REFINEMENT OF JENSEN S INEQUALITY FOR OPERATOR CONVEX FUNCTIONS

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1 Avaiabe oie at Adv. Iequa. App. 204, 204:26 ISSN: REFINEMENT OF JENSEN S INEQUALITY FOR OPERATOR CONVEX FUNCTIONS L. HORVÁTH, K.A. KHAN 2, J. PEČARIĆ 3,4, Departmet o Mathematics, Uiversity o Paoia, Egyetem u.0, 8200 Veszprém, Hugary 2 Departmet o Mathematics, Uiversity o Sargodha, Sargodha 4000, Paista 3 Abdus Saam Schoo o Mathematica Scieces, GC Uiversity, 68-B, New Musim Tow, Lahore 54600, Paista 4 Facuty o Textie Techoogy, Uiversity o Zagreb, 0000 Zagreb, Croatia Copyright c 204 Horváth, Kha Pečarić. This is a ope access artice distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, reproductio i ay medium, provided the origia wor is propery cited. Abstract. I this paper, we give a reiemet o discrete Jese s iequaity or the operator covex uctios. We auch the correspodig mixed symmetric meas or positive se-adjoit operators deied o Hibert space aso estabish the reiemet o iequaity betwee power meas o stricty positive operators. Keywords: se-adjoit operators; operator covex uctios; operator meas; symboic cacuus. 200 AMS Subject Cassiicatio: 26D07, 46E63, 46E64.. INTRODUCTION-PRELIMINARIES H wi rom ow o deote a compex Hibert space. SI meas the cass o a se-adjoit bouded operators o H whose spectra are cotaied i a iterva I R. The spectrum o a bouded operator A o H is deoted by SpA. Correspodig author E-mai addresses: horvath@amos.ui-pao.hu L. Horváth, huramsms@gmai.com K.A. Kha, pecar ic@mahazu.hazu.hr J. Pečarić Received February 27, 203

2 2 L. HORVÁTH, K.A. KHAN, J. PEČARIĆ Let : D R R be a uctio et I D be a iterva. is said to be operator mootoe o I i is cotiuous o I A, B SI, A B i.e. B A is a positive operator impy A B. The uctio is said to be operator covex o I i is cotiuous o I sa +tb s A +t B or a A, B SI or a positive umbers s t such that s + t =. The uctio is caed operator cocave o J i is operator covex o J. Theorem.. Jese s operator iequaity: Let I R be a iterva, et : I R be a operator covex uctio o I. I C i SI i =,...,, w i > 0 i =,..., such that i= w i =, the i=w i C i i= w i C i. I is a operator cocave uctio o I, the the iequaity i is reversed. Some iterpoatios o are give i [3] as oows. Theorem.2. Uder the coditios o the Jese s operator iequaity 2 where or i= 3, := w i C i =,...,..., = i <...<i w i j j= i= j= w i C i, w i j C i j. w i j j= Theorem.3. I the coditios o the Jese s operator iequaity are satisied, the 4 where or i= w i C i... +,,..., = i= w i C i,

3 REFINEMENT OF JENSEN S INEQUALITY FOR OPERATOR CONVEX FUNCTIONS 3 5, = + i... i w i j j= j= w i j C i j. w i j j= A se-adjoit bouded operator A o H is caed stricty positive i it is positive ivertibe, or equivaety, SpA [m, M] or some 0 < m < M. The power meas or stricty positive operators C := C,...,C with positive weights w := w,...,w are deied i [3] as oows: 6 M r C,w = M r C,...,C ;w,...,w := W i= w i C r i r, where r R \ {0} W : = w i. The oowig resut about the mootoicity o power i= meas is aso give i [3]: 7 M s C,w M r C,w hods i either s r, s /,, r /, or /2 s r or s r /2. Some symmetric mixed meas, correspodig to the expressios 3 5 are itroduced i [3]: or r,s R \ {0} or W =, deie 8 M s,r; := i <...<i s w i j M rc s i,...,c i ;w i,...,w i, j= where, 9 M s,r; := + i... i s w i j M rc s i,...,c i ;w i,...,w i, j= where. The oowig resut rom [3] gives some reiemets o 7.

4 4 L. HORVÁTH, K.A. KHAN, J. PEČARIĆ Theorem.4. Let C be a -tupe o stricty positive operators, et w i > 0 i =,..., such that W =. The the oowig iequaities are vaid 0 M s C,w = M s,r;... M s,r;... M s,r; = M r C,w, M s C,w = M s,r;... M s,r;... M r C,w, i either i s r or ii r s or iii s, r s 2r; whie the reverse iequaities are vaid i either iv r s or v s r or vi s, r s 2r. I this paper, we geeraize the above resuts by usig a ew reiemet o the Jese s iequaity rom [2]. First, we give the otatios itroduced i [2]: Let X be a set. The power set o X is deoted by PX. X meas the umber o eemets i X. The usua symbo N is used or the set o atura umbers icudig 0. Let u v 2 be ixed itegers. Deie the uctios by S v,w : {,...,u} v {,...,u} v, w v, S v : {,...,u} v P {,...,u} v, T v : P{,...,u} v P {,...,u} v S v,w i,...,i v := i,i 2,...,i w,i w+,...,i v, w v,

5 REFINEMENT OF JENSEN S INEQUALITY FOR OPERATOR CONVEX FUNCTIONS 5 S v i,...,i v := T v I := i,...,i v I v {S v,w i,...,i v }, w=, i I = S v i,...,i v, i I. Next, et the uctio α v,i : {,...,u} v N, i u, be give by: α v,i i,...,i v meas the umber o occurreces o i i the sequece i,...,i v. For each I P{,...,u} v et α I,i := α v,i i,...,i v, i u. i,...,i v I It is easy to see that the depedece o the uctios S v,w, S v, T v α v,i o u does ot pay a importat roe, so we ca use simpiied otatios. The oowig hypotheses wi give the basic cotext o our resuts. H Let 2 be ixed itegers, et I be a subset o {,...,} such that 2 α I,i, i. H 2 Let I R be a iterva, et C i SI i. H 3 Let w,...,w be positive umbers such that H 4 Let the uctio : I R be operator covex. j= w j =. H 5 Let h, g : I R be cotiuous stricty operator mootoe uctios. We eed some urther preparatios. Startig rom I, we itroduce the sets I {,...,} iductivey by I := T I, 2. Obviousy, I = {,...,}, by 2, this isures that α I,i = i. From 2 agai, we have that α I,i, i.

6 6 L. HORVÁTH, K.A. KHAN, J. PEČARIĆ For ay 2 or ay j,..., j I, et H I j,..., j := { i,...,i,m I {,...,} S,m i,...,i = j,..., j }. Usig these sets we deie the uctios t I, : I N iductivey by 3 4 t I, i,...,i :=, i,...,i I ; t I, j,..., j := t I, i,...,i. i,...,i,m H I j,..., j 2. MAIN RESULTS The mai resuts o this paper ivove some specia expressios, which we ow describe. Suppose H -H 4. For ay et 5 A, = A, I,C,...,C,w,...,w := i,...,i I s= associate to each the operator 6 := α I,i s A, = A, I,C,...,C,w,...,w... t I, i,...,i i,...,i I s= With these preparatios out o the way we come to s= α I,i s α C I i,is s w i s α I,is s=, s= α C I i,is s w i s α I,is s=. Theorem 2.. Assume H -H 4. The a 7 r=w r C r A, A,... A,2 A, = r= w r C r.

7 REFINEMENT OF JENSEN S INEQUALITY FOR OPERATOR CONVEX FUNCTIONS 7 b Suppose H I j,..., j = β or ay j,..., j I 2. The 8 A, = A, = I thus r=w r C r i,...,i I s= C is,, s= s= A, A,... A 2,2 A, = r= w r C r. To prove these resuts we ca use the same method as i the proo o the mai resut Theorem i [2], so we omit the proos. 3. DISCUSSION, AND APPLICATIONS Throughout Exampes based o the exampes i [2] the coditios H 2 -H 4 wi be assumed. Theorem 2. cotais Theorem.2, as the irst exampe shows. Exampe 3.. Let I := {i,...,i {,...,} i <... < i },. The α I,i = i =,..., esurig H with =. It is easy to chec that T I = I = 2,...,, I = =,...,, or every = 2,..., H I j,..., j =, j,..., j I, thereore, thas to Theorem 2. b, 9 A, = r=w r C r i <...<i s= C is, s= s= A, A,... A 2,2 A, = =,...,. r= w r C r.

8 8 L. HORVÁTH, K.A. KHAN, J. PEČARIĆ I w =... = w =, the A, = Ci C i, =,...,, i <...<i thus 9 gives Theorem.2. The ext exampe iustrates that Theorem.3 is a aso specia case o Theorem 2.. Exampe 3.2. Let I := {i,...,i {,...,} i... i },. Obviousy, α I,i i =,...,, thereore H is satisied. It is ot hard to see that T I = I = 2,..., I = + =,..., or each = 2,..., H I j,..., j =, j,..., j I. Cosequety, by appyig Theorem 2. b, we deduce that A, = 20 + r=w r C r i... i By taig w =... = w =, we obtai that A, = + thus 20 gives Theorem.3. s= C is,, s= s=... A,... A, = i... i r= w r C r. Ci C i,, The oowig two exampes are particuar cases o Theorem 2. b. Exampe 3.3. Let I := {,...,},.

9 REFINEMENT OF JENSEN S INEQUALITY FOR OPERATOR CONVEX FUNCTIONS 9 Triviay, α I,i i =,...,, hece H hods. It is evidet that T I = I = 2,..., I = =,..., or every = 2,..., thereore Theorem 2. b eads to A, = r=w r C r H I j,..., j =, j,..., j I, i,...,i I s=... A,... A, = Especiay, or w =...w = we id that A, = Ci C i i,...,i I C is,, s= s= r= w r C r,., =,...,. Exampe 3.4. For et I cosist o a sequeces i,...,i o distict umbers rom {,...,}. The α I,i i =,...,, hece H is vaid. It is immediate that T I = I = 2,...,, I =... + =,...,, or each = 2,..., H I j,..., j =, j,..., j I. rom them, o accout o Theorem 2. b, oows A, =... + i,...,i I s= r=w r C r I we set w =... = w =, the A, =... + C is, s= s= A,... A,... A, = i,...,i I Ci C i =,..., r= w r C r., =,...,.

10 0 L. HORVÁTH, K.A. KHAN, J. PEČARIĆ I the seque two iterestig cosequeces o Theorem 2. a are give. Exampe 3.5. Let c i be a iteger i =,...,, et := i= c i, et I = P c,...,c cosist o a sequeces i,...,i i which the umber o occurreces o i {,...,} is c i i =,...,. Evidety, H is satisied. A simpe cacuatio shows that I = P c,...,c i,c i,c i+,...,c!, α I,i = c i=!...c! c i, i =,...,, t I, i,...,i =, i i,...,i P c,...,c i,c i,c i+,...,c, i =,...,, r=w r C r = A, Accordig to Theorem 2. a = c!...c!! i,...,i I s= r=w r C r c is A, r= s= c C i i s s w i s c i s s= w r C r,. where A, = i= c i w i r= w r C r w i w i c i c i C i. Exampe 3.6. Let I 2 := {i,i 2 {,...,} 2 i i 2 }. The otatio i i 2 meas that i divides i 2. Sice i i i =,...,, H hods. I this case α I2,i = [ i ] + di, i =,...,,

11 REFINEMENT OF JENSEN S INEQUALITY FOR OPERATOR CONVEX FUNCTIONS where [ i ] is the argest atura umber that does ot exceed i, di deotes the umber o positive divisors o i. By Theorem 2. a, we have r C r w i w [ ] + [ ] i2 r=w i i,i 2 I 2 + di i2 + di 2 w [ ] i C i +di i + w [ ] i 2 C i 2 i2 +di 2 w r C r. [ i ] w i + w [ i 2 +di i2 ]+di 2 r= 4. SYMMETRIC MEANS Assume H -H 3. The power meas correspodig to i := i,...,i I =,..., are give as: 2 M r I,i := s= α Cr I,is i s w i s α I,is s= r, r 0. Next, we itroduce the mixed symmetric meas correspodig to the expressios 5 6 as oows: 22 Ms,rI, := or i =i,...,i I j= w i j α I,i j M r I,i s s, s 0, 23 M s,ri, :=... t I,i i =i,...,i I j= w i j α I,i j Mr I,i s s, s 0. The oowig resut is a comprehesive geeraizatio o Theorem.4. Theorem 4.. Assume H -H 3 or a -tupe C o stricty positive operators. The 24 M s C,w = M s,ri,... M s,ri, M r C,w. hods i either

12 2 L. HORVÁTH, K.A. KHAN, J. PEČARIĆ i s r or ii r s or iii s, r s 2r; whie the reverse iequaities hod i 24 i either iv r s or v s r or vi s, r s 2r. Proo. It is we ow see [] that the uctio : D R R, x = x p is operator covex o 0, i either p 2 or p 0, operator cocave o 0, i 0 p, whie is operator mootoe o 0, i 0 p. It is aso true that is operator mootoe o 0, i p 0. By usig these acts, we ca appy Theorem 2. a to the uctio x = x s r, the operators C r i i =,...,. Assume H -H 3 H 5. The we deie the quasi-arithmetic meas with respect to 5 6 as oows: 25 Mh,g I, := h or i,...,i I s= α I,i s h g s= α gc I i,is s, s= α I,is 26 Mh,g I, := h... t I,i i =i,...,i I s= α I,is h g s= α gc I is,is s= α I,is. The mootoicity o these geeraized meas is obtaied i the ext coroary. Coroary 4.2. Assume H -H 3 H 5. For a cotiuous stricty operator mootoe uctio q : I R we deie M q := q i=w i qc i.

13 REFINEMENT OF JENSEN S INEQUALITY FOR OPERATOR CONVEX FUNCTIONS 3 The 27 M h = Mh,g I,... Mh,g I, M g, i either h g is operator covex h is operator mootoe or h g is operator cocave h is operator mootoe; 28 M g = Mg,h I,... Mg,h I, M h, i either g h is operator covex g is operator mootoe or g h is operator cocave g is operator mootoe. Proo. First, we appy Theorem 2. a to the uctio h g repace C i to gc i, the we appy h to the iequaity comig rom 7. This gives 27. A simiar argumet gives 28: g h, C i = hc i g ca be used. Assume H -H 3, suppose H II j,..., j = β or ay j,..., j I 2. I this case the power meas correspodig to i := i,...,i I =,..., has the orm M r I,i = M r I,i = s= w i j s= r w i j Ci r j, r 0. Now, or we itroduce the mixed symmetric meas reated to 8 as oows: 29 Ms,rI 2 := s I,i s, s 0. I w i j M r i =i,...,i I j= Coroary 4.3. Assume H -H 3, suppose H II j,..., j = β or ay j,..., j I 2. The 30 M s C,w = Ms,rI 2... Ms,rI 2 M r C,w. hods i either i s r or ii r s or

14 4 L. HORVÁTH, K.A. KHAN, J. PEČARIĆ iii s, r s 2r; whie the reverse iequaities hod i 30 i either iv r s or v s r or vi s, r s 2r. Proo. It comes rom Theorem 4.. Assume H -H 3 H 5, suppose H II j,..., j = β or ay j,..., j I 2. We deie or the quasi-arithmetic meas with respect to 8 as oows: 3 Mh,g 2 I := h I h g gc is s=. i,...,i I s= s= Coroary 4.4. Assume H -H 3 H 5, suppose H II j,..., j = β or ay j,..., j I 2. The 32 M h = M 2 h,g I... M 2 h,g I M g, where either h g is operator covex h is operator mootoe or h g is operator cocave h is operator mootoe; 33 M g = M 2 g,h I... M 2 g,h I M h, where either g h is operator covex g is operator mootoe or g h is operator cocave g is operator mootoe. Proo. Simiar to the proo o Coroary 4.2. Fiay, we appy the resuts o this sectio i some specia cases. Throughout Remars , which are based o exampes i [2], the coditios H 2 -H 3 i the mixed symmetric meas H 5 i the quasi-arithmetic meas wi be assumed. Remar 4.5. I the case o Exampe 3., or 29 becomes 34 M 2 s,ri = i <...<i s s w i j M r I,i, s 0. j=

15 3 has the orm REFINEMENT OF JENSEN S INEQUALITY FOR OPERATOR CONVEX FUNCTIONS 5 35 Mh,g 2 I = h i <...<i h g s= s= Remar 4.6. Uder the settig o Exampe 3.2, or 29 becomes 36 Ms,rI 2 = 3 has the orm 37 Mh,g 2 I = h + + i... i i... i gc is. s= s s w i j M r I,i, s 0. j= h g s= s= gc is. s= represets mixed symmetric meas as give i [3]. Thereore Coroary 4.3 is a geeraizatio o resuts give i [3]. Remar 4.7. Uder the settig o Exampe 3.3, or, 29 eads to 38 Ms,rI 2 = s s w i j M r I,i, s 0. j= 3 gives 39 Mh,g 2 I = h respectivey. i =i,...,i I i =i,...,i I s= i =i,...,i I h g s= gc is, s= Remar 4.8. Uder the settig o Exampe 3.4, or =,...,, 29 gives 40 Ms,rI 2 = s s... + w i j M r I,i, s 0. j= 3 has the orm 4 M 2 h,g I = h respectivey i =i,...,i I h g s= s= gc is s=,

16 6 L. HORVÁTH, K.A. KHAN, J. PEČARIĆ Remar 4.9. Uder the costructio o Exampe 3.5, 23 is writte as 42 Ms,rI, = i= c i w i w j C r j w i c i Ci r j= w i c i s r s, s 0, r 0, whie 26 becomes 43 Mh,g I, = h i= c i w i h g r= w r gc r w i c i gc i w i. c i Remar 4.0. Uder the costructio o Exampe 3.6, 22 gives 44 Ms,rI 2,2 = whie 25 gives i 2 =i,i 2 I 2 2 j= w i j [ i j ] M r I 2,i 2 s + di j s, s 0, 45 Mh,g I 2,2 = h i,i 2 I 2 2 s= [ is ]+di s h g 2 s= [ is ] gc is +dis. 2 s= [ is ] +dis Coict o Iterests The authors decare that there is o coict o iterests. Acowedgemets The irst author was supported by Hugaria Natioa Foudatios or Scietiic Research Grat No. K027, the secod by Higher Educatio Commissio Paista the third by the Croatia Miistry o Sciece, Educatio Sports uder the Research Grat REFERENCES [] T. Furuta, J. M. Hot, J. Pečarić, Y. Seo, Mod-Pečarić method i operator iequaities, Eemet, Zagreb [2] L. Horváth, J. Pečarić, A reiemet o the discrete Jese s iequaity, Math. Ieq. App. 4 20,

17 REFINEMENT OF JENSEN S INEQUALITY FOR OPERATOR CONVEX FUNCTIONS 7 [3] B. Mod, J. Pečarić, Remars o Jese s iequaity or operator covex uctios, A. Uiv. Mariae Curie- Sodowsa Sec. A ,