concave, i.e., for all x, y ∈ supp f and 0 ≤ λ ≤
1, we have f ( λx + (1 − λ ) y ) ≥ h λf ( x ) s + (1 − λ ) f ( y ) s i s .s -concave function has been studied by Avriel [3], Borell [6], Brascamp and Lieb [6, 7], Rotem[23, 24].For any function f : R n → [0 , ∞ ) and any integer s >
0, we define K f = n ( x, y ) ∈ R n + s = R n × R s : x ∈ supp f, | y | ≤ f ( x ) s o . (2.1)where, for given x ∈ R n and y ∈ R s , ( x, y ) are coordinates in R n + s . This set K f is nonemptyand convex if and only if f is s -concave. The volume of K f can be computed as V n + s ( K f ) = Z supp f κ s · (cid:16) f s ( x ) (cid:17) s = κ s Z f, (2.2)where κ s = π s/ Γ ( s +1 ) is the volume of the s -dimensional Euclidean unit ball.For positive number s , two functions f, g : R n → [0 , ∞ ) with nonempty supports andnonempty set M ⊂ R with nonnegative coordinates, we define the function f ⊕ M,s g as[ f ⊕ M,s g ] ( z ) = sup n(cid:16) af ( x ) s + bg ( y ) s (cid:17) s : x ∈ supp( f ) , y ∈ supp( g ) , z = ax + by, ( a, b ) ∈ M o when z ∈ supp f ⊕ M supp g. If z / ∈ supp f ⊕ M supp g , we set [ f ⊕ M,s g ] ( z ) = 0. This definitionis motivated by [13] and Lemma 3.2.For s >
0, two functions f, g : R n → [0 , ∞ ) with nonempty supports, p ≥ q , we define f ⊕ p,s g as f ⊕ M,s g by taking M = { ( a, b ) : a q + b q = 1 , a, b ≥ } , i.e.,[ f ⊕ p,s g ] ( z ) = sup n (cid:16) (1 − λ ) q f ( x ) s + λ q g ( y ) s (cid:17) s : x ∈ supp( f ) , y ∈ supp( g ) ,λ ∈ [0 , , z = (1 − λ ) q x + λ q y o when z ∈ supp f + p supp g. If z / ∈ supp f + p supp g , we set [ f ⊕ p,s g ] ( z ) = 0.For s > p ≥ λ > f : R n → [0 , ∞ ), we define the function λ × p,s f : R n → [0 , ∞ )as [ λ × p,s f ]( x ) = λ sp f ( λ − p x ) . (2.3)Note that, condition (1.7) implies h ≥ [ µ × p,s f ] ⊕ p,s [ ω × p,s g ] (2.4)pointwise.If s is an integer, it is easy to see that K λ × p,s f = λ p K f = { λ p y : y ∈ K f } . Thus, V n + s ( K λ × p,s f ) = λ n + sp V n + s ( K f ) . (2.5) PR´EKOPA-LEINDLER TYPE INEQUALITY OF THE L p BRUNN-MINKOWSKI INEQUALITY 5 some useful lemmas Let p ≥
1. If f is an s-concave function, so is λ × p,s f for λ >
0. In addition, if f, g are s -concave functions containing the origin in their supports, the function f ⊕ p,s g is also s -concave and contain the origin in its support, which can be deduced from Lemma 3.1 bytaking M = { ( a, b ) : a q + b q = 1 , a, b ≥ } . Lemma 3.1.
Let M ⊂ R be a nonempty compact set with nonnegative coordinates and M = { (0 , } . Let f, g be s-concave functions where s > . Then f ⊕ M,s g is s-concave if oneof the following conditions holds: (i) M is convex. (ii) supp f and supp g contain the origin.Proof. Set h = [ f ⊕ M,s g ] . Since M = { (0 , } , we get that h is not identically zero. This givesthat supp h is nonempty.We turn to prove the compactness of supp h . It is equivalent to proving that { z : h ( z ) > } is bounded. By the definition of h , we have { z : h ( z ) > } ⊂ supp f ⊕ M supp g. (3.1)Since supp f, M and supp g are all compact, supp f ⊕ M supp g is compact. Therefore, weobtain that { z : h ( z ) > } is bounded. Therefore, supp h is compact.We will prove that supp h = supp f ⊕ M supp g. (3.2)The compactness of supp f, M and supp g gives that supp f ⊕ M supp g is the closure of { x : f ( x ) > } ⊕ M { y : g ( y ) > } . It follows from our assumption of M that { x : f ( x ) > } ⊕ M { y : g ( y ) > } ⊂ { z : h ( z ) > } . Taking closure on both side givessupp f ⊕ M supp g ⊂ { z : h ( z ) > } = supp h. Now, (3.1) implies (3.2).Let z , z ∈ supp h and θ ∈ (0 , . For given ε > , there exist ( a , b ) , ( a , b ) ∈ M, x , x ∈ supp f, y , y ∈ supp g with z = a x + b y , z = a x + b y (3.3)such that h ( z ) s − ε ≤ a f ( x ) s + b g ( y ) s ,h ( z ) s − ε ≤ a f ( x ) s + b g ( y ) s . (3.4) Y. WU
It remains to prove that supp h is convex and h s is concave in its support. By (3.2), theformer is equivalent to proving that there exist ( a, b ) ∈ M, x ∈ supp f, y ∈ supp g such that(1 − θ ) z + θz = ax + by. (3.5)The latter is equivalent to proving h ((1 − θ ) z + θz ) s ≥ (1 − θ ) h ( z ) s + θh ( z ) s . (3.6)(i) Since supp f and supp g are convex, let x = (1 − λ ) x + λx , y = (1 − µ ) y + µy , (3.7)where λ, µ ∈ [0 ,
1] are to be determined. Then, by (3.3), (3.5) becomes (1 − θ ) a = (1 − λ ) a,θa = λa, (1 − θ ) b = (1 − µ ) b,θb = µb. (3.8)This is equvalent to solve the system ( (1 − θ ) a + θa = a, (1 − θ ) b + θb = b. Since M is convex and ( a , b ) , ( a , b ) ∈ M , one can find ( a, b ) ∈ M that satisfies this system.Thus, (3.5) holds.Therefore, by (3.5), the condition that a, b ≥
0, (3.7), (3.8), (3.3) and (3.4), we get h ((1 − θ ) z + θz ) s ≥ af ( x ) s + bg ( y ) s ≥ a (1 − λ ) f ( x ) s + aλf ( x ) s + b (1 − µ ) f ( y ) s + bµf ( y ) s =(1 − θ ) a f ( x ) s + (1 − θ ) b f ( y ) s + θa f ( x ) s + θb f ( y ) s ≥ (1 − θ ) h ( z ) s + θh ( z ) s − ε. Since ε is arbitrary, (3.6) holds.(ii) Since supp f and supp g are convex and contain the origin, let x = α ((1 − λ ) x + λx ) , y = β ((1 − µ ) y + µy ) , (3.9) PR´EKOPA-LEINDLER TYPE INEQUALITY OF THE L p BRUNN-MINKOWSKI INEQUALITY 7 where λ, µ, α, β ∈ [0 ,
1] are to be determined. Similarly, (3.5) becomes (1 − θ ) a = (1 − λ ) αa,θa = λαa, (1 − θ ) b = (1 − µ ) βb,θb = µβb. (3.10)This is equvalent to solve the system ( (1 − θ ) a + θa = αa, (1 − θ ) b + θb = βb. Set a = max { a , a } , b = max { b , b } , then one can find α, β ∈ [0 ,
1] that satisfy this system.Thus, (3.5) holds.It follows from f, g are s-concave and the assumption of (ii) that f ( αx ) s ≥ αf ( x ) s + (1 − α ) f ( o ) s ≥ αf ( x ) s , and g ( βy ) s ≥ βg ( y ) s + (1 − β ) g ( o ) s ≥ βg ( y ) s . Together with (3.5), the condition that a, b ≥
0, (3.9), (3.10), (3.3) and (3.4), we obtain h ((1 − θ ) z + θz ) s ≥ af ( x ) s + bg ( y ) s ≥ (1 − λ ) af ( αx ) s + λaf ( αx ) s + (1 − µ ) bf ( βy ) s + µbf ( βy ) s ≥ (1 − λ ) aαf ( x ) s + λaαf ( x ) s + (1 − µ ) bβf ( y ) s + µbβf ( y ) s =(1 − θ ) a f ( x ) s + (1 − θ ) b f ( y ) s + θa f ( x ) s + θb f ( y ) s ≥ (1 − θ ) h ( z ) s + θh ( z ) s − ε. Since ε is arbitrary, (3.6) holds. (cid:3) Lemma 3.2.
Let s > be an integer and let M ⊂ R be a nonempty set with nonnegativecoordinates. Then, for any two functions f, g : R n → [0 , ∞ ) with nonempty supports, K f ⊕ M K g ⊂ K f ⊕ M,s g . In addition, if M ⊂ { ( a, b ) : 0 ≤ a ≤ , ≤ b ≤ } , then int ( K f ⊕ M K g ) = int K f ⊕ M,s g , where int A denotes the interior of a set A . Y. WU
Proof.
First, we prove K f ⊕ M K g ⊂ K f ⊕ M,s g . (3.11)Let ( x, x ′ ) ∈ K f , ( y, y ′ ) ∈ K g , ( a, b ) ∈ M. Then | x ′ | ≤ f ( x ) s , | y ′ | ≤ g ( y ) s , a, b ≥ . By the definition of f ⊕ M,s g, we obtain[ f ⊕ M,s g ] ( ax + by ) s ≥ af ( x ) s + bg ( y s ) ≥ a | x ′ | + b | y ′ |≥ | ax ′ + by ′ | . That is, ( ax + by, ax ′ + by ′ ) ∈ K f ⊕ M,s g . Thus, (3.11) holds.It remains to prove int K f ⊕ M,s g ⊂ K f ⊕ M K g . (3.12)Without loss of generality, we can assume that int K f ⊕ M,s g is nonempty. Let ( z, z ′ ) ∈ int K f ⊕ M,s g . Then, there exists ε > , such that[ f ⊕ M,s g ] ( z ) s − ε > | z ′ | . By the definition of f ⊕ M,s g, there exist ( a, b ) ∈ M, x ∈ supp f, y ∈ supp g with z = ax + by such that af ( x ) s + bg ( y ) s > [ f ⊕ M,s g ] ( z ) s − ε. Therefore, we get af ( x ) s + bg ( y ) s > | z ′ | . (3.13)Set x ′ = af ( x ) s af ( x ) s + bg ( y ) s z ′ , y ′ = bg ( y ) s af ( x ) s + bg ( y ) s z ′ . Since 0 ≤ a, b ≤ , ( x, x ′ ) ∈ K f , ( y, y ′ ) ∈ K g . Thus ( z, z ′ ) = a ( x, x ′ ) + b ( y, y ′ ) ∈ K f ⊕ M K g . Therefore, (3.12) holds. (cid:3)
Remark . Let p ≥
1. By Lemma 3.2, we can conclude that for any µ, ω > s > K [ µ × p,s f ] + p K [ ω × p,s g ] ⊂ K [ µ × p,s f ] ⊕ p,s [ ω × p,s g ] and int (cid:0) K [ µ × p,s f ] + p K [ ω × p,s g ] (cid:1) = int K [ µ × p,s f ] ⊕ p,s [ ω × p,s g ] . (3.14) PR´EKOPA-LEINDLER TYPE INEQUALITY OF THE L p BRUNN-MINKOWSKI INEQUALITY 9 proof of main theorems We turn to prove Theorem 1.5. proof of Theorem 1.5.
First assume that s is an integer.The L p Brunn-Minkowski inequality (1.6) for ( n + s )-dimensional sets, (2.5) and Remark3.1 implies V ∗ n + s ( K [ µ × p,s f ] ⊕ p,s [ ω × p,s g ] ) pn + s ≥ V n + s ( K [ µ × p,s f ] + p K [ ω × p,s g ] ) pn + s ≥ µV n + s ( K f ) pn + s + ωV n + s ( K g ) pn + s , where V ∗ n + s stands for outer Lebesgue measure (the set K [ µ × p,s f ] ⊕ p,s [ ω × p,s g ] may be non-measurable).By (2.2), this is equivalent to (cid:18)Z ∗ R n [ µ × p,s f ] ⊕ p,s [ ω × p,s g ] (cid:19) pn + s ≥ µ (cid:18)Z R n f (cid:19) pn + s + ω (cid:18)Z R n g (cid:19) pn + s , (4.1)where R ∗ is the outer integral. Note that R h = R ∗ h . Thus, it follows from (2.4) and (4.1)that (1.8) holds.Next assume that s = lt is rational.Note that, by H¨older’s inequality (See [15]) and (1.7), for any x , · · · , x t , y , · · · , y t ∈ R n ,(1 − λ ) q µ p t Y i =1 f ( x i ) ts + λ q ω p t Y i =1 g ( y i ) ts ≤ t Y i =1 (cid:16) (1 − λ ) q µ p f ( x i ) s + λ q ω p g ( y i ) s (cid:17)! t ≤ t Y i =1 h (cid:16) (1 − λ ) q µ p x i + λ q ω p y i (cid:17) ts . (4.2)For a function r : R n → [0 , ∞ ) , we define ˜ r : R nt → [0 , ∞ ) by˜ r ( x ) = ˜ r ( x , . . . , x t ) = t Y i =1 r ( x i )where x = ( x , · · · , x t ) ∈ ( R n ) t are coordinates in R nt . Thus, (4.2) implies that for x, y ∈ R nt ,˜ h ((1 − λ ) q µ p x + λ q ω p y ) ts ≥ (1 − λ ) q µ p ˜ f ( x ) ts + λ q ω p ˜ g ( y ) ts . Now, ts = l is integer. This gives that (cid:18)Z h (cid:19) pn + s = (cid:18)Z ˜ h (cid:19) pt ( n + s ) ≥ µ (cid:18)Z ˜ f (cid:19) pt ( n + s ) + ω (cid:18)Z ˜ g (cid:19) pt ( n + s ) = µ (cid:18)Z f (cid:19) pn + s + ω (cid:18)Z g (cid:19) pn + s . The case that s is irrational follows by a standard approximation argument. (cid:3) Using Theorem 1.5, we will prove a functional L p Minkowski inequality.For p ≥ , s > f, g : R n → [0 , ∞ ) with nonempty support, we define˜ S p,s ( f ; g ) = pn + s lim ε → + R [ f ⊕ p,s ( ε × p,s g )] − R fε whenever the integrals are defined and the limit exists. The motivation of this definition isfrom the defintion of L p mixed volume, see [19]. Corollary 4.1.
Let s > and f, g : R n → [0 , ∞ ) be integrable functions with nonemptysupport such that ˜ S p,s ( f ; g ) exists. Then, ˜ S p,s ( f ; g ) ≥ (cid:18)Z f (cid:19) − pn + s (cid:18)Z g (cid:19) pn + s . (4.3) If s is an integer, f = λ × p,s g, where λ > , and g is s-concave such that supp g has nonemptyinterior and contains the origin, then equality holds.Proof. By (4.1), Z [ f ⊕ p,s ( ε × p,s g )] ≥ (cid:18)Z f (cid:19) pn + s + ε (cid:18)Z g (cid:19) pn + s ! n + sp (4.4) ≥ (cid:18)Z f (cid:19) + ε · n + sp (cid:18)Z f (cid:19) − pn + s (cid:18)Z g (cid:19) pn + s . Since ˜ S p,s ( f ; g ) exists, the definition of ˜ S p,s ( f ; g ) implies the desired inequality.If s is an integer, f = λ × p,s g and g is s-concave with supp g such that supp g has nonemptyinterior and contains the origin, then K f and K [ ε × p,s g ] are convex bodies containing the originfor ε >
0, and K f = λ p K g .By (1.4) and the homogeneity of support function, h p K f + p K [ ε × p,sg ] ( x ) = h p K f ( x ) + p h p K [ ε × p,sg ] ( x ) = h pλ p K g ( x ) + h pε p K g ( x ) = ( λ + ε ) h p K g ( x )for all x ∈ R n , which shows that K f + p K [ ε × p,s g ] = ( λ + ε ) p K g is a convex body. It followsfrom Remark 3.1 thatint K f ⊕ p,s [ ε × p,s g ] = int (cid:0) K f + p K [ ε × p,s g ] (cid:1) = int (cid:16) ( λ + ε ) p K g ) (cid:17) . Since these sets are convex, we get that V n + s (cid:0) K f ⊕ p,s [ ε × p,s g ] (cid:1) = ( λ + ε ) n + sp V n + s ( K g ) . PR´EKOPA-LEINDLER TYPE INEQUALITY OF THE L p BRUNN-MINKOWSKI INEQUALITY 11
Therefore, by (2.2), Z [ f ⊕ p,s ( ε × p,s g )] − Z f = V n + s (cid:0) K f ⊕ p,s [ ε × p,s g ] (cid:1) − V n + s ( K f ) κ s = (cid:16) ( λ + ε ) n + sp − λ n + sp (cid:17) V n + s ( K g ) κ s = (cid:16) ( λ + ε ) n + sp − λ n + sp (cid:17) Z g. Now, by the definition ˜ S p,s ( f ; g ) and the condition that f = λ × p,s g = λ sp f ( xλ p ), we have˜ S p,s ( f ; g ) = pn + s lim ε → + ( λ + ε ) n + sp − ε n + sp ε Z g = λ n + sp − Z g = (cid:18)Z f (cid:19) − pn + s (cid:18)Z g (cid:19) pn + s . (cid:3) References [1] D. Alonso-Guti´errez, B. Gonz´alez Merino, C. H. Jim´enez, and R. Villa. Rogers-Shephard inequality forlog-concave functions.
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School of Mathematics science, East China Normal University, Shanghai 200241,China; Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, Shanghai200241, China
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