Levinson s type generalization of the Jensen inequality and its converse for real Stieltjes measure

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1 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 DOI 0.86/s y R E S E A R C H Open Access Levinson s type generliztion of the Jensen inequlity nd its converse for rel Stieltjes esure Rozrij Mikić, Josip PečrićndMirnRodić * * Correspondence: rodic@ttf.hr Fculty of Textile Technology, University of Zgreb, Prilz brun Filipović8, Zgreb, 0000, Croti Abstrct We derive the Levinson type generliztion of the Jensen nd the converse Jensen inequlity for rel Stieltjes esure, not necessrily positive. As consequence, lso the Levinson type generliztion of the Herite-Hdrd inequlity is obtined. Siilrly, we derive the Levinson type generliztion of Giccrdi s inequlity. The obtined results re then pplied for estblishing new en-vlue theores. The results fro this pper represent generliztion of severl recent results. MSC: 6D5; 6A5; 6A4 Keywords: Jensen s inequlity; converse Jensen s inequlity; Herite-Hdrd s inequlity; Giccrdi s inequlity; Levinson s inequlity; Green function; en-vlue theores Introduction nd preliinry results The well-known Jensen inequlity sserts tht for convex function ϕ : I R R we hve ( ϕ p i x i p i ϕ(x i, (. where x i I for i =,...,n,ndp i re nonnegtive rel nubers such tht = n p i >0. Steffensen [] showed tht inequlity (. lsoholdsinthecsewhen(x,...,x n is onotonic n-tuple of nubers fro the intervl I nd (p,...,p n is n rbitrry rel n-tuple such tht 0 P k (k =,...,n, >0,whereP k = k p i. His result is clled the Jensen-Steffensen inequlity. Bos [] gve the integrl nlog of the Jensen-Steffensen inequlity. Theore. ([] Let ϕ : I R be continuous convex function, where I is the rnge of the continuous onotonic function (either incresing or decresing f :[, b] R, nd let λ: [, b] R be either continuous or of bounded vrition stisfying λ( λ(x λ(b for ll x [, b], λ(<λ(b. The Author(s 07. This rticle is distributed under the ters of the Cretive Coons Attribution 4.0 Interntionl License ( which perits unrestricted use, distribution, nd reproduction in ny ediu, provided you give pproprite credit to the originl uthor(s nd the source, provide link to the Cretive Coons license, nd indicte if chnges were de.

2 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge of 5 Then ( b f (x ϕ b b ϕ(f (x b. (. The generliztion of this result is lso given by Bos in []. It is the so-clled Jensen- Bos inequlity (see lso [3]. Theore. ([] If λ: [, b] R is either continuous or of bounded vrition stisfying λ( λ(x λ(y λ(x λ(y n λ(x n λ(b for ll x k y k, y k, y 0 =, y n = b, nd λ(<λ(b, nd if f is continuous nd onotonic (either incresing or decresing in ech of the n intervls y k, y k, then inequlity (. holds. The following theore sttes the well-known Levinson inequlity. Theore.3 ([4] Let f : 0, c R stisfy f 0 nd let p i, x i, y i, i =,...,nbesuch tht p i >0, n p i =,0 x i cnd x + y = x + y = = x n + y n. (.3 Then the following inequlity is vlid: p i f (x i f( x p i f (y i f(ȳ, (.4 where x = n p ix i nd ȳ = n p iy i denote the weighted rithetic ens. Nuerous ppers hve been devoted to extensions nd generliztions of this result, s well s to wekening the ssuptions under which inequlity (.4 is vlid (see for instnce [5 8], nd [9]. Afunctionf : I R is clled k-convex if [x 0,...,x k ]f 0 for ll choices of k + distinct points x 0, x,...,x k I. Ifthekth derivtive of convex function exists, then f (k 0, but f (k y not exist (for properties of divided differences nd k-convex functions see [3]. Rerk.4 (i Bullen [6] rescled Levinson s inequlity to generl intervl [, b] nd showed tht if function f is 3-convex nd p i, x i, y i, i =,...,n,resuchthtp i >0, n p i =, x i, y i b,(.3 holds for soe c, b nd x{x,...,x n } x{y,...,y n }, (.5 then (.4 holds.

3 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 3 of 5 (ii Pečrić [8]provedthtinequlity(.4 is vlid when one wekens the previous ssuption (.5to x i + x n i+ c nd p i x i + p n i+ x n i+ p i + p n i+ c, for i =,,...,n. (iii Mercer [7] de significnt iproveent by replcing condition (.3with weker one,i.e. he proved thtinequlity (.4 holds under the following conditions: f 0, p i >0, p i =, x i, y i b, x{x,...,x n } x{y,...,y n }, p i (x i x = p i (y i ȳ. (.6 (iv Witkowski [9] showedthtit is enough to ssue tht f is 3-convex in Mercer s ssuptions. Furtherore, Witkowski wekened the ssuption (.6ndshowed tht equlity cn be replced by inequlity in certin direction. Furtherore, Bloch, Pečrić, nd Prljk in their pper [5] introducednewclssof functions K c (, b tht extends 3-convex functions nd cn be interpreted s functions tht re 3-convex t point c, b. They showed tht K c (, b is the lrgest clss of functions for which Levinson s inequlity (.4 holds under Mercer s ssuptions, i.e. tht f K c(, b if nd only if inequlity (.4 holds for rbitrry weights p i >0, n p i =nd sequences x i nd y i tht stisfy x i c y i for i =,,...,n. We give the definition of the clss K c (, b extended to n rbitrry intervl I. Definition.5 Let f : I R nd c I, where I is the interior of I. We sy tht f K c(i (f K c(i if there exists constnt D such tht the function F(x=f (x D x is concve (convex on, c] I nd convex (concve on [c,+ I. Rerk.6 For the clss K c (, b the following useful results hold (see [5]: ( If f Ki c(, b, i =,,ndf (c exists, then f (c=d. ( The function f :(, b R is 3-convex (3-concve, if nd only if f K c (, b (f K c (, b for every c (, b. Jkšetić, Pečrić, nd Prljk in [0] gve the following Levinson type generliztion of the Jensen-Bos inequlity. Theore.7 ([0] Let c I nd let f :[, b ] R nd g :[, b ] R be continuous onotonic functions (either incresing or decresing with rnges I, c] nd I [c,+, respectively. Let λ: [, b ] R nd μ: [, b ] R be continuous or of bounded vrition stisfying λ( λ(x λ(y λ(x λ(y n λ(x n λ(b

4 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 4 of 5 for ll x k y k, y k, y 0 =, y n = b, nd λ( <λ(b, nd μ( μ(u μ(v λ(u μ(v n μ(u n μ(b for ll u k v k, v k, v 0 =, v n = b, nd μ( <μ(b. If ϕ K c (I is continuous nd if f (x = holds, then ( b f (x g ( (x dμ(x b dμ(x ϕ(f (x ( b ϕ g(x dμ(x dμ(x f (x ( ϕ(g(x dμ(x b ϕ dμ(x g(x dμ(x dμ(x. (.7 On the other hnd, in [] Pečrić, Perić, nd Rodić Lipnović generlized the Jensen inequlity (. for rel Stieltjes esure. They considered the Green function G defined on [α, β] [α, β]by G(t, s= (t β(s α β α for α s t, (s β(t α β α for t s β, which is convex nd continuous with respect to both s nd t. The function G is continuous under s nd continuous under t, nd it cn esily be shown by integrting by prts tht ny function ϕ :[α, β] R, ϕ C ([α, β], cn be represented by ϕ(x= β x β α ϕ(α+ x α β α ϕ(β+ β α (.8 G(x, sϕ (s ds. (.9 Using tht fct, the uthors in [] gve the conditions under which inequlity (.holds for rel Stieltjes esure, which is not necessrily positive nor incresing. This result is stted in the following theore. Theore.8 ([] Let g :[, b] R be continuous function nd [α, β] intervl such tht the ige of g is subset of [α, β]. Let λ :[, b] R be continuous function or the function of bounded vrition, such tht λ( λ(b nd b g(x b [α, β]. Then the following two stteents re equivlent:

5 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 5 of 5 ( For every continuous convex function ϕ :[α, β] R ( b g(x ϕ b holds. ( For ll s [α, β] ( b g(x G b, s b ϕ(g(x b (.0 b G(g(x, s b (. holds, where the function G :[α, β] [α, β] R is defined in (.8. Furtherore, the stteents (nd (re lsoequivlentifwechngethesignofinequl- ity in both (.0 nd (.. Note tht for every continuous concve function ϕ :[α, β] R inequlity (.0 isre- versed, i.e. the following corollry holds. Corollry.9 ([] Let the conditions fro the previous theore hold. Then the following two stteents re equivlent: ( For every continuous concve function ϕ :[α, β] R the reverse inequlity in (.0 holds. ( For ll s [α, β] inequlity (. holds, where the function G is defined in (.8. Moreover, the stteents ( nd ( relsoequivlentifwechngethesignofinequlity in both stteents ( nd (. The in i of our pper is to give Levinson type generliztion of the result fro Theore.8. In tht wy, generliztion of Theore.7 for rel Stieltjes esure, not necessrily positive nor incresing, will lso be obtined. Min results In order to siplify the nottion, throughout this pper we use the following nottion: f = f (x nd ḡ = g(x dμ(x. dμ(x The following theore sttes our in result. Theore. Let f :[, b ] R nd g :[, b ] R be continuous functions, [α, β] R n intervl nd c α, β such tht f ([, b ] [α, c] nd g([, b ] [c, β]. Let λ: [, b ] R nd μ: [, b ] R be continuous functions or functions of bounded vrition such tht λ( λ(b nd μ( μ(b nd such tht f [α, c] nd ḡ [c, β]

6 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 6 of 5 nd C := f b (x f = g (x dμ(x dμ(x holds. If for ll s [α, c] nd for ll s [c, β] we hve ḡ (. G( f, s G(f (x, s nd G(ḡ, s G(g(x, s dμ(x, (. dμ(x where the function G is defined in (.8, then for every continuous function ϕ K c ([α, β] we hve ϕ(f (x ϕ( f D C ϕ(g(x dμ(x ϕ(ḡ. (.3 dμ(x Thestteentlsoholdsifwereversellsignsofinequlitiesin(. nd (.3. Proof Let ϕ K c([α, β] be continuous function on [α, β]ndletφ(x=ϕ(x D x,where D is the constnt fro Definition.5. Since the function φ is continuous nd concve on [α, c]ndforlls [α, c](.holds, fro Corollry.9 it follows tht φ( f φ(f (x. When we rerrnge the previous inequlity, we get ϕ(f (x ϕ( f D [ b f (x b f ]. (.4 Since the function φ is continuous nd convex on [c, β] ndforlls [c, β] (. holds, fro Theore.8 it follows tht φ(ḡ φ(g(x dμ(x, dμ(x nd fter rerrnging we get [ D b g ] (x dμ(x b ḡ ϕ(g(x dμ(x dμ(x dμ(x ϕ(ḡ. (.5 Inequlity (.3 follows directly by cobining inequlities (.4nd(.5, nd tking into ccount the condition (..

7 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 7 of 5 Corollry. Let the conditions fro the previous theore hold. (i If for ll s [α, c] nd s [c, β] inequlities (. hold, where the function G is defined in (.8, then for every continuous function ϕ K c ([α, β] the reverse inequlities in (.3 hold. (ii If for ll s [α, c] nd s [c, β] the reverse inequlities in (. hold, then for every continuous function ϕ K c ([α, β] (.3 holds. Rerk.3 It is obvious fro the proof of the previous theore tht if we replce the equlity (. by weker condition C := D then (.3becoes [ b f ] (x f C := D [ b g (x dμ(x b dμ(x ϕ(f (x ϕ( f C C ϕ(g(x dμ(x dμ(x ḡ ], (.6 ϕ(ḡ. Since the function ϕ belongs to clss K c ([α, β], we hve ϕ (c D ϕ + (c (see[5], so if dditionlly ϕ is convex (resp. concve, the condition (.6 cn be furtherwekened to f b (x f g (x dμ(x dμ(x ḡ. Rerk.4 It is esy to see tht Theore. further generlizes the Levinson type generliztion of the Jensen-Bos inequlity given in Theore.7. Nely, if in Theore. we set the functions f nd g to be onotonic, nd the functions λ nd μ to stisfy λ( λ(x λ(y λ(x λ(y n λ(x n λ(b for ll x k y k, y k, y 0 =, y n = b,ndλ( <λ(b, nd μ( μ(u μ(v λ(u μ(v n μ(u n μ(b for ll u k v k, v k, v 0 =, v n = b,ndμ( <μ(b, then since the function G is continuous nd convex in both vribles, we cn pply the Jensen inequlity nd see tht for ll s [α, c]nds [c, β] inequlities (. hold, so we get exctly Theore.7. 3 Discrete cse Inthissectionwegivetheresultsforthediscretecse.Theproofsresiilrtothosein the integrl cse given in the previous section, so we will stte these results without the proofs. In Levinson s inequlity (.3 nd its generliztions (see [5] we see tht p i (i =,...,n re positive rel nubers. Here, we will give generliztion of tht result, llowing p i to lso be negtive, with the su not equl to zero, but with suppleentry dend on p i nd x i given by using the Green function G defined in (.8.

8 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 8 of 5 Here we use the coon nottion: for rel n-tuples (x,...,x n nd(p,...,p n weset P k = k p i, P k = P k (k =,...,n nd x = n p ix i. Anlogously, for rel - tuples (y,...,y nd(q,...,q wedefineq k, Q k (k =,...,ndȳ. We lredy know fro the first section tht we cn represent ny function ϕ :[α, β] R, ϕ C ([α, β], in the for (.9, where the function G is defined in (.8, nd by soe clcultion it is esy to show tht the following holds: ϕ( x ( β p i ϕ(x i = G( x, s p i G(x i, s ϕ (s ds. α Using tht fct the uthors in [] derived the nlogous results of Theore.8 nd Corollry.9 for discrete cse, nd here, siilrly s in the previous section, we get the following results. Theore 3. Let [α, β] R be n intervl nd c α, β. Let x i [, b ] [α, c], p i R (i =,...,n be such tht 0nd x [α, c], nd let y j [, b ] [c, β], q j R (j =,..., be such tht Q 0nd ȳ [c, β] nd let C := p i x i x = Q If for ll s [α, c] nd for ll s [c, β] we hve G( x, s q j y j ȳ. (3. p i G(x i, s nd G(ȳ, s Q q j G(y j, s, (3. where the function G is defined in (.8, then for every continuous function ϕ K c ([α, β] we hve p i ϕ(x i ϕ( x D C Q q j ϕ(y j ϕ(ȳ, (3.3 where D is the constnt fro Definition.5. Inequlity (3.3 is reversed if we chnge the signs of inequlities in (3.. Corollry 3. Let the conditions fro the previous theore hold. (i If for ll s [α, c] nd s [c, β] the inequlities in (3. hold, where the function G is defined in (.8, then for every continuous function ϕ K c ([α, β] the reverse inequlities in (3.3 hold. (ii If for ll s [α, c] nd s [c, β] the reversed inequlities in (3. hold, then for every continuous function ϕ K c ([α, β] (3.3 holds. Rerk 3.3 Theore 3. is the generliztion of Levinson s type inequlity given in [5]. Nely, since the function G is convex in both vribles, in the cse when ll p i >0nd q j > 0 we cn pply the Jensen inequlity nd we see tht for ll s [α, c] nds [c, β] inequlities (3. hold. Now fro Theore 3. nd Corollry 3. we get the result fro [5].

9 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 9 of 5 Rerk 3.4 We cn replce the equlity fro the condition (3. by weker condition in the nlogous wy s in Rerk.3 fro the previous section. 4 Converses of the Jensen inequlity The Jensen inequlity for convex functions iplies whole series of other clssicl inequlities. One of the ost fous ones ongst the is the so-clled Edundson-Lh- Ribrič inequlity which sttes tht, for positive esure μ on [0,] nd convex function φ on [, M] ( < < M <+, if f is ny μ-esurble function on [0, ] such tht f (x M for x [0, ], one hs 0 φ(f dμ 0 dμ M f M φ(+ f M φ(m, (4. where f = 0 fdμ/ 0 dμ. It ws obtined in 973. by Lh nd Ribrič in their pper []. Since then, there hve been ny ppers written on the subject of its generliztions nd converses (for instnce, see [3]nd[3]. In [4] the uthors gve Levinson type generliztion of inequlity (4. forpositive esures. In this section we will obtin siilr result involving signed esures, with suppleentry dend by using the Green function G defined in (.8. In order to do so, we first need to stte result fro [], which gives us version of the Edundson-Lh- Ribrič inequlity for signed esures. Theore 4. ([] Let g :[, b] R be continuous function nd [α, β] be n intervl such tht the ige of g is subset of [α, β]. Let, M [α, β] ( M be such tht g(t Mforllt [, b]. Let λ :[, b] R be continuous function or the function of bounded vrition, nd λ( λ(b. Then the following two stteents re equivlent: ( For every continuous convex function ϕ :[α, β] R b ϕ(g(x b M ḡ M ϕ(+ ḡ ϕ(m (4. M holds, where ḡ = ( For ll s [α, β] b g(x b. b G(g(x, s b M ḡ ḡ G(, s+ G(M, s (4.3 M M holds, where the function G :[α, β] [α, β] R is defined in (.8. Furtherore, the stteents ( nd ( re lso equivlent if we chnge the sign of inequlity in both (4. nd (4.3. Note tht for every continuous concve function ϕ :[α, β] R inequlity (4. isre- versed, i.e. the following corollry holds.

10 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 0 of 5 Corollry 4. ([] Let the conditions fro the previous theore hold. Then the following two stteents re equivlent: ( For every continuous concve function ϕ :[α, β] R the reverse inequlity in (4. holds. ( For ll s [α, β] inequlity (4.3 holds, where the function G is defined in (.8. Moreover, the stteents ( nd ( relsoequivlentifwechngethesignofinequlity in both stteents ( nd (. In the following theore we give the Levinson type generliztion of the upper result, nd we use siilr ethod to Section of this pper. Theore 4.3 Let f :[, b ] R nd g :[, b ] R be continuous functions, [α, β] R n intervl nd c α, β such tht f ([, b ] = [, M ] [α, c] nd g([, b ] = [, M ] [c, β], where M nd M. Let λ: [, b ] R nd μ: [, b ] R be continuous functions or functions of bounded vrition such tht λ( λ(b nd μ( μ(b nd C := M f M + f M M f (x = M ḡ M + g M M If for ll s [α, c] we hve g (x dμ(x. (4.4 dμ(x G(f (x, s M f G(, s+ f G(M, s (4.5 M M nd for ll s [c, β] we hve G(g(x, s dμ(x M ḡ G(, s+ ḡ G(M, s, (4.6 dμ(x M M where the function G is defined in (.8, then for every continuous function ϕ K c ([α, β] we hve M f ϕ( + f ϕ(m ϕ(f (x M M b D C M ḡ ϕ( + g ϕ(m ϕ(g(x dμ(x M M b. (4.7 dμ(x Thestteentlsoholdsifwereversellsignsofinequlitiesin(4.5, (4.6 nd (4.7. Proof Let ϕ K c ([α, β] be continuous function on [α, β]ndletφ(x=ϕ(x D x,where D is the constnt fro Definition.5.

11 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge of 5 Since the function φ is continuous nd concve on [α, c]ndforlls [α, c](4.5holds, fro Corollry 4. it follows tht φ(f (x M f φ( + f φ(m. M M When we rerrnge the previous inequlity, we get M f ϕ( + f ϕ(m ϕ(f (x M M b D [ M f M + f M M f (x ]. (4.8 Since the function φ is continuous nd convex on [c, β] ndforlls [c, β] (4.6 holds, fro Theore 4. it follows tht φ(g(x dμ(x M ḡ φ( + ḡ φ(m, dμ(x M M nd fter rerrnging we get [ D M ḡ M + ḡ M M g ] (x dμ(x dμ(x M ḡ ϕ( + ḡ ϕ(m ϕ(g(x dμ(x M M b. (4.9 dμ(x Inequlity (4.7 follows directly by cobining inequlities (4.8 nd(4.9, nd tking into ccount the condition (4.4. Corollry 4.4 Let the conditions fro the previous theore hold. (i If for ll s [α, c] the inequlity in (4.5 holds, nd for ll s [c, β] the inequlity in (4.6 holds, then for every continuous function ϕ K c ([α, β] the reverse inequlities in (4.7 hold. (ii If for ll s [α, c] the reversed inequlity in (4.5 holds, nd for ll s [c, β] the reversed inequlity in (4.6 holds, then for every continuous function ϕ K c ([α, β] the inequlities in (4.7 hold. Rerk 4.5 It is obvious fro the proof of the previous theore tht if we replce the equlity (4.4 by weker condition C := D [ M f M + f M M f ] (x C := D [ M ḡ M + g M M g (x dμ(x dμ(x ], (4.0

12 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge of 5 then (4.7becoes M f ϕ( + f ϕ(m ϕ(f (x M M b C C M ḡ ϕ( + g ϕ(m M M ϕ(g(x dμ(x. dμ(x Since ϕ (c D ϕ + (c(see[5], if dditionlly ϕ is convex (resp. concve, the condition (4.0 cn be further wekened to M f M + f M M f (x M ḡ M + g M M g (x dμ(x. dμ(x 5 Discrete for of the converses of the Jensen inequlity In this section we give the Levinson type generliztion for converses of Jensen s inequlity in discrete cse. The proofs re siilr to those in the integrl cse given in the previous section, so we give these results with the proofs oitted. As we cn represent ny function ϕ :[α, β] R, ϕ C ([α, β], in the for (.9, where the function G is defined in (.8, by soe clcultion it is esy to show tht the following holds: p i ϕ(x i b x b ( β = α ϕ( x b ϕ(b p i G(x i, s b x b x G(, s G(b, s ϕ (s ds. b Usingthtfcttheuthorsin[] derived nlogous results of Theore 4. nd Corollry 4. for discrete cse. In [4] the uthors obtined the following Levinson type generliztion of the discrete Edundson-Lh-Ribrič inequlity. Theore 5. ([4] Let < A c b B <+. If x i [, A], y j [b, B], p i >0, q j >0for i =,...,nnd,...,resuchtht n p i = q j =nd A x A + x A A p i x i = B ȳ B b b + ȳ b B b B q j y j, where x = n p ix i nd ȳ = q jy j, then for every f K c (, B we hve A x x f (+ A A f (A p i f (x i B ȳ B b f (b+ ȳ b B b f (B q j f (y j.

13 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 3 of 5 Ourfirstresultisgenerliztionoftheresultfro[4] sttedbove,inwhichitisllowed for p i, q j to lso be negtive, with the su not equl to zero, but with suppleentry dends on p i, q j nd x i, y j given by using the Green function G defined in (.8. Theore 5. Let [α, β] R be n intervl nd c α, β. Let x i [, b ] [α, c], b, p i R (i =,...,n, nd y j [, b ] [c, β], b, q j R (j =,..., be such tht 0 nd Q 0nd C := b x b + x b b = b ȳ b + ȳ b b Q If for ll s [α, c] we hve p i x i q j y j. (5. p i G(x i, s b x b G(, s+ x b G(b, s, (5. nd for ll s [c, β] we hve Q q j G(y j, s b ȳ b G(, s+ ȳ b G(b, s, (5.3 where x = n p ix i, ȳ = Q q jy j nd the function G is defined in (.8, then for every continuous function ϕ K c ([α, β] we hve b x b ϕ( + x b ϕ(b p i ϕ(x i D C b ȳ b ϕ( + ȳ b ϕ(b Q q j ϕ(y j. (5.4 Thestteentlsoholdsifwereversellsignsoftheinequlitiesin(5., (5.3, nd (5.4. Rerk 5.3 If we set ll p i, q j to be positive, then Theore 5. becoes the result fro [4] which is stted bove in Theore 5.. Corollry 5.4 Let the conditions fro the previous theore hold. (i If for ll s [α, c] inequlity (5. holds nd for ll s [c, β] inequlity (5.3 holds, then for every continuous function ϕ K c ([α, β] the reverse inequlities in (5.4 hold. (ii If for ll s [α, c] the reversed inequlity in (5. holds nd for ll s [c, β] the reversed inequlity in (5.3 holds, then for every continuous function ϕ K c ([α, β] (5.4 holds. Rerk 5.5 We cn replce the equlity fro the condition (5. by weker condition in the nlogous wy s in Rerk 4.5 fro the previous chpter.

14 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 4 of 5 6 The Herite-Hdrd inequlity The clssicl Herite-Hdrd inequlity sttes tht for convex function ϕ :[, b] R the following estition holds: ( + b ϕ b b ϕ(x dx ϕ(+ϕ(b. (6. ItsweightedforisprovedbyFejérin[5]: If ϕ :[, b] R isconvexfunctionnd p: [, b] R nonnegtive integrble function, syetric with respect to the iddle point ( + b/, then the following estition holds: ( + b b ϕ p(x dx b ϕ(xp(x dx ϕ(+ϕ(b b p(x dx. (6. Fink in [6] discussed the generliztion of (6. by seprtely lookingthe left ndright side of the inequlity nd considering certin signed esures. In their pper [7], the uthors gve coplete chrcteriztion of the right side of the Herite-Hdrd inequlity. Rodić Lipnović, Pečrić, nd Perić in [] obtined the coplete chrcteriztion for the left nd the right side of the generlized Herite-Hdrd inequlity for the rel Stieltjes esure. In this section Levinson type generliztion of the Herite-Hdrd inequlity for signed esures will be given s consequence of the results given in Sections nd 4. Here we use the following nottion: x = x nd ỹ = ydμ(y dμ(y. Corollry 6. Let [α, β] R be n intervl nd c α, β nd let [, b ] [α, c] nd [, b ] [c, β]. Let λ: [, b ] R nd μ: [, b ] R be continuous functions or functions of bounded vrition such tht λ( λ(b, μ( μ(b nd x [α, c], ỹ [c, β], nd such tht C := x b x = y dμ(y dμ(y If for ll s [α, c] nd for ll s [c, β] the inequlities ỹ. (6.3 G( x, s G(x, s nd G(ỹ, s G(y, s dμ(y dμ(y (6.4 hold, where the function G is defined in (.8, then for every continuous function ϕ K c ([α, β] we hve ϕ(x ϕ( x D C ϕ(y dμ(y ϕ(ỹ. (6.5 dμ(y Thestteentlsoholdsifwereversellsignsoftheinequlitiesin(6.4 nd (6.5.

15 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 5 of 5 Rerk 6. Let the conditions fro the previous corollry hold. (i If for ll s [α, c] nd s [c, β] inequlities (6.4 hold, then for every continuous function ϕ K c ([α, β] the reverse inequlities in (6.5hold. (ii If for ll s [α, c] nd s [c, β] thereversedinequlitiesin(6.4 hold, then for every continuous function ϕ K c ([α, β] (6.5 holds. Note tht for the Levinson type generliztion of the left-side inequlity of the generlized Herite-Hdrd inequlity it is necessry to dend tht x [α, c]ndỹ [c, β]. Rerk 6.3 If in Rerk.3 we put f (x=x nd g(x=x, we cn obtin weker conditions insted of equlity (6.3 under which inequlity (6.5 holds. Siilrly, fro the results given in the fourth section we get the Levinson type generliztion of the right-side inequlity of the generlized Herite-Hdrd inequlity. Here we llow tht the en vlue x goes outside of the intervl [α, c]ndỹ outside of the intervl [c, β]. Corollry 6.4 Let [α, β] R be n intervl nd c α, β, nd let [, b ] [α, c] nd [, b ] [c, β]. Let λ: [, b ] R nd μ: [, b ] R be continuous functions or functions of bounded vrition such tht λ( λ(b nd μ( μ(b ndsuchtht C := b x b + x b b x = b ỹ b + ỹ b b y dμ(y dμ(y. (6.6 If for ll s [α, c] we hve G(x, s b x G(, s+ x G(b, s, (6.7 b b nd for ll s [c, β] we hve G(y, s dμ(y b ỹ G(, s+ ỹ G(b, s, (6.8 dμ(y b b where the function G is defined in (.8, then for every continuous function ϕ K c ([α, β] we hve b x ϕ( + x ϕ(b ϕ(x b b b D C b ỹ ϕ( + ỹ ϕ(b ϕ(y dμ(y b b b. (6.9 dμ(y Thestteentlsoholdsifwereversellsignsoftheinequlitiesin(6.7, (6.8 nd (6.9.

16 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 6 of 5 Rerk 6.5 Let the conditions fro the previous theore hold. (i If for ll s [α, c] inequlity (6.7 holds nd for ll s [c, β] inequlity (6.8 holds, then for every continuous function ϕ K c ([α, β] the reverse inequlities in (6.9 hold. (ii If for ll s [α, c] thereversedinequlityin(6.7 holds nd for ll s [c, β] the reversed inequlity in (6.8 holds, then for every continuous function ϕ K c ([α, β] (6.9 holds. Rerk 6.6 If in Rerk 4.5 we put f (x=xnd g(x=x, we cn obtin nlogous weker conditions insted of equlity (6.6 under which inequlity (6.9 holds. It is esy to see tht for λ(x =x nd μ(x =x the conditions (6.4, (6.7 nd(6.8 re lwys fulfilled. In tht wy we cn obtin Levinson type generliztion of both sides in the clssicl weighted Herite-Hdrd inequlity. Corollry 6.7 Let [α, β] R be n intervl nd c α, β, nd let [, b ] [α, c] nd [, b ] [c, β]. (i If C := (b = (b holds, then for every continuous function ϕ K c ([α, β] b D C b ( + b ϕ(x dx ϕ ( + b ϕ(x dx ϕ. (ii If C := 6 (b = 6 (b holds, then for every continuous function ϕ K c ([α, β] ϕ( +ϕ(b b D C ϕ(+ϕ(b ϕ(x dx b ϕ(x dx. If ϕ K c ([α, β], then the inequlities in (i nd (ii re reversed. 7 The inequlities of Giccrdi nd Petrović The well-known Petrović inequlity [8] for convex function f :[0,] R is given by ( f (x i f x i +(n f(0, (7. where x i (i =,...,n re nonnegtive nubers such tht x,...,x n, n x i [0, ]. The following generliztion of (7.isgivenbyGiccrdi(see[3]nd[9].

17 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 7 of 5 Theore 7. (Giccrdi, [9] Let p =(p,...,p n be nonnegtive n-tuple nd x = (x,...,x n be rel n-tuple such tht ( (x i x 0 p j x j x i 0, i =,...,n, x 0, p i x i [, b] nd p i x i x 0. If f :[, b] R is convex function, then where ( ( p i f (x i Af p i x i + B p i f (x 0, (7. A = n p i(x i x 0 n n p, B = p ix i ix i x n 0 p. ix i x 0 In this section we will use n nlogous technique s in the previous sections to obtin Levinson type generliztion of the Giccrdi inequlity for n-tuples p of rel nubers which re not necessrily nonnegtive. As siple consequence, we will obtin Levinson type generliztion of the originl Giccrdi inequlity (7..In ordertodoso,wefirstneed to stte two results fro []. Theore 7. ([] Let x i [, b] [α, β], b, p i R (i =,...,n be such tht 0. Then the following two stteents re equivlent: ( For every continuous convex function f :[α, β] R p i f (x i b x x f (+ f (b (7.3 b b holds. ( For ll s [α, β] p i G(x i, s b x x G(, s+ G(b, s (7.4 b b holds, where the function G :[α, β] [α, β] R is defined in (.8. Moreover, the stteents (nd (re lsoequivlentifwechngethesignoftheinequl- ity in both inequlities, in (7.3 nd in (7.4. Corollry 7.3 ([] Under the conditions fro the previous theore, the following two stteents re lso equivlent: ( For every continuous concve function ϕ :[α, β] R the reverse inequlity in (7.3 holds.

18 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 8 of 5 ( For ll s [α, β] inequlity (7.4 holds. Moreover, the stteents ( nd ( relsoequivlentifwechngethesignofinequlity in both stteents ( nd (. Our first result is Levinson type generliztion of the Giccrdi inequlity for n-tuples p nd -tuples q of rbitrry rel nubers insted of nonnegtive rel nubers. Theore 7.4 Let [α, β] R be n intervl nd c α, β such tht [, b ] [α, c] nd [, b ] [c, β]. Let p nd x be n-tuples of rel nubers, nd let q nd y be -tuples of rel nubers such tht = n p i 0,Q = q i 0,nd ( (x j x 0 p i x i x j 0 (j =,...,n; x 0, p i x i [, b ]; p i x i x 0 ; ( (y j y 0 q i y i y j 0 (j =,...,; y 0, q i y i [, b ]; q i y i y 0. (7.5 Let ( ( C := A p i x i + B p i x 0 p i x i ( ( = A q j y j + B q j y 0 q j y j, (7.6 where A = A = n p i(x i x 0 n n p, B = p ix i ix i x n 0 p, ix i x 0 q j(y j y 0 q, B = q jy j jy j y 0 q. jy j y 0 (7.7 If Q >0nd for ll s [α, β] nd the function G defined in (.8 the inequlities ( ( p i G(x i, s A G p i x i, s + B p i G(x 0, s, (7.8 ( ( q j G(y j, s A G q j y j, s + B q j G(y 0, s, (7.9

19 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 9 of 5 hold, then for every continuous function ϕ K c ([α, β] we hve ( ( A ϕ p i x i + B p i ϕ(x 0 p i ϕ(x i ( D ( C A ϕ q j y j + B q j ϕ(y 0 q j ϕ(y j. (7.0 The stteent lso holds if we reverse ll signs of the inequlities in (7.8, (7.9, nd (7.0. Proof We follow the se ide s in the proof of Theore 4.3 fro Section 4. We pply Theore 7. nd Corollry 7.3 to the function φ(x=ϕ(x D x,whichisconcveon[α, c] nd convex on [c, β]. We set = in{x 0, n p ix i }, b = x{x 0, n p ix i } on [α, c], nd then we set = in{y 0, q jy j } nd b = x{y 0, q jy j } on [c, β], s well s consider the signs of nd Q. We oit the detils. Corollry 7.5 Let the ssuptions fro the previous theore hold. (i If >0nd Q <0nd if for ll s [α, β] inequlity (7.8 holds nd inequlity (7.9 is reversed, then (7.0 holds. (ii If <0nd Q >0nd if for ll s [α, β] inequlity (7.8 is reversed nd inequlity (7.9 holds, then (7.0 holds. Stteents (i nd (ii lso hold if we reverse the signs in ll of the inequlities. Corollry 7.6 Let the ssuptions fro the previous theore hold. (i If Q >0nd if for ll s [α, β] inequlities (7.8 nd (7.9 hold, then for every continuous function ϕ K c ([α, β] the reversed inequlity in (7.0 holds. (ii If Q >0nd if for ll s [α, β] inequlities (7.8 nd (7.9 re reversed, then for every continuous function ϕ K c ([α, β] (7.0 holds. (iii If >0nd Q <0nd if for ll s [α, β] inequlity (7.8 holds nd inequlity (7.9 is reversed, then for every continuous function ϕ K c ([α, β] the reversed inequlity in (7.0 holds. (iv If <0nd Q >0nd if for ll s [α, β] inequlity (7.8 is reversed nd inequlity (7.9 holds, then for every continuous function ϕ K c ([α, β] the reversed inequlity in (7.0 holds. Stteents (iii nd (iv lso hold if we reverse the signs in ll of the entioned inequlities. Rerk 7.7 Oneneedstonoticethtifwesetp i (i =,...,n ndq j (j =,..., tobe positive, Theore 7.4 becoes the Levinson type generliztion of the originl Giccrdi inequlity (7.. Rerk 7.8 As in the previous sections, we cn replce the equlity (7.6 byweker condition [ ( C := D ( ] A p i x i + B p i x 0 p i x i [ ( C := D ( ] A q j y j + B q j y 0 q j y j, (7.

20 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 0 of 5 nd then (7.0becoes ( ( A ϕ p i x i + B p i ϕ(x 0 p i ϕ(x i C ( ( C A ϕ q j y j + B q j ϕ(y 0 q j ϕ(y j. Since ϕ (c D ϕ + (c(see[5], if dditionlly ϕ is convex (resp. concve, the condition (7. cn be further wekened to ( ( A p i x i + B p i x 0 p i x i ( ( A q j y j + B q j y 0 q j y j. 8 Men-vlue theores Let f :[, b ] R nd g :[, b ] R be continuous functions, [α, β] R n intervl nd c α, β such tht f ([, b ] = [, M ] [α, c] ndg([, b ] = [, M ] [c, β]. Let λ: [, b ] R nd μ: [, b ] R be continuous functions or functions of bounded vrition such tht λ( λ(b ndμ( μ(b, nd let ϕ K c ([α, β] be continuous function. Motivted by the results obtined in previous sections, we define the following liner functionls which, respectively, represent the difference between the right nd the left side of inequlities (.3nd(4.7: Ɣ J (ϕ= ϕ(g(x dμ(x ϕ(ḡ ϕ(f (x dμ(x where f [α, c], ḡ [c, β]; Ɣ ELR (ϕ= M ḡ ϕ( + g ϕ(m ϕ(g(x dμ(x M M b dμ(x M f ϕ( f ϕ(m + M M + ϕ( f, (8. ϕ(f (x, (8. where M nd M. We hve: (i Ɣ J (ϕ 0,when(. holds nd for ll s [α, c], s [c, β] (. holds; (ii Ɣ ELR (ϕ 0,when(4.4holds,ndforlls [α, c] (4.5 holds nd for ll s [c, β] (4.6 holds. In the following two theores we give the en-vlue theores of the Lgrnge nd Cuchy type, respectively. Theore 8. Let f :[, b ] R nd g :[, b ] R be continuous functions, [α, β] R n intervl nd c α, β such tht f ([, b ] = [, M ] [α, c] nd g([, b ] =

21 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge of 5 [, M ] [c, β]. Let λ: [, b ] R nd μ: [, b ] R be continuous functions or functions of bounded vrition such tht λ( λ(b, μ( μ(b. Let Ɣ J nd Ɣ ELR be liner functionls defined bove, nd let ϕ C 3 ([α, β]. (i If (. holds nd for ll s [α, c], s [c, β] (. holds, then there exists ξ [α, β] such tht [ Ɣ J (ϕ= ϕ (ξ b g 3 b (x dμ(x 6 b f 3 (x dμ(x + f 3 ḡ 3 ]. (8.3 (ii If (4.4 holds, nd for ll s [α, c] (4.5 holds nd for ll s [c, β] (4.6 holds, then there exists ξ [α, β] such tht [ Ɣ ELR (ϕ= ϕ (ξ M ḡ 3 6 M + g M 3 M g 3 (x dμ(x dμ(x M f 3 M f M 3 M + f 3 (x ]. (8.4 Proof Since ϕ (x is continuous on [α, β], it ttins its iniu nd xiu vlue on [α, β], i.e. there exist = in x [α,β] ϕ (x ndm = x x [α,β] ϕ (x. The functions ϕ, ϕ : [α, β] R defined by ϕ (x=ϕ(x 6 x3 nd ϕ (x= M 6 x3 ϕ(x re 3-convex becuse ϕ (x 0ndϕ (x 0, so fro Rerk.6 it follows tht they belong to the clss K c([α, β]. Fro Theore. it follows tht Ɣ J(ϕ 0ndƔ J (ϕ 0, nd fro Theore 4.3 it follows tht Ɣ ELR (ϕ 0ndƔ ELR (ϕ 0ndsoweget 6 Ɣ J( ϕ Ɣ J (ϕ M 6 Ɣ J( ϕ, (8.5 6 Ɣ ELR( ϕ Ɣ ELR (ϕ M 6 Ɣ ELR( ϕ, (8.6 where ϕ(x =x 3. Since the function ϕ is 3-convex, we hve ϕ K c ([α, β], so by pplying Theore. (resp. Theore 4.3 wegetɣ J ( ϕ 0(resp.Ɣ ELR ( ϕ 0. If Ɣ J ( ϕ =0 (resp. Ɣ ELR ( ϕ=0,then(8.5 ipliesɣ J (ϕ=0(resp.(8.6 ipliesɣ ELR (ϕ=0,so(8.3 (resp. (8.4 holds for every ξ [α, β]. Otherwise, dividing (8.5 byɣ J ( ϕ>0(resp.(8.6 by Ɣ ELR ( ϕ>0weget 6 Ɣ J(ϕ Ɣ J ( ϕ M 6 ( resp. 6 Ɣ ELR(ϕ Ɣ ELR ( ϕ M, 6 nd continuity of ϕ ensures the existence of ξ [α, β] stisfying(8.3 (resp.ξ [α, β] stisfying (8.4. Theore 8. Let the conditions fro Theore 8. hold. Let ϕ, ψ C 3 ([α, β]. If Ɣ J (ψ 0 nd Ɣ ELR (ψ 0,then there exist ξ, ξ [α, β] such tht Ɣ J (ϕ Ɣ J (ψ = ϕ (ξ ψ (ξ or ϕ (ξ =ψ (ξ =0

22 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge of 5 nd Ɣ ELR (ϕ Ɣ ELR (ψ = ϕ (ξ ψ (ξ or ϕ (ξ =ψ (ξ =0. Proof Let us define function χ C 3 ([α, β] by χ(x=ɣ J (ψϕ(x Ɣ J (ϕψ(x. Due to the linerity of Ɣ J we hve Ɣ J (χ=0.theore8. iplies tht there exist ξ, ξ [α, β] such tht Ɣ J (χ= χ (ξ Ɣ J ( ϕ, 6 Ɣ J (ψ= ψ (ξ Ɣ J ( ϕ, 6 where ϕ(x =x 3.NowwehveƔ J ( ϕ 0,becuseotherwisewewouldhveƔ J (ψ =0, which is contrdiction with the ssuption Ɣ J (ψ 0.Sowehve χ (ξ =Ɣ J (ψϕ (ξ Ɣ J (ϕψ (ξ =0, nd this gives us the first cli of the theore. The second cli is proved in n nlogous nner, by observing the liner functionl Ɣ ELR insted of Ɣ J. Rerk 8.3 Note tht if in Theore 8. we set the function ψ to be ψ(x =x 3,weget exctly Theore 8.. Rerk 8.4 Note tht if we set the functions f, g, λ, ndμ fro our theores to fulfill the conditions fro Jensen s integrl inequlity or Jensen-Steffensen s, or Jensen-Brunk s, or Jensen-Bos inequlity, then - pplying tht inequlity on the function G which is continuous nd convex in both vribles - we see tht in these cses for ll s [α, c], s [c, β] inequlities in (. hold, nd so froour resultswe directly get the results frothe pper [0]. Rerk 8.5 If in the definition of the functionl Ɣ J (resp. Ɣ ELR wesetf (x=x nd g(x= x, then we get functionl tht represents the difference between the right nd the left side of the left-hnd prt (resp. right-hnd prt of the generlized Herite-Hdrd inequlity. In the se nner, dequte results of Lgrnge nd Cuchy type for those functionls cn be derived directly fro Theore 8. nd Theore Discrete cse Let [α, β] R nd c α, β.letx i [, b ] [α, c], p i R (i =,...,nbesuchtht 0, nd let y j [, b ] [c, β], q j R (j =,..., besuchthtq 0.Letϕ K c ([α, β] be continuous function. As before, otivted by the discrete results obtined in previous sections, we define the following liner functionls which, respectively, represent the difference between the right nd the left side of inequlities (3.3, (5.4, nd (7.0: Ɣ JD (ϕ= Q q j ϕ(y j p i ϕ(x i +ϕ( x ϕ(ȳ, (8.7

23 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 3 of 5 where x [α, c], ȳ [c, β]; Ɣ ELRD (ϕ= b ȳ b ϕ( + ȳ b ϕ(b Q where b nd b ; b x b ϕ( x b ϕ(b + q j ϕ(y j ( ( Ɣ G (ϕ=a ϕ q j y j + B q j ϕ(y 0 q j ϕ(y j p i ϕ(x i, (8.8 ( ( A ϕ p i x i B p i ϕ(x 0 + p i ϕ(x i, (8.9 where the conditions (7.5holdndA, B, A, B re defined in (7.7. We hve: (i Ɣ JD (ϕ 0,when(3. holds nd for ll s [α, c], s [c, β] (3. holds; (ii Ɣ ELRD (ϕ 0,when(5.holds,ndforlls [α, c] (5. holds nd for ll s [c, β] (5.3 holds; (iii Ɣ G (ϕ 0,when Q >0nd (7.6holds,ndforlls [α, β] (7.8nd(7.9 hold. The following two results re en-vlue theores of the Lgrnge nd Cuchy type, respectively, nd they re obtined in n nlogous wy to the theores of the se type in the previous sections, so we oit the proof. Theore 8.6 Let [α, β] R be n intervl nd c α, β. Let x i [, b ] [α, c], p i R (i =,...,n be such tht 0nd let y j [, b ] [c, β], q j R (j =,..., be such tht Q 0.Let Ɣ JD, Ɣ ELRD, nd Ɣ G be the liner functionls defined bove, nd let ϕ C 3 ([α, β]. (i If (3. holds nd for ll s [α, c], s [c, β] (3. holds, then there exists ξ [α, β] such tht [ Ɣ D (ϕ= ϕ (ξ 6 Q q j y 3 j p i x 3 i + x3 ȳ ]. 3 (8.0 (ii If (5. holds, nd for ll s [α, c] (5. holds nd for ll s [c, β] (5.3 holds, then there exists ξ [α, β] such tht [ Ɣ ELRD (ϕ= ϕ (ξ b ȳ 3 6 b + ȳ b 3 b Q b f b 3 f b b 3 + q j y 3 j ] p i x 3 i. (8.

24 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 4 of 5 (iii If Q >0nd (7.6 holds, nd for ll s [α, β] (7.8 nd (7.9 hold, then there exists ξ 3 [α, β] such tht [ ( 3 ( Ɣ G (ϕ= ϕ (ξ 3 A q j y j + B q j y q j y 3 j A ( p i x i 3 B ( ] p i x p i x 3 i. (8. Theore 8.7 Let the conditions of Theore 8.6 hold nd let ϕ, ψ C 3 ([α, β]. If Ɣ JD (ψ 0, Ɣ ELRD (ψ 0,nd Ɣ G (ψ 0,then there exist ξ, ξ, ξ 3 [α, β] such tht ll of the following stteents hold: Ɣ JD (ϕ Ɣ JD (ψ = ϕ (ξ or ϕ (ξ ψ =ψ (ξ =0, (8.3 (ξ Ɣ ELRD (ϕ Ɣ ELRD (ψ = ϕ (ξ or ϕ (ξ ψ =ψ (ξ =0, (8.4 (ξ Ɣ G (ϕ Ɣ G (ψ = ϕ (ξ 3 or ϕ (ξ ψ 3 =ψ (ξ 3 =0. (8.5 (ξ 3 Rerk 8.8 Note tht if in Theore 8.7 we set the function ψ to be ψ(x =x 3,weget exctly Theore 8.6. As consequence of the previous two theores, we now give soe further results in which we give explicit conditions on p i, x i (i =,...,nndq j, y j (j =,...,for(8.0nd (8.3 to hold, where using the properties of the function G we cn skip the suppleentry conditions on tht function. Corollry 8.9 Let x i [α, c], p i R + (i =,...,n nd y j [c, β], q j R + (j =,...,, nd let ϕ, ψ :[α, β] R. (i If (3. holds nd ϕ C 3 ([α, β], then there exists ξ [α, β] such tht (8.0 holds. (ii If (3. holds nd ϕ, ψ C 3 ([α, β], then there exists ξ [α, β] such tht (8.3 holds. Proof Note tht p i, q j > 0 iplies tht x [α, c] ndȳ [c, β], so we cn set the intervl [, b ]tobe[α, c] nd[, b ]tobe[c, β]. The function G isconvex,sobyjensen sinequlity we see tht the inequlities in (3. hold for ll s [α, c], s [c, β]. Now we cn pply Theore 8.6 nd Theore 8.7 to get the stteents of this corollry. Corollry 8.0 Let (x,...,x n be onotonic n-tuple, x i [α, c](i =,...,n nd (y,...,y be onotonic -tuple, y j [c, β](j =,...,. Let (p,...,p n be rel n-tuple such tht 0 P k (k =,...,n, >0, nd (q,...,q be rel -tuple such tht 0 Q k Q (k =,...,, Q >0. Let ϕ, ψ :[α, β] R.

25 Mikićetl.Journl of Inequlities nd Applictions (07 07:4 Pge 5 of 5 (i If (3. holds nd ϕ C 3 ([α, β], then there exists ξ [α, β] such tht (8.0 holds. (ii If (3. holds nd ϕ, ψ C 3 ([α, β], then there exists ξ [α, β] such tht (8.3 holds. Proof Suppose tht x x x n.wehve (x x= p i (x x i = i= (x j x j ( P j 0 j= so it follows tht x x.furtherore, n n ( x x n = p i (x i x n = (x j x j+ P j 0, so x x n. We see tht we hve obtined x n x x,thtis, x [α, c]. In n nlogous wy we cn get ȳ [c, β]. Therefore, s well s in the proof of the previous corollry, we cn set the intervl [, b ]tobe[α, c] nd[, b ]tobe[c, β]. By the Jensen-Steffensen inequlity we see tht for the convex function G the inequlities in (3. hold for ll s [α, c], s [c, β]. Now the stteents of this corollry follow directly fro Theore 8.6 nd Theore 8.7. Copeting interests The uthors declre tht they hve no copeting interests. Authors contributions All uthors contributed eqully nd significntly in writing this rticle. All uthors red nd pproved the finl nuscript. Acknowledgeents This reserch is supported by Crotin Science Foundtion under the project Received: 3 Noveber 06 Accepted: 6 Deceber 06 References. Steffensen, JF: On certin inequlities nd ethods of pproxition. J. Inst. Actur. 5, (99. Bos, RP: The Jensen-Steffensen inequlity. Publ. Elektroteh. Fk. Univ. Beogr., Ser. Mt. Fiz , -8 ( Pečrić, JE, Proschn, F, Tong, YL: Convex Functions, Prtil Orderings, nd Sttisticl Applictions. Mthetics in Science nd Engineering, vol. 87. Acdeic Press, Sn Diego (99 4. Levinson, N: Generlistion of n inequlity of Ky Fn. J. Mth. Anl. Appl. 8, ( Bloch, IA,Pečrić, J, Prljk, M: Generliztion of Levinson s inequlity. J. Mth. Inequl. 9(, (05 6. Bullen,PS:AninequlityofN.Levinson.Publ.Elektroteh.Fk.Univ.Beogr.,Ser.Mt.Fiz.4-460, 09- ( Mercer, AMcD: Short proof of Jensen s nd Levinson s inequlity. Mth. Gz. 94, (00 8. Pečrić, J: On n inequlity of N. Levinson. Publ. Elektroteh. Fk. Univ. Beogr., Ser. Mt. Fiz , 7-74 ( Witkowski, A: On Levinson s inequlity. RGMIA Reserch Report Collection 5, Art. 68 (0 0. Jkšetić, J, Pečrić, J, Prljk, M: Generlized Jensen-Steffensen inequlity nd exponentil convexity. J. Mth. Inequl. 9(4, (05. Pečrić, J, Perić, I,RodićLipnović, M: Unifor tretent of Jensen type inequlities. Mth. Rep. 6(66(,83-05 (04. Lh, P, Ribrič, M: Converse of Jensen s inequlity for convex functions. Publ. Elektroteh. Fk. Univ. Beogr., Ser. Mt. Fiz , 0-05 ( Edundson, HP: Bounds on the expecttion of convex function of rndo vrible. The Rnd Corportion, Pper No. 98 ( Jkšić, R, Pečrić, J: Levinson s type generliztion of the Edundson-Lh-Ribrič inequlity. Mediterr. J. Mth. 3(, (06 5. Fejér, L: Über die Fourierreihen, II. Mth. Nturwiss. Anz. Ungr. Akd. Wiss. 4, ( Fink, AM: A best possible Hdrd inequlity. Mth. Inequl. Appl., 3-30 ( Flore, A, Niculescu, CP: A Herite-Hdrd inequlity for convex-concve syetric functions. Bull. Mth. Soc. Sci. Mth. Rou. 50(98( ( Petrović, M: Sur une fonctionnelle. Publ. Mth. Univ. Belgrde, (93 9. Vsić, PM, Pečrić, JE: On the Jensen inequlity for onotone functions I. An. Univ. Vest. Tiiş., Ser. Mt.-Infor., (979