Inequalities of Aleksandrov body

Transcription

1 RESEARCH Ope Access Iequalities of Aleksadrov body Hu Ya 1,2* ad Jiag Juhua 1 * Correspodece: huya12@126. com 1 Departmet of Mathematics, Shaghai Uiversity, Shaghai , Chia Full list of author iformatio is available at the ed of the article Abstract A ew cocept of p-aleksadrov body is firstly itroduced. I this paper, p-bru- Mikowski iequality ad p-mikowski iequality o the p-aleksadrov body are established. Furthermore, some pertiet results cocerig the Aleksadrov body ad the p-aleksadrov body are preseted Mathematics Subject Classificatio: 52A20 52A40 Keywords: Aleksadrov body, p-aleksadrov body, Bru-Mikowski iequality, Mikowski iequality 1 Itroductio TheotioofAleksadrovbodywasfirstlyitroducedbyAleksadrovtosolve Mikowski problem i 1930s i [1]. The Aleksadrov body establishes the relatioship betwee the covex body cotaiig the origi ad the positive cotiuous fuctios ad characterizes the covex body by meas of the positive cotiuous fuctios. The Aleksadrov body ot oly be used to solve Mikowski problem but also has a wide rage of applicatios i other areas of Covex Geometric Aalysis. The, the Aleksadrov body is a essetial matter i the Bru-Mikowski theory ad plays a importat role i Covex Geometric Aalysis. I recet years, Ball [2], Garder [3,4], Lutwak [5-10], Klai [11], Hug [12], Haberl [13], Scheider [14], Stacu [15], Umaskiy [16] ad Zhag [17] have give cosiderable attetio to the Bru-Mikowski theory ad their various geeralizatios. The purpose of this paper is to study comprehesively the Aleksadrov body, ad most importatly, the L p aalogues of Aleksadrov body become a major goal. Here, a ew geometric body is firstly itroduced, called p-aleksadrov body. Meawhile, p-bru-mikowski iequality ad p-mikowski iequality for the p-aleksadrov bodies associated with positive cotiuous fuctios are established. Furthermore, some related results, icludig of the uiqueess results, the covergece results for the Aleksadrov bodies ad the p-aleksadrov bodies associated with positive cotiuous fuctios, are preseted. Let K deote the set of covex bodies (compact, covex subsets with o-empty iteriors) i Euclidea space R, K 0 deote the set of covex bodies cotaiig the origi i their iteriors. Let V (K) deote the -dimesioal volume of body K, for the stadard uit ball B i R, deote ω = V (B), ad let S -1 deote the uit sphere i R. Let C + (S -1 ) deote the set of positive cotiuous fuctios o S -1,edowedwith the topology derived from the max orm. Give a fuctio f ÎC + (S -1 ), the set 2011 Ya ad Juhua; licesee Spriger. This is a Ope Access article distributed uder the terms of the Creative Commos Attributio Licese ( which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited.

2 Page 2 of 13 {K K 0 : h K f } has a uique maximal elemet, the we deoted the Aleksadrov body associated with the fuctio f ÎC + (S -1 )by K(f )=max{k K 0 : h K f }. ThevolumeofbodyK(f) is deoted by V (K(f)). Followig Aleksadrov (see [18]), defie the volume V (f) ofafuctiof as the volume of the Aleksadrov body associated with the positive cotiuous fuctio f. I this paper, we geeralize ad improve Bru-Mikowski iequality ad Mikowski iequality for the Aleksadrov bodies associated with positive cotiuous fuctios ad establish p-mikowski iequality ad p-bru-mikowski iequality for the Aleksadrov bodies ad the p-aleksadrov bodies associated with positive cotiuous fuctios as follows. Theorem 1 If Q K0, f ÎC + (S -1 ), ad p 1, the V p (Q, f ) V(Q) ( p)/ V(f ) p/, (1:1) with equality if ad oly if there exists a costat c >0 such that h Q = cf, almost everywhere with respect to S(Q, )o S -1. Theorem 2 If p 1, f, g ÎC + (S -1 ), ad l, μ ÎR +, the V(λ f + p μ g) p λv(f ) p + μv(g) p, (1:2) with equality if ad oly if there exists a costat c >0 such that f = cg, almost everywhere with respect to S(K(f ), ) o S -1. The other aim of this paper is to establish the followig iequality for the Aleksadrov bodies ad the p-aleksadrov bodies associated with positive cotiuous fuctios. Theorem 3 If K(f ), K(g) Ke, are the Aleksadrov bodies associated with the fuctios f, g ÎC + (S -1 ), ad p 1, the V(f + p g) p V(f ) p + V(g) p, (1:3) with equality if ad oly if there exists a costat c >0 such that f = cg, almost everywhere with respect to S(K(f ), ) o S -1. More iterrelated otatios, defiitios, ad their backgroud materials are exhibited i the ext sectio. 2 Defiitio ad otatio The settig for this paper is -dimesioal Euclidea space R.LetK deote the set of covex bodies (compact, covex subsets with o-empty iteriors), K 0 deote the subset of K that cotais the origi i their iteriors, ad K e deote the subset of K

3 Page 3 of 13 that are cetered i R. We reserve the letter u for uit vector ad the letter B for the uit ball cetered at the origi. The surface of B is S -1,adthevolumeofB deotes ω. For ÎGL(), let t, -1,ad -t, deote the traspose, iverse, ad iverse of the traspose of, respectively. If K Î K, the support fuctio of K, h K = h(k, ):R (0, ), is defied by h(k, u) =max{u x : x K}, u S 1, where u x deotes the stadard ier product of u ad x. The set K will be viewed as equipped with the usual Hausdorff metric, d, defied by d(k, L) = h K -h L, where is the sup (or max) orm o the space of cotiuous fuctios o the uit sphere, C(S -1 ). For K, L Î K,ada, b 0 (ot both zero), the Mikowski liear combiatio, ak + bl Î K is defied by h(αk + βl, ) = αh(k, ) + βh(l, ). (2:1) Firey itroduced, for each real p 1, ew liear combiatios of covex bodies: For K, L K 0,ada, b 0 (ot both zero), the Firey combiatio, α K+ p β L K 0 whose support fuctio is defied by (see [19]) h(α K+ p β L, ) p = αh(k, ) p + βh(l, ) p. (2:2) Obviously, a K= a 1/p K. For K, L Î K,ada, b 0 (ot both zero), by the Mikowski existece theorem (see [3,14]), there exists a covex body a K + b L Î K, such that S(α K + β L, ) = αs(k, ) + βs(l, ), (2:3) where S(K, ) deotes the surface area measure of K, ad the liear combiatio a K + b Lis called a Blaschke liear combiatio. Lutwak geeralized the otio of Blaschke liear combiatio i [5]: For K, L Ke, ad p 1, defie K+ p L Ke by S p (K+ p L, ) =S p (K, )+S p (L, ). (2:4) The existece ad uiqueess of K + p L are guarateed by Mikowski s existece theorem i [5]. 2.1 Mixed volume ad p-mixed volume If K i Î K (I = 1, 2,..., r) adl i (i = 1, 2,..., r) are oegative real umbers, the of fudametal importace is the fact that the volume of r i=1 λ ik i is a homogeeous polyomial i l i give by r V( λ i K i )= λ i1...λ i V(K i1...k i ), (2:5) i=1 i 1,...,i where the sum is take over all -tuples (i 1,... i ) of positive itegers ot exceedig r. The coefficiet V(K i1...k i ), which is called the mixed volume of K i1...k i, depeds oly o the bodies K i1...k i ad is uiquely determied by (2.5). If K 1 =... K -i = K ad K -i +1 =...=K = L, the the mixed volume V (K 1... K ) is usually writte as V i (K, L).

4 Page 4 of 13 Let r = 1 i (2.5), we see that V(λ 1 K 1 )=λ 1 V(K 1). Further, from (2.5), it follows immediately that V(K + εl) V(K) V 1 (K, L) = lim. ε 0 ε Aleksadrov (see [1]) ad Fechel ad Jesse (see [20]) have show that correspodig to each K Î K, there is a positive Borel measure, S(K, )os -1, called the surface area measure of K, such that V 1 (K, Q) = 1 S 1 h(q, u)ds(k, u), (2:6) for all Q Î K. For p 1, the p-mixed volume V p (K, L) ofk, L K 0, was defied by (see [5]) p V V(K+ p ε L) V(K) p(k, L) = lim. ε 0 ε That the existece of this limit was demostrated i [5]. It was also show i [5], that correspodig to each K K0, there is a positive Borel measure, S p (K, )os -1 such that V p (K, Q) = 1 h(q, u) p ds p (K, u), (2:7) S 1 for all Q K 0.IttursoutthatthemeasureS p (K, ) is absolutely cotiuous with respect to S(K, ) ad has Rado-Nikodym derivative, ds p (K, ) ds(k, ) = h(k, )1 p. (2:8) From (2.7) ad (2.8), we have V p (K, Q) = 1 h(q, u) p h(k, u) 1 p ds(k, u), (2:9) S 1 where S(K, )=S 0 (K, ) is the surface area measure of K. Obviously, for each K K 0, p 1, V p (K, K) =V(K). (2:10) 2.2 Aleksadrov body If a fuctio f Î C + (S -1 ) (deoted the set of positive cotiuous fuctios o S -1 ad edowed with the topology derived from the max orm), the set {K K 0 : h K f } has a uique maximal elemet, the the Aleksadrov body associated with the fuctio f is deoted by

5 Page 5 of 13 K(f )=max{k K 0 : h K f }. (2:11) From (2.11) ad (2.1), we have: If f, g Î C+(S -1 ), ad l, μ 0 (ot both zero), the K(λf + μg) λk(f ) + μk(g). (2:12) Obviously, if f is the support fuctio of a covex body K, the the Aleksadrov body associated with f is K.V (K(f )) deotes the volume of body K(f ). Followig Aleksadrov (see [18]), defie the volume V (f ) of a fuctio f as the volume of the Aleksadrov body associated with the positive fuctio f. For Q K0, f Î C + (S -1 ), ad p 1, V p (Q, f ) is defied by (see [5]) V p (Q, f )= 1 f (u) p h(q, u) 1 p ds(q, u), (2:13) S 1 Obviously, V p (K, h K )=V (K), for all K K p-aleksadrov body Defiitio 1 Let f, g Î C + (S -1 ), p 1, ad defie ε> mi{f (u) p /g(u) p, u S 1 }, f + p ε g =(f p + εg p ) 1/p. (2:14) The, the set {Q K 0 : h(q, ) (f p + g p ) 1/p }, has a uique maximal elemet. We deote the p-aleksadrov body associated with the fuctio f + p g Î C + (S -1 )by K p (f + p g)=max{q K 0 : h(q, ) (f p + g p ) 1/p }, (2:15) for p 1. The volume of body K p (f + p g) is deoted by V (K p (f + p g)), ad defie the volume V (f + p g)ofthefuctiof + p g as the volume of the p-aleksadrov body associated with the positive fuctio f + p g. From (2.2), we have the followig result: If f, g Î C + (S -1 ), ad p 1, the K p (f + p g) K(f )+ p K(g). (2:16) We ote that the equality coditio i (2.16) is clearly holds, whe f ad g are the support fuctios of K(f )adk(g), respectively. Also, the case p = 1 of (2.16) is just (2.12). 3 of the mai results The followig Lemmas will be required to prove our mai theorems. Lemma 1 [5]If K(f ) is the Aleksadrov body associated with f Î C + (S -1 ), the h K(f) = falmost everywhere with respect to the measure S(K(f ), ) o S -1.

6 Page 6 of 13 Obviously, if K(f ) is the Aleksadrov body correspodig to a give fuctio f Î C + (S -1 ), its support fuctio has the property that 0 <h K f ad V (f )=V (h K(f ) ). Lemma 2 [5]If p 1, K(f ) is the Aleksadrov body associated with f Î C + (S -1 ), the V (f )=V (K(f )) = V p (K(f ), f ), i.e V(f )= 1 S 1 h(k(f ), u)ds(k(f ), u). Lemma 3 [5]If K K 0, f Î C + (S -1 ), the, for p 1, p V V(h K + p ε f ) V(h K ) p(k, f ) = lim. (3:1) ε 0 ε We get the followig Bru-Mikowski iequality for the Aleksadrov bodies associated with positive cotiuous fuctios. Lemma 4 If f, g Î C + (S -1 ), ad l, μ ÎR +, the V(λf + μg) 1/ λv(f ) 1/ + μv(g) 1/, (3:2) with equality if ad oly if there exist a costat c >0 ad t 0, such that f = cg + t, almost everywhere with respect to S(K(f ), ) o S -1. Sice f, g Î C + (S -1 ), from (2.11), (2.12) ad the Bru-Mikowski iequality (see [21]), we get V(K(λf + μg)) 1/ V(λK(f )+μk(g)) 1/ λv(k(f )) 1/ + μv(k(g)) 1/. (3:3) The equality coditio i (3.3) is that f, g are the support fuctios of K(f ) ad K(g) which are homothetic, respectively. From Lemma 1 ad Lemma 2, we get the followig result V(λf + μg) 1/ λv(f ) 1/ + μv(g) 1/, (3:4) with equality if ad oly if there exist a costat c>0 ad t 0, such that f = cg + t, almost everywhere with respect to S(K(f ), ) o S -1. A immediate cosequece of the defiitio of a Firey liear combiatio, ad the itegral represetatio (2.13), is that for Q K 0, the p-mixed volume V p (Q, ) :C + (S 1 ) (0, ) is Firey liear.

7 Page 7 of 13 Lemma 5 If p 1, Q K 0, f, g Î C + (S -1 ), ad l, μ Î R +, the V p (Q, λ f + p μ g) =λv p (Q, f )+μv p (Q, g). (3:5) From (2.13), (2.14), we obtai V p (Q, λ f + p μ g) = 1 (λ f + p μ g) p h(q, u) 1 p ds(q, u) S 1 = 1 S 1 (λf p + μg p )h(q, u) 1 p ds(q, u) = λv p (Q, f )+μv p (Q, g). Ithefollowig,wewillprovethep-Mikowski iequality for the Aleksadrov bodies associated with positive cotiuous fuctios. of Theorem 1. Firstly, let p = 1 i Lemma 3, we get let ε = Let V(h Q + εf ) V(h Q ) V 1 (Q, f ) = lim, ε 0 ε t 1 t, we have V 1 (Q, f ) = lim t 0 V((1 t)h Q + tf ) (1 t) V(h Q ) t(1 t) 1 V((1 t)h Q + tf ) V(h Q ) (1 (1 t) )V(h Q ) = lim + lim t 0 t t 0 t V((1 t)h Q + tf ) V(h Q ) = lim + V(h Q ). t 0 t f (t) =V((1 t)h Q + tf ) 1/, 0 t 1, we see that f (0) = V 1(Q, f ) V(h Q ) V(h Q ) 1. From Lemma 4, we kow that f is cocave, i.e. Thus, V 1 (Q, f ) V(h Q ) V(h Q ) 1 V(f ) 1 V(h Q ) 1. V 1 (Q, f ) V(Q) 1 V(f ) 1. (3:6) Accordig to the equality coditio i iequality (3.3), ad usig Lemma 1 ad Lemma 2, we have the equality holds i iequality (3.6), if ad oly if there exist a

8 Page 8 of 13 costat c>0adt 0, such that h Q = cf + t, almost everywhere with respect to S (Q, )os -1. Secodly, from the Hölder iequality (see [22]), together with the itegral represetatios (2.13) ad (2.6), we obtai V p (Q, f )= 1 f (u) p h(q, u) 1 p ds(q, u) S 1 V 1 (Q, f ) p V(Q) 1 p, whe this combied with iequality (3.6), we have V p (Q, f ) V(Q) p V(f ) p. (3:7) To obtai the equality coditios, we ote that there is equality i Hölder s iequality precisely whe V 1 (Q, f )h Q = V (Q)f, almost everywhere with respect to the measure S(Q, )os -1. Combiig the equality coditios i (3.6), ad usig Lemma 1, it shows that the equality holds if ad oly if there exists a costat c>0suchthath Q = cf, almost everywhere with respect to S(Q, )os -1. Usig the above Lemmas ad Theorem 1, we ca get the followig Corollaries describig the uiqueess results. Corollary 1 Suppose K, L K 0, ad F C + (S -1 ) is a class of fuctios such that h K,h L Î F. (i) If p>1, ad V p (K, f )=V p (L, f ), for all f Î F, the K = L. (ii) If p =, ad V p (K, f ) V p (L, f ), for all f Î F, the K ad L are dilates, ad hece V p (K, f )=V p (L, f ), for all f C + (S 1 ). If p>1, take f = h K, ad from (2.13), Lemma 2 ad Theorem 1, we get Hece, V p (K, f )=V p (K, h K )=V p (L, h K ) V(L) p V(h K ) p. V(K) V(L). Similarly, take f = h L, we get V(L) V(K). I view of the equality coditios of Theorem 1, we obtai that K = L. If = p, the hypothesis together with Theorem 1, we have V p (K, f ) V p (L, f ) V(L) p V(f ) p, with equality i the right iequality implyig that L ad K(f ) are dilates. Take f = h K, sice = p, the terms o the left ad right are idetical, ad thus, K ad L must dilates; hece, V p (K, f )=V p (L, f ), for all f C + (S 1 ).

9 Page 9 of 13 Corollary 2 Suppose f, g Î C + (S -1 ), ad F C + (S -1 ) is a class of fuctios such that f, g Î F. If p >1, ad V p (Q, f )=V p (Q, g), for all h Q F, the f = g almost everywhere o S -1. Sice f, g Î C + (S -1 ), accordig to (2.11), we deote two Aleksadrov bodies K(f )ad K(g). From the hypothesis, takig Q = K(f ), ad usig Lemma 2 ad Theorem 1, we get the, V p (K(f ), f )=V(f )=V p (K(f ), g) V(K(f )) p V(g) p, V(f ) V(g). Similarly, take Q = K(g), we get V(g) V(f ). From the equality coditios of Theorem 1, we obtai K(f ) = K(g). I view of the defiitio of Aleksadrov body, ad usig Lemma 1, the f = g, almost everywhere o S 1. Corollary 3 Suppose p>1, ad f, g Î C + (S -1 ), such that S p (K(f ), ) S p (K(g), ). (i) If V (f ) V (g), ad p <, the f = g almost everywhere o S -1. (ii) If V (f ) V (g), ad p >, the f = g almost everywhere o S -1. Suppose a fuctio Î C + (S -1 ), ad p>1, sice S p (K(f ), ) S p (K(g), ), it follows from the itegral represetatio (2.13) ad (2.8) that V p (K(f ), φ) V p (K(g), φ), for all φ C + (S 1 ). As before, take = h K(g), from Lemma 1, Lemma 2, ad Theorem 1, we get V(f ) p V(g) p. Applyig the hypothesis, ad from the defiitio of the Aleksadrov body ad Lemma 1, we obtai the desired results. Corollary 4 Suppose p 1, f, g Î C + (S -1 ), ad F C + (S -1 ) is a class of fuctios such that f, g Î F. If

10 Page 10 of 13 V p (K, f ) V(f ) = V p(k, g), for all h K F, V(g) the f = g almost everywhere o S -1. Accordig to (2.11), we deote two Aleksadrov bodies K(f )adk(g). From the hypothesis, takig K = K(f ) ad K = K(g), ad combiig with Lemma 2 ad Theorem 1, respectively, we obtai V(g) V(f ) ad V(f ) V(g), Hece, i view of the equality coditios of Theorem 1, the defiitio of Aleksadrov body, ad Lemma 1, we get the desired result f = g, almost everywhere o S 1. Now, the p-bru-mikowski iequality for the p-aleksadrov bodies ad the Aleksadrov bodies associated with positive cotiuous fuctios is established as followig. of Theorem 2. From Lemma 5 ad Theorem 1, we get V p (Q, λ f + p μ g) =λv p (Q, f )+μv p (Q, g) V(Q) p [λv(f ) p + μv(g) p ], with equality if ad oly if K(f) ad K(g) are dilates of Q. Now, take Q = K p (l f + p μ g), use (2.10), ad recall V (f )=V (K(f )) = V p (K(f ), f ), we have V(λ f + p μ g) p λv(f ) p + μv(g) p. Also, we ote that the equality holds, if ad oly if K(f )adk(g) are dilates. Usig Lemma 1, we get the coditio of equality holds if ad oly if there exists a costat c >0 such that f = cg, almost everywhere with respect to S(K(f ), ) os -1. The, we will prove Theorem 3 by usig the geeralized Blaschke liear combiatio. of Theorem 3. Suppose a fuctio Î C + (S -1 ), ad p 1, from the itegral represetatio (2.13), (2.8), ad (2.4), it follows that for K(f ), K(g) K e, V p (K(f )+ p K(g), φ) =V p (K(f ), φ)+v p (K(g), φ), (3:8) which together with Theorem 1, yields V p (K(f )+ p K(g), φ) V(φ) p [V(K(f )) p with equality if ad oly if K(f); K(g) ad K( ) are dilates. + V(K(g)) p ], (3:9) Now, take φ = h K(f )+p K(g), recall Vp(K, h K )=V (K ), ad from Lemma 2, we get V(K(f )+ p K(g)) p V(f ) p + V(g) p.

11 Page 11 of 13 I view of (2.16), we have V(K p (f + p g)) V(K(f )+ p K(g)). Hece, we get V(K p (f + p g)) p V(K(f )+ p K(g)) p V(f ) p + V(g) p. From Lemma 2 agai, we obtai V(f + p g) p V(f ) p + V(g) p. I view of the equality coditio (3.9), ad from Lemma 1, we get the equality holds if ad oly if there exists a costat c>0 such that f = cg, almosteverywherewith respect to S(K(f ), ) o S -1. Remark 1 The case p = 1 of the iequality of Theorem 3 is V(f + g) 1 V(f ) 1 + V(g) 1, (3:10) with equality if ad oly if there exists a costat c>0 such that f = cg, almost everywhere with respect to S(K(f ), ) o S -1. Theaboveiequality(3.10)isjusttheKeser-Süss iequality type for the Aleksadrov bodies associated with positive cotiuous fuctios. Actually, from these above proofs, we see Bru-Mikowski iequality, Mikowski iequality, ad Kesser-Süss iequality are equivalet. 4 Covergece of Aleksadrov body I this sectio, we establish a coverget result about the Aleksadrov bodies associated with positive cotiuous fuctios. The followig Lemmas will be required to prove our mai result. Lemma 6 [7]If p 1, ad K i is a sequece of bodies i K 0, such that K i K 0 K 0,theS p (K i, ) S p (K 0, ), weakly. Lemma 7 Suppose K i K K 0, ad f i f Î C + (S -1 ).Ifp 1, the V p (K i,f i ) V p (K, f ). Sice f i f Î C + (S -1 ), the f i are uiformly bouded o S -1. Hece, f p i f p, uiformly o S 1. By Lemma 6, K i K implies that S p (K i, ) S p (K, ), weakly o S 1.

12 Page 12 of 13 Hece, S 1 f i (u) p ds p (K i, u) S 1 f (u) p ds p (K, u). I view of the itegral represetatio (2.13) ad (2.8), we get the desired result. The covergece result will be established as followig. Theorem 4 Suppose p >1, f Î C + (S -1 ).Iff i is a sequece of fuctios i C + (S -1 ), such that V p (Q, f i ) V p (Q, f ), for all Q K 0, the f i f. Firstly, sice f i is a sequece i C + (S -1 ), f i are uiformly bouded o S -1. Applyig the Blaschke selectio theorem (see [3]), it guaratees the existece of a subsequece of the f i, which is agai deoted by f i, covergig to a positive cotiuous fuctio f 0 o S -1. Secodly, sice f i are uiformly bouded o S -1, 1+ p f i 1+ p f 0, uiformly o S 1. Defie fi C + (S 1 ),by fi =1+ p f i. Sice fi f 0, it follows from Lemma 7 that V p (Q, f i ) V p (Q, f 0 ), for allq K 0. However, sice V p (Q, f i ) V p (Q, f ), for all Q K 0, ad fi =1+ p f i, for all i>0, it follows from Lemma 5 that Thus, V p (Q, f i ) V p (Q,1+ p f ), for all Q K 0. V p (Q, f 0 )=V p (Q,1+ p f ), for all Q K 0. By Corollary 2, this meas f0 =1+ p f, which shows f 0 = f. Thus, every subsequece of f i has a subsequece that coverges to f. Ackowledgemets The authors express their deep gratitude to the referees for their may very valuable suggestios ad commets. The research of Hu-Ya ad Jiag-Juhua was supported by Natioal Natural Sciece Foudatio of Chia ( ), Shaghai Leadig Academic Disciplie Project (S30104), ad the research of Hu-Ya was partially supported by Iovatio Program of Shaghai Muicipal Educatio Commissio (10yz160). Author details 1 Departmet of Mathematics, Shaghai Uiversity, Shaghai , Chia 2 Departmet of Mathematics, Shaghai Uiversity of Electric Power, Shaghai , Chia Authors cotributios HY ad JJH joitly cotributed to the mai results Theorem 1, Theorem 2, Theorem 3 ad Theorem 4. HY drafted the mauscript ad made the text file. Both authors read ad approved the fial mauscript. Competig iterests The authors declare that they have o competig iterests.

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