Order generalised gradient and operator inequalities

Transcription

1 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 DOI.86/s y R E S E A R C H Open Access Order generalised gradien and operaor inequaliies Sever S Dragomir and Eder Kikiany * * Correspondence: ekikiany@uj.ac.za; eder.kikiany@gmail.com Deparmen of Pure and Applied Mahemaics, Universiy of Johannesburg, PO Box 54, Auckland Park, 6, Souh Africa Full lis of auhor informaion is available a he end of he aricle Absrac We inroduce he noion of order generalised gradien, a generalisaion of he noion of subgradien, in he conex of operaor-valued funcions. We sae some operaor inequaliies of Hermie-Hadamard and Jensen ypes. We discuss he connecion beween he noion of order generalised gradien and he Gâeaux derivaive of operaor-valued funcions. We sae a characerisaion of operaor convexiy via an inequaliy concerning he order generalised gradien. MSC: 47A63; 46E4 Keywords: subgradien inequaliy; operaor convex funcion; operaor inequaliy Background Convex funcions play a crucial role in many fields of mahemaics, mos prominenly in opimisaion heory. There are wo main imporan inequaliies which characerise convex funcions, namely Jensen s and Hermie-Hadamard s inequaliies. In 95 96, Jensen defined convex funcions as follows: f : I R R isaconvexfuncionifandonlyif a + b f a+fb f for any a, b I. Inequaliy is referred o as Jensen s inequaliy. Hermie-Hadamard s inequaliy provides a refinemen for Jensen s inequaliy, namely, for a convex funcion f : I R R, a + b f b a b a f x dx f a+f b for any a, b I. We refer he reader o Secion for furher deails regarding hese inequaliies. Similarly o he case of real-valued funcions, he operaor convexiy can be characerised by some operaor inequaliies. Hansen and Pedersen [] characerise operaor convexiy via a non-commuaive generalisaion of Jensen s inequaliy. If f is a real coninuous funcion on an inerval I,andAH ishese of boundedself-adjoin operaorson a Hilber space H wih specra in I,henf is operaor convex if and only if f a i x ia i a i f x ia i 5 Dragomir and Kikiany; licensee Springer. This is an Open Access aricle disribued under he erms of he Creaive Commons Aribuion License hp://creaivecommons.org/licenses/by/4., which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly credied.

2 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page of 3 for x,...,x n AH anda,...,a n BH wih n a i a i =. We refer he reader o Secion for furher deails regarding his characerisaion. One of he useful differenial properies of convex funcions is he fac ha heir onesided direcional derivaives exis universally [, p.3]. Jus as he ordinary wo-sided direcional derivaives of a differeniable funcion can be described in erms of gradien vecors, he one-sided direcional derivaives can be described in erms of subgradien vecors [, p.3]. A vecor x is said o be a subgradien of a convex funcion f : K R n R a poin x if f x f y x x y for all y K. 3 This condiion is referred o as he subgradien inequaliy [,p.4].if3holdsforevery x K,hen3 characerises he convexiy of f cf. Eisenberg [3, Theorem ]. In his paper, we inroduce he noion of order generalised gradien cf. Secion 3 for operaor-valued funcions, which is a generalisaion of 3 wihou he assumpion of convexiy in he seings of bounded self-adjoin operaors on a Hilber space. Furhermore, we sae some inequaliies of Hermie-Hadamard and Jensen ypes for he order generalised gradien in Secion 4. Finally, insecion5, we sae he connecion beween he order generalised gradien and Gâeaux derivaive of operaor-valued funcions. We sae a characerisaion of convexiy analogues o 3 inheconexofoperaor-valued funcions. Inequaliies for convex funcions This secion serves as a poin of reference for known resuls regarding some inequaliies relaed o convex funcions boh real-valued and operaor-valued funcions.. Jensen s inequaliy Jensen s inequaliy for convex funcions plays a crucial role in he heory of inequaliies due o he fac ha oher inequaliies, such as he arihmeic-geomeric mean, Hölder, Minkowski and Ky Fan s inequaliies, can be obained as paricular cases of i. Le C be a convex subse of he linear space X and f be a convex funcion on C. Ifp = p,...,p n is a probabiliy sequence and x =x,...,x n C n,hen f p i x i p i f x i. 4 This inequaliy is referred o as Jensen s inequaliy. Recenly, Dragomir [4] obainedhe following refinemen of Jensen s inequaliy: f [ p j x j min p k f k {,...,n} k= n p jx j p k x k p k [ n p k f p jx j p k x k + n p k ] + p k f x k ] p k f x k k=

3 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page 3 of 3 [ n max p k f p ] jx j p k x k + p k f x k p k p j f x j, 5 k {,...,n} where f, x k and p k are as defined above. For oher refinemens of Jensen s inequaliy, we refer he reader o Pečarić and Dragomir [5] and Dragomir [6]. The above resul provides a differen approach o he one ha Pečarić and Dragomir [5] obained in 989 f p i x i i,...,i k+ = i,...,i k = xi + + x ik+ p i...p ik+ f k + xi + + x ik p i...p ik f k p i f x i 6 for k, and p, x are as defined above. If q,...,q k wih k q j =, hen he following refinemen obained in 994 by Dragomir [6]alsoholds: f p i x i i,...,i k = i,...,i k = xi + + x ik p i...p ik f k p i...p ik f q x i + + q k x ik+ p i f x i, 7 where k n and p, x are as defined above. For more refinemens and applicaions relaed o he generalised riangle inequaliy, he arihmeic-geomeric mean inequaliy, he f -divergence measures, Ky Fan s inequaliy, ec., we refer he readers o [6 ]and[4].. Hermie-Hadamard s inequaliy The followinginequaliyalsoholdsforanyconvexfuncion f defined on R: a + b b af b a f x dx b a f a+fb, a, b R. 8 IwasfirsdiscoveredbyHermiein88inhejournalMahesis []. However, his resul was nowhere menioned in he mahemaical lieraure and was no widely known as Hermie s resul []. Beckenbach, a leading exper on he hisory and he heory of convex funcions wroe ha his inequaliy was proven by Hadamard in 893 [3]. In 974, Mirinović found Her-

4 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page 4 of 3 mie s noe in Mahesis []. Since 8 was known as Hadamard sinequaliy, he inequaliy is now commonly referred o as Hermie-Hadamard s inequaliy []. Hermie-Hadamard s inequaliy has been exended in many differen direcions. One of he exensions of his inequaliy is in he vecor space seings. Firsly, we sar wih he following definiions and noaion: Le X be a vecor space and x, y be wo disinc vecors in X. We define he segmen generaed by x and y o be he se [x, y]:= { x + y, [, ] }. For any real-valued funcion f defined on he segmen [x, y], here exiss an associaed funcion g x,y :[,] R wih g x,y =f [ x + y ]. We remark ha f is convex on [x, y]ifandonlyifg is convex on [, ]. For any convex funcion defined on a segmen [x, y] X, wehavehehermie-hadamard inegral inequaliy cf. Dragomir [4, p.] and Dragomir [5, p.]: x + y f f [ x + y ] d f x+fy, x, y X, 9 which can be derived by he classical Hermie-Hadamard inequaliy 8 for he convex funcion g x,y :[,] R. Consider he funcion f x= x p x X and p <, which is convex on X, hen we have he following norm inequaliy derived from 9 [6, p.6]: x + y p for any x, y X. x + y p x p + y p d.3 Non-commuaive generalisaion of Jensen s inequaliy Hansen [7] discussed Jensen s operaor inequaliy for operaor monoone funcions. Moivaed by Aujla s work [8] on he marix convexiy of funcions of wo variables, Hansen [9] characerised operaor convex funcions of wo variables in erms of a noncommuaive generalisaion of Jensen s inequaliy cf. [9, Theorem 3.]. A simplified proofofhisresulformulaedformaricesisgiveninaujla[].thecaseforseveral variables is given in Hansen []. The case for self-adjoin elemens in he algebra M n of n-square marices is given in Hansen and Pedersen []. Finally, Hansen and Pedersen [] presened a generalisaion of he above resuls for self-adjoin operaors defined on a Hilber space. Theorem We denoe by BH he Banach algebra of all bounded linear operaors on he Hilber space H. If f is a real coninuous funcion on an inerval I, and AH is he se of bounded self-adjoin operaors on a Hilber space H wih specra in I, hen he following condiions are equivalen: i f is operaor convex; ii f n a i x ia i n a i f x ia i for x,...,x n AH and a,...,a n BH wih n a i a i = ;

5 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page 5 of 3 iii f v xv v f xv for any x AH and any isomery v BH; iv pf pxp + s pp pf xp for x AH, projecion p BH, every self-adjoin operaor x wih specrum in I and s I..4 Subgradien inequaliy Recall he following definiion of a subgradien []. Definiion Avecorx is said o be a subgradien of a convex funcion f : K R n R a poin x if f x f y x x y for all y K. The following heorem is a useful characerisaion of convexiy cf. Eisenberg [3,Theorem ]. Theorem 3 If U is a nonempy open subse of R n, f : U R is a differeniable funcion on U, and K is a convex subse of U, hen f is convex on K if and only if f x f y x y T f y for all x, y K where f y denoes he gradien of f a y. This heorem has been generalised and employed in obaining opimaliy condiions of a non-differeniable minimax programming problem in complex spaces cf. Lai and Liu [3]. Noe ha x y T f y canbewrienasf y x y, which can be inerpreed as he direcional derivaive of f a poin y in x y direcion. 3 Order generalised gradien Throughouhepaper,weusehefollowingnoaion.WedenoebyBH hebanachalgebra of all bounded linear operaors on he Hilber space H,,, and by AHhelinear subspace of all self-adjoin operaors on H.WedenoebyP + H AH he convex cone of all posiive definie operaors defined on H,hais,P P + Hifandonlyif Px, x, and for all x H, Px, x = implies x =. This gives a parial ordering we refer o i as he operaor order on AH, where wo elemens A, B AHsaisfyA B if and only if B A P + H. Definiion 4 Le C be a convex se in AH. A funcion f : C AH hashefuncion f : C AH AHasanorder generalised gradien if f A f B f B, A B for any A, B C in he operaor order of AH. Remark 5 We remark ha in, if f is a real-valued differeniable funcion on an open se U R n,and f is he gradien of f,henbecomes. We also noe ha here is no assumpion of convexiy a his poin. We discuss he convexiy case in Secion 5.

6 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page 6 of 3 Proposiion 6 If Q AH and f : AH AH, f A=QA Q, hen f B, X:=QBX + XBQ 3 is an order generalised gradien for f. Proof Observe ha BX + XB AHandifP AHhenPBX + XBP AH. We need o prove ha f A f B f B, A B for any A, B AH, ha is, QA Q QB Q Q [ BA B+A BB ] Q. 4 Since Q [ BA B+A BB ] Q = QBAQ QB Q + QABQ QB Q, hence 4is equivalen o QA Q QB Q QBAQ QB Q + QABQ QB Q which is also equivalen o QA B Q which always holds. This complees he proof. We denoe by PH AH he convex cone of all nonnegaive operaors defined on H. Proposiion 7 If P PH, hen he funcion f : AH AH, f A=APA has f B, X:=XPB + BPX 5 as an order generalised gradien. Proof Observe ha XPB + BPX AH. We need o prove ha APA BPB A BPB + BPA B = APB BPB + BPA BPB, ha is, APA APB BPA + BPB.

7 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page 7 of 3 Bu APA APB BPA + BPB =A BPA B and since A BPA B, and his complees he proof. Recall P + H AH he convex cone of all posiive definie operaors defined on H, ha is, P P + Hifandonlyif Px, x, and for all x H, Px, x = implies x =. Proposiion 8 Le f : P + H AH defined by f A=QA Q, where Q AH. The funcion f : P + H P + H AH wih f B, X= QB XB Q is an order generalised gradien for f. Proof For B P + H, B P + H henb XB P + H for any X P + H andhus QB XB Q P + Hshowingha f B, X AH. We need o prove ha QA Q QB Q QB A BB Q, ha is, QA B AB Q + QB A BB Q or equivalenly QA B AB Q QB B AB Q or Q A B B AB Q. Bu Q A B B AB Q = Q A B AA B AB Q = Q A B A A B Q, which is rue since for A P + Hwehaveha A B A A B and Q AH.

8 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page 8 of 3 4 Inequaliies involving order generalised gradiens We sar his secion by he following definiion. Definiion 9 An order generalised gradien f : C AH AHis i operaor convex if f B, αx + βy α f B, X+β f B, Y for any B C, X, Y AH and α, β wih α + β =; ii operaor sub-addiive if f B, X + Y f B, X+ f B, Y for any B C and X, Y AH; iii posiive homogeneous if f B, αx=α f B, X for any B C, X AH and α ; iv operaor linear if f B, αx + βy =α f B, X+β f B, Y for any B C, X, Y AH and α, β R. I can be seen ha if f, is operaor linear, hen i is posiive homogeneous and subaddiive. If f, is posiive homogeneous and sub-addiive, hen i is operaor convex. Theorem Le f : C AH be operaor convex and f : C AH AH be an order generalised gradien for f. Then, for any A, B, C and [, ], we have he inequaliies f B, B A f A, B A f A+ fb f A + B f A + B,. 6 Proof If we wrie he definiion of f for B insead of A,wege f B f A f A, B A, which is equivalen o f A, B A f A f B. Therefore, for any A, B C, we have he gradien inequaliies f A, B A f A f B f B, A B. 7

9 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page 9 of 3 Since C isaconvexse,henceby7wehave f A, B A f A f A + B f A + B, B A 8 and f B, B A f B f A + B f A + B, B A 9 for any,. If we muliply 8by and 9by and add he obained inequaliies, hen we ge f A, B A f B, B A f A+ fb f A + B f A + B, B A + f A + B, B A. Since f, is operaor convex, we also know ha f A + B, B A + f A + B, B A f A + B, B A+ B A f A + B,, which complees he proof. Corollary Under he assumpions of Theorem, If f, is posiive homogeneous, hen we have [ f B, A B+ f A, B A ] f A+ fb f A + B. If f, is operaor linear, hen [ f B, B A f A, B A ] f A+ f B f A + B. 4. Hermie-Hadamard ype operaor inequaliies In his subsecion, we will sae inequaliies of Hermie-Hadamard ype for order generalised gradiens.

10 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page of 3 Corollary Under he assumpions of Theorem, if f is posiive homogeneous, hen we have he following inequaliy: [ f B, A B+ f A, B A ] 6 f A+f B We obain by inegraing over [, ]. f A + B d. Example 3. We consider he funcion f A=QA Q wih Q AH.Wenoehaheorder generalised gradien f B, X=QBX + XBQ is operaor linear. Then f B, X f A, X=QBX + XBQ QAX + XAQ = Q [ B AX + XB A ] Q. For X = B A,wehenge f B, B A f A, B A=QB A Q. Applying inequaliy we have QB A Q Q [ A + B A + B ] Q 3 for any A, B AH and Q AH.. We consider he funcion f A=APA wih P PH. We noe hahe order generalised gradien f B, X=XPB + BPX is operaor linear. Then f B, X f A, X=XPB + BPX XPA APX = XPB A+B APX. If X = B A,wehenge f B, B A f A, B A=B APB A.

11 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page of 3 Applying inequaliy we have B APB A APA + BPB A + B P A + B for any A, B AH and P PH. 3. For f A=QA Q wih Q AH and A P + H, we noe ha he order generalised gradien f B, X= QB XB Q is operaor linear. Then f B, X f A, X= QB XB Q + QA XA Q. For X = B A,wege f B, B A f A, B A = QB B AB Q + QA B AA Q = Q B B AB Q + Q A BA A Q = QA BA Q + QB AB Q QB Q QA Q. By we have he inequaliy [ QA BA Q + QB AB Q QB Q QA Q ] QA Q + QB Q Q A + B Q for any A, B P + H and Q AH. 4. Jensen ype operaor inequaliies In his subsecion, we will sae inequaliies of Jensen ype for order generalised gradiens. Theorem 4 Le f : C AH AH be a funcion ha possesses f : C AH AH as an order generalised gradien. Then, for any A i C, i {,...,n} and p i wih := n p i >,we have he inequaliies p j f A j, A j p j f A j f p j f, A j. 4

12 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page of 3 Proof From he definiion of an order generalised gradien we have f A, B A f A f B f B, A B. 5 Now, if we choose A = A j, j {,...,n} and B =/ n p ia i in 5, hen we ge f A j, A j f A j f f, A j 6 for any j {,...,n}. We obain he desired inequaliies 4 by muliplying he inequaliies in 6byp j and aking he sum over j from o n; and divide he resuled inequaliies by. Corollary 5 Under he assumpions of Theorem 4, we have he following resuls: If f : C AH is convex, hen p j f A j f f,. 7 If f is linear, hen f B,=for any B, andwegehejensen sinequaliy p j f A j f. 8 3 If f is linear, we have p j f A j, A j P n p j f A j f. 9 Theorem 6 Under he assumpions of Theorem 4, we have he following resuls: p j f A j p j f A, A j A f A p j f A j + p j f A j, A A j.

13 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page 3 of 3 Proof From 5wealsohave f A, A j A f A f A j f A j, A A j. 3 If we muliply 3 byp j andakehesumoverj from o n and divide he resuled inequaliies by,hen p j f A, A j A f A p j f A j p j f A j, A A j, which complees he proof. Remark 7 If f is linear in Theorem 6, hen we ge simpler inequaliies such as f A p j f A j + p j f A j, A p j f A j, A j and f A p j f A j p j f A, A j + p j f A, A. Therefore, if A AH issuchha p j f A j, A p j f A j, A j, hen we have he Slaer ype inequaliy cf. Slaer [4]andPečarić[5] f A p j f A j. 5 Connecion wih Gâeaux derivaives In his secion, we consider he connecion beween order generalised gradiens and Gâeaux derivaives. We refer he reader o Dragomir [6]for some inequaliies of Jensen ype, involving Gâeaux derivaives of convex funcions in linear spaces. Le C AHbeaconvexse.Thenf : C AHissaidobeoperaor convex if for all [, ] and A, B C,wehave f [ A + B ] f A+f B. We sar wih he following lemmas.

14 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page 4 of 3 Lemma 8 Le F : R BH be a funcion such ha lim ± F exiss. Then lim ± F is a bounded linear operaor and [ ] lim F ± x, y for all nonzero x, y H. = lim ± Fx, y Proof We provide he proof for he righ-sided limi, as he proof for he lef-sided limi follows similarly. Le ε > and for x, y H, wherex, y,seε = ε/ x H y H. Since lim + F=L,hereexissδ such ha F L BH < ε when < < δ.noehal BH sincebh is a Banach space, hence F L is also a bounded linear operaor. Now, we have Fx, y Lx, y F L x H y H F L BH x H y H < ε x H y H = ε, which complees he proof. Lemma 9 Le f : AH AH be operaor convex and A AH. Then, for all B AH, boh limis and f A B= lim f A + B f A + f A B= lim + f A + B f A exis and are bounded self-adjoin operaors. Proof Fix an arbirary B AH, and le G= f A + B f A, R \{}. We wan o show ha G is nondecreasing. Le < <,hen [ f A + B f A =f A + B+ ] A f A f A + B+ = [ f A + B f A ]. f A f A

15 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page 5 of 3 Thus, Also, f A + B f A f A + B f A. f A B f A = f [A + B] f A Noe also ha A + B B f A=f f [A + B] f A = f A B f A. [ = f A + B+ ] A B f A + B+ f A B, which implies ha f A f A + B+f A B; and hus f A + B f A [ f A B f A ], which implies ha f A + B f A f A B f A. By he above exposiions, we conclude ha G is nondecreasing on R\{}. This proves ha boh f AB and + f AB exis and are bounded linear operaors by Lemma 8. Noe ha for all R, anda, B AH, f [B + A B] f B is a self-adjoin operaor. If x, y H, hen Lemma 8 gives us [ ] f [B + A B] f B lim x, y ± [ ] f [B + A B] f B = lim x, y ± = lim ± = x, x, lim ± [ f [B + A B] f B [ f [B + A B] f B ] y ] y, which complees he proof.

16 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page 6 of 3 Theorem Le C AH be a convex se and f : C AH be operaor convex. Then ± fdefinedby ± f A f A + B f A B= lim, A, B C 3 ± is an order generalised gradien. Proof Le, and A, B C.Sincef is operaor convex, we have f [B + A B] f B This is equivalen o K := f A f B Noe ha for all x H, [ ] lim K ± x, x = f [ B + A] f B f B+f A fb f [B + A B] f B = lim x ± Kx, P + H. = f A f B. by Lemma 8. SinceK P + H, Kx, x, hence [lim ± K]x, x, which implies ha [ lim f A fb ± Therefore, ] f [B + A B] f B P + H. + f B f [B + A B] f B A B= lim f A fb. ± Lemma 9 gives us f B A B + f B A B, which implies ha f B A B f A f B. Thus boh + f and f are order generalised gradiens. Proposiion Le f : AH AH be operaor convex and A AH. The righ Gâeaux derivaive of f is sub-addiive, i.e. + f A B + C + f A B+ + f A C

17 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page 7 of 3 for any B, C AH. The lef Gâeaux derivaive of f is super-addiive, i.e. f A B + C f A B+ f A C for any B, C AH. Proof Since f is operaor convex, we have he following for any B, C A and >: f [A + B + C] f A = f [ A +B+ A +C] f A f A +B f A + f A +C f A. By a similar argumen o he proof of Theorem,weconcludeha + f A f [A + B + C] f A B + C = lim + f A +B f A f A +C f A lim + lim + + = + f A B+ + f A C as desired. The proof for he lef Gâeaux derivaive of f followssimilarly. Remark We remark ha he Gâeaux laeral derivaives is always posiive homogeneous wih respec o he second variable, i.e. for any funcion f : AH AH and fixed A AH, ± f A αb=α ± f A B for all α andb AH. The Gâeaux derivaive, on he oher hand, is always homogeneous wih respec o he second variable, i.e. for any funcion f : AH AHandfixed A AH, f A αb=α f A B for all α C and B AH. The following resul resaes Theorem 3 in he seing of operaor-valued funcions. Corollary 3 Le C AH be a convex se and f : C AH be a Gâeaux differeniable funcion. Then f is operaor convex if and only if f definedby f A + B f A f A B=lim, A, B C 3 is an order generalised gradien.

18 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page 8 of 3 Proof For any A, B C,iff is operaor convex, hen by Theorem ± f A B= lim ± f A + B f A are order generalised gradiens. Since f is assumed o be Gâeaux differeniable, boh limis are equal, hence f A B=lim f A + B f A is an order generalised gradien for any A, B C. Conversely, we have he following inequaliy: f B A B f A f B for any A, B C.LeC, D C,,, and choose A = C and B = C + D.Thenwe have f [ C + D ] C D f C f [ C + D ]. 33 Le A = D and B = C + D.Thenwehave f [ C + D ] C D f D f [ C + D ]. 34 Muliply 33 by and 34 by, and add he resuling inequaliies o obain f [ C + D ] f C+ f D, which complees he proof. The following resul follows by Corollary and employing he fac ha he Gâeaux laeral derivaives are posiive homogenous. Corollary 4 Hermie-Hadamard ype inequaliy Le f : C AH AH be operaor convex. The following inequaliy holds: [ ± f B A B+ ± f A B A ] 6 f A+f B f A + B d. The above inequaliyalsoholds for f when f is Gâeaux differeniable. Example 5 We noe hahefuncion f x= logx is operaor convex. The log funcionis operaor Gâeaux differeniable wih he following explici formula for he derivaive cf. Pedersen [7,p.55]: loga B= si + A BsI + A ds

19 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page 9 of 3 for A, B AH and I he ideniy operaor. Thus, we have he following inequaliy: [ si + B A BsI + B ds 6 ] + si + A B AsI + A ds loga+logb + log A + B d. We consider he operaor convex funcion f x=xlogx, and using he following represenaion cf. Pedersen [7,p.55] log= s + s + ds, and noing he fac ha d d log=log,wehave f A B= s + si + A A IBds. Then we have he following inequaliies: [ 6 s + si + B B IA B ds ] + s + si + A A IB A ds A loga+b logb [ A + B ] log A + B d. The following resuls follow by Theorems 4 and 6. Corollary 6 Jensen ype inequaliy Le f : C AH AH be operaor convex. Then, for any A i C, i {,...,n} and p i wih := n p i >,we have he inequaliies p j ± f A j p j f A j f p j ± f A j A j.

20 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page of 3 We also have p j f A j p j ± f A A j A f A p j f A j + p j ± f A j A A j. The above inequaliies alsoholdfor f when f is Gâeaux differeniable. Example 7 We have he following inequaliies for he operaor convex funcion f x= logx: p j si + A j A j si + A j ds and p j loga j +log p j si + A j si + loga p j loga j + ds p j si + A A j AsI + A ds p j loga j p j si + A j A A j si + A j ds. We have he following inequaliies for he operaor convex funcion f x=x logx: p j s + si + A j A j I p j A j loga j p j I A j ds log si + s + A j ds

21 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page of 3 and p j A j loga j p j s + si + A A IA j A ds A loga p j A j loga j + p j s + si + A j A j IA A j ds. 6 Conclusions For a funcion f : C AH definedonaconvexsec AH, he funcion f : C AH AH isanorder generalised gradien if f A f B f B, A B for any A, B C in he operaor order of AH. We have he followingoperaorinequaliies. Operaor inequaliies of Hermie-Hadamard ype: [ f B, A B+ f A, B A ] 6 f A+f B Operaor inequaliies of Jensen ype: and p j f A j, f A + B d for any A, B C. A j p j f A j f p j f, A j ; p j f A j p j f A, A j A f A p j f A j + p j f A j, A A j for any A C, A i C, i {,...,n} and p i wih := n p i >. 3 Operaor inequaliies of Slaer ype: if f is linear and A AH is such ha p j f A j, A p j f A j, A j,

22 Dragomir and Kikiany Journal of Inequaliies and Applicaions 5 5:49 Page of 3 hen f A p j f A j for any A C, A i C, i {,...,n} and p i wih := n p i >. Order generalised gradiens exend he noion of subgradiens, wihou he assumpion of convexiy, for operaor-valued funcions. This noion is also conneced o he Gâeaux laeral derivaives. If f is operaor convex, hen ± f defined by ± f A f A + B f A B= lim, A, B C ± is an order generalised gradien. Furhermore, if f : C AH is a Gâeaux differeniable funcion, we have he following characerisaion: f is operaor convex if and only if f defined by f A + B f A f A B=lim, A, B C is an order generalised gradien. This characerisaion of convexiy is a generalised version of Theorem 3 of Secion cf. Eisenberg [3, Theorem ]. Compeing ineress The auhors declare ha hey have no compeing ineress. Auhors conribuions SSD and EK conribued equally in all sages of wriing he paper. All auhors read and approved he final manuscrip. Auhor deails School of Engineering and Science, Vicoria Universiy, PO Box 448, Melbourne, Vicoria 8, Ausralia. Deparmen of Pure and Applied Mahemaics, Universiy of Johannesburg, PO Box 54, Auckland Park, 6, Souh Africa. Acknowledgemens The research of E Kikiany is suppored by he Claude Leon Foundaion. The auhors would like o hank he anonymous referees for valuable suggesions ha have been incorporaed in he final version of he manuscrip. Received: 5 November 4 Acceped: 3 January 5 References. Hansen, F, Pedersen, GK: Jensen s operaor inequaliy. Bull. Lond. Mah. Soc. 354, Rockafellar, RT, Tyrrell, R: Convex Analysis, Princeon Landmarks in Mahemaics. Reprin of he 97 original, Princeon paperbacks. Princeon Universiy Press, Princeon Eisenberg, E: A gradien inequaliy for a class of nondiffereniable funcions. Oper. Res. 4, Dragomir, SS: A refinemen of Jensen s inequaliy wih applicaions for f -divergence measures. Taiwan. J. Mah. 4, Pečarić, JE, Dragomir, SS: A refinemen of Jensen inequaliy and applicaions. Sud. Univ. Babeş Bolyai, Mah. 4, Dragomir, SS: A furher improvemen of Jensen s inequaliy. Tamkang J. Mah. 5, Dragomir, SS: An improvemen of Jensen s inequaliy. Bull. Mah. Soc. Sci. Mah. Roum. 3484, Dragomir, SS: Some refinemens of Ky Fan s inequaliy. J. Mah. Anal. Appl. 63, Dragomir, SS: A new improvemen of Jensen s inequaliy. Indian J. Pure Appl. Mah. 6, Dragomir, SS: Semi-Inner Producs and Applicaions. Nova Science Publishers, New York 4. Mirinović, DS, Lacković,IB:Hermieandconvexiy.Aequ.Mah.8, Pečarić, JE, Proschan, F, Tong, YL: Convex Funcions, Parial Orderings, and Saisical Applicaions. Academic Press, San Diego Beckenbach, EF: Convex funcions. Bull. Am. Mah. Soc. 54, Dragomir, SS: An inequaliy improving he firs Hermie-Hadamard inequaliy for convex funcions defined on linear spaces and applicaions for semi-inner producs. J. Inequal. Pure Appl. Mah. 3, Aricle ID 3 5. Dragomir, SS: An inequaliy improving he second Hermie-Hadamard inequaliy for convex funcions defined on linear spaces and applicaions for semi-inner producs. J. Inequal. Pure Appl. Mah. 33,Aricle ID35

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