Research Article Application of Functionals in Creating Inequalities

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1 Journ of Funton Spes Voume 016, Arte ID , 6 pges Reserh Arte Appton of Funtons n Cretng Inequtes Ztko PvT, 1 Shnhe Wu, nd Vedrn Novose 1 1 Mehn Engneerng Futy n Svonsk Brod, Unversty of Osjek, Trg Ivne BrćMžurnć, Svonsk Brod, Crot Deprtment of Mthemts, Longyn Unversty, Longyn, Fujn 36401, Chn Correspondene shoud e ddressed to Shnhe Wu; shnhewu@gm.om Reeved 14 Juy 016; Aepted 7 Septemer 016 Adem Edtor: Cogero Vetro Copyrght 016 Ztko Pvć et. Ths s n open ess rte dstruted under the Cretve Commons Attruton Lense, whh permts unrestrted use, dstruton, nd reproduton n ny medum, provded the orgn work s propery ted. The pper des wth the fundment nequtes for onvex funtons n the ounded osed nterv. The mn nequty nudes onvex funtons nd postve ner funtons extendng nd refnng the funton form of Jensen s nequty. Ths nequty mpes the Jensen, Fejér, nd, thus, Hermte-Hdmrd nequty, s we s ther refnements. 1. Introduton In our reserh, we ppy the theory of postve ner funtons to onvex nyss. Let us rememer the nt notons reted to postve ner funtons on the spe of re funtons. Let S e nonempty set, nd et F = F(S) e suspe of the ner spe of re funtons on domn S. We ssume tht spe F ontns unt funton u defned y u(s) = 1 for every s S.SuhspeF ontnseveryreonstntκwthn the menng of κ=κund every omposte funton f(g) of funtong F nd n ffne funton f:r R.Atuy, f f(x) = κ 1 x+κ, then the omposton f (g) =κ 1 g+κ u (1) eongs to F. Let L = L(F) e the spe of ner funtons on spe F. FuntonL Ls sd to e unt (normzed) f L(u) = 1. Suh funton hs property L(κu) = κ for every re onstnt κ.ifg F s funton nd f L Ls unt funton,thenffnefuntonf:r R stsfes equty f(l(g))=l(f(g)). () Funton L L s sd to e postve (nonnegtve) f nequty L(g) 0 hods for every nonnegtve funton g F. If pr of funtons g 1,g F stsfes nequty g 1 (s) g (s) for every s S,thentfoowstht L (g 1 ) L(g ). (3) If g F s funton wth the mge n nterv [, ] (.e., u g u), then every postve unt funton L L meets nuson L(g) [, ] (.e., L(g) ). The sme s true for eh osed nterv I R. Introdung ontnuous onvex funton, we n expose the funton form of Jensen s nequty. Theorem A. Let g F efuntonwththemgenosed nterv I R,nd et L L e postve unt funton. Then eh ontnuous onvex funton f:i R suh tht f(g) F stsfes nequty f(l(g)) L(f(g)). (4) We w onsder onvex funtons n ounded osed nterv [, ] wth endponts <.Ehpontx [,]n e represented y the unque nom onvex omnton where x=α x +β x, (5) α x = x, β x = x. (6)

2 Journ of Funton Spes Convex funton f:[,] R s ounded y two nes. The sent ne of funton f psses through grph ponts A(, f()) nd B(, f()),ndtsequtons f se x (x) = x f () + f (). (7) Let (, ) e n nteror pont. The support nes of funton f pss through grph pont C(, f()).ehsupport ne s spefed y sope oeffent κ [f ( ), f (+)],nd ts equton s (x) =κ(x ) +f(). (8) The support-sent ne nequty (x) f(x) f se (x) (9) hods for every x [,]. In 1931, Jessen (see [1, ]) stted the funton form of Jensen s nequty for onvex funtons n nterv I R. In 1988, I. Rs nd I. Rş (see [3]) ponted out tht I must e osed otherwse t oud hppen tht L(g) I nd tht f must e ontnuous otherwse t oud hppen tht the nequty n formu (4) does not ppy. In Theorem A, we hve tken nto ount I. Rs nd I. Rş s remrks. Some generztons of the funton form of Jensen s nequty nefoundn[4]. A onse ook on funton nyss, whh ontns n essent overvew of opertor theory nd ndtes the mportne of postve ner funtons, s ertny the ook n [5].. Mn Resuts We frsty present the extenson of the nequty n formu (4) onernng nterv [, ]. Lemm 1. Let g F e funton wth the mge n [, ], nd et L L e postve unt funton. Then eh ontnuous onvex funton f:[,] R suh tht f(g) F stsfes doue nequty f(l(g)) L(f(g)) f se (L (g)). (10) Proof. The pont =L(g)s n nterv [, ]. Weskeththe proof n two steps dependng on the poston of. If (, ),wetkesupportne of f t. By ppyng postve funton L to the support-sent nequty n formu (9) wth x=g(s),wheres S,weget L( (g)) L (f (g)) L (f se (g)). (11) By utzng the ffnty of funtons (), the ove nequty tkes the form where the frst term nd f se v formu (L (g)) L(f (g)) f se (L (g)), (1) (L (g)) = f (L (g)). (13) If {, },wereyontheontnutyoff usng support ne t pont of open nterv (, ) tht s ose enough to. Gven ε>0,wenfnd (, ) so tht f () ε< (). (14) By omnng the ove nequty nd the nequty n formu (1) wth the support ne t,weotn f () ε< () L(f (g)) f se () =f(). (15) Lettng ε pproh zero, we reh the onuson L(f(g)) = f(). Inthsse,trvnequtyf() f() f() represents formu (10). Formu (10) n e expressed n the form whh nudes the onvex omnton of nterv endponts nd. The respetve form of Lemm 1 s s foows. Corory. Let g F e funton wth the mge n [, ], nd et L L e postve unt funton. Let =L(g) =α +β. (16) Then eh ontnuous onvex funton f:[,] R suh tht f(g) F stsfes doue nequty f(α +β ) L (f (g)) α f () +β f (). (17) Proof. As regrds to the st terms of formue (10) nd (17), we hve f se (L (g)) = α f se () +β f se () =α f () +β f () (18) euse of the ffnty of f se nd ts ondene wth f t endponts. In order to refne the nequty n formu (10), we w omne the sent nes of onvex funton f wth postve unt funtons. Lemm 3. Let (, ) e pont. Then eh onvex funton f:[,] R stsfes the sent nes nequty for every x [,]. mn {f se mx {f se (x),fse (x)} f(x) (x),fse (x)} (19) Proof. Cses x [,]nd x [,]shoud e onsdered. Suppose tht funton g F hs the mge n [, ] nd s not denty equ to or. Suh funton stsfes nequty g(s 1 ) g(s ) for some numer (, ) nd some pr of ponts s 1,s S.Inthtse,wenfndpr of funtons L 1,L L meetng reted nequty L 1 (g) L (g). (0)

3 Journ of Funton Spes 3 For exmpe, we n tke the pont evutons t s 1 nd s, tht s, the funtons defned y L 1 (q) = q(s 1 ) nd L (q) = q(s ) for every funton q F. In the mn theorem, we use funtons L 1 nd L stsfyngthenequtynformu(0). Theorem 4. Let (, ) e pont. Let g F e funton wth the mge n [, ], ndetl 1,L L e postve unt funtons suh tht L 1 (g) [, ] nd L (g) [, ]. Let L=λ 1 L 1 +λ L e onvex omnton of L 1 nd L. Then eh ontnuous onvex funton f:[,] R suh tht f(g) F stsfes the seres of nequtes f(l(g)) λ 1 f(l 1 (g)) + λ f(l (g)) L (f (g)) λ 1 f se f se (L (g)). (L 1 (g)) + λ f se (L (g)) (1) Proof. By ppyng the onvexty of f to onvex omnton L(g) = λ 1 L 1 (g) + λ L (g),weget f (L (g)) λ 1 f (L 1 (g)) +λ f (L (g)). () By ppyng the eft-hnd sde of formu (10) to L 1 nd L,weotn λ 1 f(l 1 (g)) + λ f(l (g)) λ 1 L 1 (f (g)) + λ L (f (g)) = L (f (g)). (3) As the rght-hnd sde of formu (19) wth x=g(s),the nequty f(g(s)) mx {f se (g (s)),fse (g (s))} (4) hods for every s S.BytngwthL 1 n the ove nequty nd usng ssumpton L 1 (g) [, ],wefnd L 1 (f (g)) L 1 (f se (g)) =fse (L 1 (g)), (5) nd smry, y tng wth L ndusngthessumpton L (g) [, ],wefnd L (f (g)) L (f se (g)) = fse (L (g)). (6) Mutpton y λ 1 nd λ ndthensummtonyed λ 1 L 1 (f (g)) + λ L (f (g)) λ 1 f se (L 1 (g)) + λ f se (L (g)). Usng mn sent f se, we reh onuson λ 1 f se (L 1 (g)) + λ f se (L (g)) λ 1 f se =f se (L (g)). (L 1 (g)) + λ f se (L (g)) (7) (8) Puttng together the nequtes n formue (), (3), (7), nd (8) nto seres, we heve the nequty n formu (1). y=f(x) L 1 (g) L(g) L (g) Fgure 1: Geometr presentton of the nequty n formu (1). The geometr presentton of the seres of nequtes n formu (1) s reted n Fgure 1. The nequty terms re represented y fve k dots ove pont L(g). To emphsze nterv endponts nd, wepresentthe foowng verson of Theorem 4. Corory 5. Let (, ) e pont. Let g F e funton wth the mge n [, ], ndetl 1,L L e postve unt funtons suh tht L 1 (g) [, ] nd L (g) [, ]. Let L=λ 1 L 1 +λ L e onvex omnton of L 1 nd L,nd et =L(g)=α +β. Then eh ontnuous onvex funton f:[,] R suh tht f(g) F stsfes the seres of nequtes f(α +β ) λ 1 f(l 1 (g)) + λ f(l (g)) where L (f (g)) αf () +βf() +γf() α f () +β f (), L α=λ 1 (g) 1, L β=λ (g), γ=λ 1 L 1 (g) L +λ (g). (9) (30) Proof. To ute oeffents α, β, ndγ, we nude onvex omntons L 1 (g) = α 1 +γ 1 nd L (g) = γ +β.then the fourth term of formu (1) tkes the form λ 1 f se (L 1 (g)) + λ f se (L (g)) =λ 1 α 1 f () +λ β f () +(λ 1 γ 1 +λ γ )f(). (31)

4 4 Journ of Funton Spes Tkng the oeffent of f() nd usng formu (6), we ute L α=λ 1 α 1 =λ 1 (g) 1. (3) Smry we determne β nd γ. Let us fnsh the seton y presentng the generzton of Theorem 4 tht uses sever sent nes. Corory 6. Let = 0 < 1 < < n 1 < n = e ponts. Let g F e funton wth the mge n [, ],ndet L L e postve unt funtons suh tht L (g) [ 1, ] for = 1,...,n.LetL= n =1 λ L e onvex omnton of funtons L. Then eh ontnuous onvex funton f:[,] R suh tht f(g) F stsfes the seres of nequtes f(l(g)) n =1 n =1 λ f(l (g)) L (f (g)) λ f se 1 (L (g)) f se (L (g)). 3. Apptons to Integr nd Dsrete Inequtes (33) We frsty utze Lemm 1 to otn very gener ntegr nequty. Corory 7. Let g : [,] R e n ntegre funton wth the mge n [, ], ndeth:[,] R e postve ntegre funton. Thenehonvexfuntonf:[,] R stsfes doue nequty f( gh dx ) f(g)hdx hdx hdx ( g) h dx f () ( ) hdx + (g )hdx ( ) hdx f (). (34) Proof. Let F ethespeofntegrefuntonsover domn S = [,]. Composton f(g) s ounded n [, ] nd ontnuous most everywhere n [, ]. Therefore f(g) s ntegre over [, ],tht s,f(g) F. The ntegrtng ner funton L defned y L(q)=L(q;h)= qh dx (35) hdx for every q F s postve nd unt. The frst term of formu (34) s equ to f(l(g)), theseondtermsequ to L(f(g)), ndthethrdtermsequtof se (L(g)). Thus, formu (34) fts nto the frme of formu (10), nd t s vd for ontnuous onvex funton f. Let us verfy tht the nequty n formu (34) ppes to onvex funton whh s not ontnuous t endponts. We oserve the poston of pont =L(g)= gh dx. (36) hdx If (, ), then we my utze ontnuous extenson f of f/(, ) to [, ] n formu (34). The frst two terms re the sme s we use f, nd the st terms stsfy nequty α f () +β f () = fse () <fse () (37) =α f () +β f (). So, formu (34) ppes to f n ths se. If {, }, then ether g(x) 0 or g(x) 0for every x [,]. We rerrnge formu (36) to e ntegr equton (g )hdx=0, (38) from whh t foows tht g(x) = for most every x [,]. Thuswehvethtf(g(x)) = f() for most every x [,], nd nequty f() f() f() represents formu (34). Respetng onsdertons, we my onude tht the nequty n formu (34) ppes to ny onvex funton f. The nequty n formu (34) s the extended verson of Jensen s nequty for the rto of ntegrs n nterv [, ], s we s the generzed form of the Fejér nd Hermte- Hdmrd nequty. Let us demonstrte the smpftons of the nequty n formu (34) retng to the dentty, unt, nd symmetr funton. Usng h(x) = 1, we get the extenson of the ss ntegr form of Jensen s nequty (see [6]) f( gdx ) f(g)dx ( g) dx ( ) f () + (g ) dx ( ) f (). (39) Usng dentty funton g(x) = x nd funton h(x), stsfyng equton h(x) = h( + x), whh represents the symmetry wth enter t mdpont ( + )/, wehvethe ss form of Fejér nequty (see [7]) f( + ) fh dx hdx f () +f(). (40)

5 Journ of Funton Spes 5 Nmey, s onsequene of the symmetry we hve euse xh dx = (x (+)/) hdx hdx hdx + ((+) /) hdx hdx = + (41) (x + )hdx=0. (4) Foowng formu (41), we n onude tht the Fejér nequty s vd for funton h stsfyng weker ondton xh dx/ h dx = ( + )/. Usng h(x) = 1 n Fejér s nequty n formu (40), we otn the ss form of Hermte-Hdmrd nequty (see [8, 9]) f( + ) fdx f () +f(). (43) To otn refnements of the nequty n formu (34), we use pont (, ) nd ppy Theorem 4. Corory 8. Let (, ) e pont. Let g:[,] R e n ntegre funton suh tht g(x) [, ] for x [,]nd g(x) [, ] for x [,],ndeth:[,] R e postve ntegre funton. Then eh onvex funton f:[,] R stsfes the seres of nequtes Proof. Just s n Corory 7, we use F s the spe of ntegre funtons n nterv [, ]. In order to ppy Theorem 4, we defne ntegrtng ner funtons L 1 (q) = L 1 (q; h) = qh dx hdx, L (q) = L (q; h) = qh dx hdx (45) for every q F.FuntonsL 1 nd L re postve nd unt. Sne g(x) [, ] for x [,],pontl 1 (g) fs nto [, ], ndsmrypontl (g) fs nto [, ]. Usng oeffents λ 1 = hdx hdx, λ = hdx hdx (46) nd funton onvex omnton L=λ 1 L 1 +λ L,weget L(g)=λ 1 L 1 (g) + λ L (g) = gh dx, hdx L(f(g))= f(g)hdx hdx. (47) By further utng the funton terms ordng to formu (1), we otn the ntegr terms of formu (44). f( gh dx ) hdx hdx f( gh dx hdx hdx ) + hdx f( gh dx ) f(g)hdx hdx hdx hdx ( g)hdx f () + (g )hdx f () ( ) hdx ( ) hdx +( (g )hdx + ( g)hdx )f() ( ) hdx ( ) hdx ( g) h dx f () + (g )hdx f (). ( ) hdx ( ) hdx (44) The seres of nequtes n formu (44) wth g(x) = x nd h(x) = 1 gves the refnement of the Hermte-Hdmrd nequty n formu (43) s foows: f( + ) f(+ fdx )+ f(+ ) f () + f () ( ) ( ) + 1 f () +f() f (). (48) The ove refnement hods for eh (, ). The verson of the ove refnement ws otned n [10] y usng the Jensen type nequty for onvex omntons wth the ommon enter. Tht nequty ws used to refne some mportnt mens. The nequty n formu (44) wth dentty funton g(x) = x nd symmetr funton stsfyng equton

6 6 Journ of Funton Spes h(x) = h(+ x)provdes the refnements of the Fejér nequty n formu (40). At the end, et us present the dsrete verson of Corory 8. Pont evutons g(x ) nd h(x ) w e shortened y g nd h,respetvey. Corory 9. Let (, ) e pont. Let g:[,] R e funton suh tht g(x) [, ] for x [,]nd g(x) [, ] for x [,],ndeth:[,] R e postve funton. Let x 1,...,x k [,]nd x k+1,...,x n [,]e ponts. Then eh onvex funton f:[,] R stsfes the seres of nequtes f( n =1 g h n =1 h ) k =1 h n =1 h f( k =1 g h k =1 h ) + n =k+1 h n =1 h k =1 ( g )h n =1 ( ) h +( k =1 (g )h n =1 ( ) h n =1 ( g )h n =1 ( ) h f( n =k+1 g h n =k+1 h ) n =1 f(g )h n =1 h f () + n =k+1 (g )h n =1 ( ) h f () + n =k+1 ( g )h n =1 ( ) h )f() f () + n =1 (g )h n =1 ( ) h f (). (49) Proof. Let F e the spe of re funtons on domn S= [, ]. We n ppy the proof of Corory 8 to summrzng ner funtons L 1 (q) = L 1 (q; h) = k =1 q h k =1 h, L (q) = L (q; h) = n =k+1 q h n =k+1 h tng to every q F, nd oeffents (50) Aknowedgments The work of the frst nd thrd uthor hs een fuy supported y Mehn Engneerng Futy n Svonsk Brod nd Crotn Sene Foundton under Projet HRZZ The work of the seond uthor hs een supported y the Ntur Sene Foundton of Fujn provne of Chn under Grnt no. 016J0103. The uthors woud ke to thnk Vemr Pvć who hs grphy prepred Fgure 1. Referenes [1] B. Jessen, Bemærknnger om konvekse Funktoner og Ugheder meem Mddeværder I, Mtemtsk Tdsskrft B, pp.17 8, [] B. Jessen, Bemærknnger om konvekse Funktoner og Ugheder meem Mddeværder II, Mtemtsk Tdsskrft B, pp , [3] I. Rs nd I. Rş, A note on Jessen s nequty, Itnernt Semnr on Funton Equtons, Approxmton nd Convexty, Unverstte Bes-Boy, n I. Rş, pp.75 80,Romn, Cuj-Npo, [4] Z. Pvć, Generztons of the funton form of Jensen s nequty, Advnes n Inequtes nd Apptons,vo.014, rte 33, 014. [5] W. Arveson, A Short Course on Spetr Theory, Sprnger, New York, NY, USA, 00. [6] J. L. W. V. Jensen, Sur es fontons onvexes et es négtés entre es veurs moyennes, At Mthemt, vo. 30, no. 1, pp ,1906. [7] L. Fejér, Üer de fourerrehen II, Mthemtsher und Nturwssenshfther Anzeger der Ungrshen Akdeme der Wssenshften, vo. 4, pp , 1906 (Hungrn). [8] Ch. Hermte, Sur deux mtes d une ntégre défne, Mthess,vo.3,p.8,1883. [9] J. Hdmrd, Étude sur es proprétés des fontons entμeres et en prtuer d une funton onsderée pr Remnn, Journ de Mthémtques Pures et Appquées,vo.58,pp ,1893. [10] Z. Pvć, Improvements of the Hermte-HADmrd nequty, Journ of Inequtes nd Apptons, vo. 015, rte, 015. [11] J. L. W. V. Jensen, Om konvekse Funktoner og Ugheder meem Mddeværder, Nyt Tdsskrft for Mtemtk B,vo.16,pp , λ 1 = k =1 h n =1 h, λ = n =k+1 h n =1 h. (51) Funtons L 1 nd L re ertny postve nd unt. The nequty n formu (49) s the extenson nd refnement of the fmous dsrete form of Jensen s nequty (see [11]). Competng Interests The uthors dere tht they hve no ompetng nterests.

7 Advnes n Opertons Reserh Advnes n Deson Senes Journ of Apped Mthemts Ager Journ of Proty nd Sttsts The Sentf Word Journ Internton Journ of Dfferent Equtons Sumt your mnusrpts t Internton Journ of Advnes n Comntors Mthemt Physs Journ of Compex Anyss Internton Journ of Mthemts nd Mthemt Senes Mthemt Proems n Engneerng Journ of Mthemts Dsrete Mthemts Journ of Dsrete Dynms n Nture nd Soety Journ of Funton Spes Astrt nd Apped Anyss Internton Journ of Journ of Stohst Anyss Optmzton