Refinements of Jensen s Inequality for Convex Functions on the Co-Ordinates in a Rectangle from the Plane

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1 Filoat 30:3 (206, DOI /FIL603803A Published by Faculty of Scieces ad Matheatics, Uiversity of Niš, Serbia Available at: Refieets of Jese s Iequality for Covex Fuctios o the Co-Ordiates i a Rectagle fro the Plae M. Adil Kha a, T. Ali a, A. Kılıça b, Q. Di c a Departet of Matheatics, Uiversity of Peshawar, Peshawar, Pakista b Departet of Matheatics ad Istitute of Matheatical Research, Uiversity Putra Malaysia (UPM, UPM, Serdag, Selagor, Malaysia c Departet of Matheatics, Uiversity of Pooch Rawalakot, Rawalakot 2350, Pakista Abstract. I this paper our ai is to give refieets of Jese s type iequalities for the covex fuctio defied o the co-ordiates of the bidiesioal iterval i the plae.. Itroductio A fuctio : [a, b] R is said to be covex if (λx + ( λy λ(x + ( λ(y ( holds for all x, y [a, b] ad 0 λ. A fuctio is said to be strictly covex if the iequality i ( is strict wheever x y ad 0 < λ <. Let : [a, b] R be a covex fuctio o [a, b]. If x i [a, b] ad p i > 0 such that = p i the p i x i p i (x i, (2 is well kow i the literature as Jese s iequality. The Jese iequality for covex fuctios plays a crucial role i the Theory of Iequalities due to the fact that other iequalities such as the arithetic-ea geoetric-ea iequality, the Hölder ad Mikowski iequalities, the Ky Fa iequality etc. ca be obtaied as particular cases of it. 200 Matheatics Subject Classificatio. Priary 26D5 Keywords. Covex fuctios o the co-ordiates, Jese s iequality Received: 9 August 205; Revised: 08 Deceber 205; Accepted: 0 Deceber 205 Couicated by Ljubiša D.R. Kočiac ad Ekre Savaş Eail addresses: adilswati@gail.co (M. Adil Kha, tahirali073@yahoo.co (T. Ali, akilic@up.edu.y (A. Kılıça, qaar.ss@gail.co (Q. Di

2 M. Adil Kha et al. / Filoat 30:3 (206, I [7], the authors have ivestigated the followig refieet of (2: p i x i p i x i i I P i I I P I + p i (x i i I \I p i x i i I 2 P I P I I I + (2 p i (x i p i x i ax I P i I I + p i (x i P I i I \I p i (x i, where : C R is a covex fuctio defied o a covex set C, x i C ad I = {I I, I I = {,..., } s.t. I 2}, i {,..., }, 3 ad P I = p i together with p i =. i I I 200 Dragoir obtaied aother refieet of Jese s iequality (see [5] : p i x i D(, p, x, I p i (x i, (3 where ad D(, p, x, I = P I p i x i P I + P Ī p i x i i I I I = {,..., }, Ī = I \ I, i {,..., } together with P I = p i, P Ī = p i ad x = (x, x 2,..., x, p = (p, p 2,..., p. Also i [6], the authors have i I i Ī proved a geeralized refieet of (2 give as uder: x i k λ j+ x i+j P Ī i Ī (x i, (4 where : [a, b] R is a covex fuctio, x := (x,..., x [a, b] such that x i+ = x i ad λ := (λ,..., λ is a positive tuple together with k λ i = for soe k, 2 k. More recetly i 205, the authors have give further geeralizatios of the results preseted i [2, 3]. I [4], the cocept of covex fuctios defied o the co-ordiates of the bidiesioal iterval of the plae of two variables was itroduced: Defiitio.. Let us cosider the bidiesioal iterval := [a, b] [c, d] i R 2 with a < b ad c < d. A fuctio : [a, b] [c, d] R is called covex o the co-ordiates if the partial appigs y : [a, b] R defied as y (t := (t, y ad x : [c, d] R defied as x (s := (x, s, are covex for all x [a, b], y [c, d]. Reark.2. Note that every covex fuctio : [a, b] [c, d] R is covex o the co-ordiates, but the coverse is ot geerally true [4]. The followig Jese s iequality for co-ordiate covex fuctios has bee give i [4].

3 M. Adil Kha et al. / Filoat 30:3 (206, Theore.3. ([4] Let : [a, b] [c, d] R be a covex fuctio o the co-ordiates o [a, b] [c, d]. If x is a -tuple i [a, b], y is a -tuple i [c, d], p is a o-egative -tuple ad w a o-egative -tuple such that = p i > 0 ad = w j > 0, the ( x, ȳ 2 p i (x i, y + w j (x, y j p i w j (x i, y j, (5 where x = p i x i, ȳ = w j y j. For other refieets ad geeralizatios of Jese s iequality ad their applicatios see [ 6, 8 3, 7 23] ad soe of the refereces give i the. I this article, we have geeralized the results give i [6], [7] ad [5] fro covex fuctios defied o the subset of R to covex fuctios defied o the co-ordiates o the bidiesioal iterval of the plae by costructig soe ew fuctioals depedig o the fuctio ad idexig sets, separatig the discrete doai of it. Furtherore the result give i [6] is exteded to co-ordiate covex fuctios. 2. Mai Results Teriologies ad otatios: Let : [a, b] [c, d] R be covex o the co-ordiates o [a, b] [c, d]. If x i [a, b], y j [c, d], ad p i, w j > 0, i {, 2,..., }, j {, 2,..., } such that, 3 with = p i ad = w j, ad let Ω = {I k : I k I = {,..., }, I k 2, I k I } ad Ω 2 = {J l : J l J = {,..., }, J l 2, J l J }, we assue I k := {, 2,..., }\I k ad J l := {, 2,..., }\ J l. Defie P I k = p i ad P = Ī p k i ad W J l = w j, i I k i I k j J l W J = w l j. For the fuctio ad the, -tuples x = (x, x 2,..., x [a, b], y = (y, y 2,..., y [c, d] j J l ad p = (p, p 2,..., p, w = (w, w 2,..., w, we defie the followig fuctioals: Ϝ(, p, x, I k = P I k p i x i, ȳ P I + ( p i xi, ȳ, k i I k i I k Ϝ(, w, y, J l = W J l x, w j y j W J l + ( w j x, yj, (6 j J l j J l D (I k, J l = Ϝ (, w, y, J l + Ϝ (, p, x, I k (7 D 2 (I k, J l = p i Ϝ (, w, y, J l, x i + w j Ϝ (, p, x, I k, y j (8

4 M. Adil Kha et al. / Filoat 30:3 (206, where Ϝ(, p, x, I k, y j = P I k P I + ( p i xi, y j, k i I k i I k Ϝ(, w, y, J l, x i = W J l x i, w j y j W J l + ( w j xi, y j, j J l j J l x = p i x i, ȳ = w j y j. Reark 2.. It is obvious that Ω = 2 2, Ω 2 = 2 2, that is, k =,..., 2 2 ad l =,..., 2 2 ad throughout the paper we will deote 2 2 by N ad 2 2 by M. The followig lea will be proved helpful i the further elaboratio of the ext refieet: Lea 2.2. Let : = [a, b] [c, d] R be a fuctio defied o. If x i [a, b], y j [c, d], ad p i, w j > 0, i {, 2,..., }, j {, 2,..., },, 3, with = p i ad = w j, the we have M N D (I k, J l l= k= = N P I k ( p i x i i I k, ȳ + (2 p i (x i, ȳ N P k= I k + M W J l ( w j y j j J x, l (2 + w j ( x, y j M W J l l= ad M l= k= N D 2 (I k, J l =. N +. M N P I k w j ( p i x i i I k (2, y j + P k= I k w M j y j j J l (2 + l= W J l p i ( x i, W J l p i w j (x i, y j p i w j (x i, y j. Proof. Sice, fro (6 we kow that D (I k, J l = Ϝ (, w, y, J l + Ϝ (, p, x, I k.

5 M. Adil Kha et al. / Filoat 30:3 (206, Therefore, we have M N D (I k, J l l= k= M N { = Ϝ(, p, x, I k + Ϝ(, w, y, J l } l= k= [ M N { PI k = p i x i, ȳ P l= k= I + ( p i xi, ȳ k i I k i I k + W J l x, w j y j W J l + ( }] w j x, yj j J l j J l = N P I k ( p i x i i I k, ȳ + (2 p i (x i, ȳ N P k= I k M W J l + ( w j y j j J x, l (2 + w j ( x, y j M W J l. l= Here it is obvious that N k= i Ī k p i (x i, y j = (2 ( p i (x i, y j, sice every p i (x i, y j appears as ay ties as there is a subset I k I, I k 2, ad that does t cotai the idex i. Siilarly we ca prove the secod part of the lea. The followig refieet of Theore.3 holds: Theore 2.3. Suppose that : = [a, b] [c, d] R is covex o the co-ordiates o. If x i [a, b], y j [c, d], ad p i, w j > 0, i {, 2,..., }, j {, 2,..., },, 3, with = p i ad = w j, the for ay I k Ω ad J l Ω 2 we have ( x, ȳ 2 i k=,...,n l=,...,m 2 i D (I k, J l 2 k=,...,n l=,...,m M l= k= p i ( x i, ȳ + D 2 (I k, J l 2 where x = M l= k= N D (I k, J l 2 ax w j ( x, y j N D 2 (I k, J l k=,...,n l=,...,m 2 ax k=,...,n l=,...,m D (I k, J l D 2 (I k, J l p i w j (x i, y j, (9 p i x i, ȳ = w j y j, k N, l M.

6 Proof. Oe-diesioal Jese s iequality gives us ( x i, ȳ M. Adil Kha et al. / Filoat 30:3 (206, w j ( x i, y j ad ( x, yj p i ( x i, y j. By Jese s iequality, we get Ϝ(, p, x, I k, y j = P I k P I + ( p i xi, y j k i I k i I k P I k p i (x i, y j + ( p i xi, y j = p i (x i, y j P I k i I k i Ī k i I k Ī k Ϝ(, p, x, I k, y j p i (x i, y j. (0 As the fuctio is covex o the first co-ordiate, so we have Ϝ(, p, x, I k, y j = P I k P I + ( p i xi, y j k i I k i I k P I k P I + P I k p k i x i, y j P Ī i I k k i I k P I k P p i x i + Ik P I k P Ī = P i I k k i Ī k I k Ī k Ϝ(, p, x, I k, y j. ( Now, fro (0 ad (, we have ( x, y j Ϝ (, p, x, I k, y j Siilarly, we ca write ( x i, ȳ Ϝ (, w, y, J l, x i p i (x i, y j. (2 w j (x i, y j. (3 Multiplyig (2 ad (3 respectively by w j ad p i ad suig over i ad j, we obtai w j ( x, y j w j Ϝ (, p, x, I k, y j p i w j (x i, y j, (4 ad p i ( x i, ȳ p i Ϝ (, w, y, J l, x i p i w j (x i, y j. (5

7 Addig (4 ad (5, oe has the followig p i ( x i, ȳ + w j ( x, y j W M. Adil Kha et al. / Filoat 30:3 (206, Now, settig x i = x ad y j = ȳ i (2, (3 ad addig we have p i Ϝ (, w, y, J l, x i + w j Ϝ (, p, x, I k, y j p i w j (x i, y j. (6 ( x, ȳ [ ( Ϝ, w, y, J l + Ϝ (, p, x, I k] 2 p i ( x i, ȳ + w j ( x, y j. Cobiig (6 ad (7 we obtai ( x, ȳ [ ( Ϝ, w, y, J l + Ϝ (, p, x, I k] p 2 i ( x i, ȳ + w j ( x, y j W p i Ϝ (, w, y, J l, x i + w j Ϝ (, p, x, I k, y j W p i w j (x i, y j. The stateet i the theore follows by takig the i ad ax of D (I k, J l ad D 2 (I k, J l over the idices k ad l with k N, l M ad together with Lea 2.2 ad usig the fact that ad i k=,...,n l=,...,m i k=,...,n l=,...,m D (I k, J l D 2 (I k, J l M l= k= M l= k= N D (I k, J l N D 2 (I k, J l This copletes the desired proof. ax k=,...,n l=,...,m ax k=,...,n l=,...,m D (I k, J l (7 D 2 (I k, J l. (8 Reark 2.4. For I k = {µ}, µ {,..., } ad J l = {ν}, ν {,..., }, the above fuctioals take the for give below Ϝ(, p, x, I k, y j = Ϝ(, p, x, {µ}, y j = p i (x i, y j, Ϝ(, w, y, J l, x i = Ϝ(, w, y, {ν}, x i = w j (x i, y j, D ({µ}, {ν} = Ϝ (, w, y, {ν} + Ϝ (, p, x, {µ}, = w j ( x, y j + p i (x i, ȳ D 2 ({µ}, {ν} = 2 p i w j (x i, y j

8 M. Adil Kha et al. / Filoat 30:3 (206, ad the refieet give i Theore 2.3 shriks to the result give i Theore.3. I the ext theore, subsets of equivalet cardiality are observed. Theore 2.5. Suppose that : = [a, b] [c, d] R is covex o the co-ordiates o. If x i [a, b], y j [c, d], ad p i, w j > 0, i {, 2,..., }, j {, 2,..., },, 3, with = p i ad = w j, the for ay I k Ω ad J l Ω 2 such that I k = s 2 ad J l = r 2 we have ( x, ȳ 2 i D (I k, J l I k =s 2 J l =r 2 i D 2 (I k, J l I k =s 2 J l =r ( r ( s l= k= ( s ( r ( s l= k= ( s D (I k, J l ( 2 ax D (I k, J l r I k =s 2 J l =r D 2 (I k, J l ( 2 ax D 2 (I k, J l r I k =s J l =r p i ( x i, ȳ + p i w j (x i, y j, w j ( x, y j where x = p i x i, ȳ = w j y j. Proof. The stateet i the theore follows by takig the i ad ax of the fuctioals give i ( ad (2, after choosig every subset I k Ω ad J l Ω 2, such that I k = s ad J l = r, with 2 s < ad 2 r <. We use the facts etioed i (7 ad (8, where ( ( s ad r represet the uber of subsets I k I ad J l J, I k = s, J l = r. Note that I k I, I k =s i Ī k p i (x i, y j = [( s ( ] p i (x i, y j, s sice every p i (x i, y j i the double su appears as ay ties as there are subsets I k I, I k = s 2 such that i I k. The subsets I k I, with I k = s ad i I k is costructed by addig s eleets fro the available oce. Algebraically, [( s ( ] p i (x i, y j = s ( p i (x i, y j. s Siilar arguets ca be give for the subsets J l J, with J l = r 2 ad oe has J l J, J l =r j J l w j (x i, y j = [( r ( ] w j (x i, y j. r Every partitio of I = {,..., } ad J = {,..., } gives the stateet obtaied i the ext theore. Theore 2.6. Suppose that : = [a, b] [c, d] R is covex o the co-ordiates o. If x i [a, b], y j [c, d], ad p i, w j > 0, i {, 2,..., }, j {, 2,..., },, 4, with = p i ad = w j. For every itegers

9 M. Adil Kha et al. / Filoat 30:3 (206, q, r, such that 4 2q ad 4 2r, there are partitios I I 2... I q = I, J J 2... J r = J with 2 I µ <, 2 J ν < for µ =, 2,..., q, ν =, 2,..., r. The we have ( x, ȳ 2 i where x = µ=,...,q ν=,...,r 2 i D (I µ, J ν 2 µ=,...,q ν=,...,r p i x i, ȳ = r ν= µ= p i ( x i, ȳ + D 2 (I µ, J ν 2 r q D (I µ, J ν (qr w j ( x, y j ν= µ= q D 2 (I µ, J ν (qr w j y j, µ q, ν r. 2 ax D (I µ, J ν µ=,...,q ν=,...,r 2 ax D 2 (I µ, J ν µ=,...,q ν=,...,r p i w j (x i, y j, Proof. Every subset I µ I ad J ν J iduced theirs copleet Ī µ ad J ν ad (9 is valid with the substitutios: I k I µ, J l J ν. For D (I µ, J ν ad D 2 (I µ, J ν we take the i ad ax over µ =,..., ad ν =,..., ad usig the facts that i µ=,...,q ν=,...,r Note: D (I µ, J ν q µ= r ν= µ= q D (I µ, J ν (qr i Ī µ p i (x i, y j = (q ax D (I µ, J ν ad µ=,...,q ν=,...,r p i (x i, y j. i µ=,...,q ν=,...,r D 2 (I µ, J ν Theore 2.3 esures the ext iproveets of Jese s differece. r ν= µ= q D 2 (I µ, J ν (qr ax D 2 (I µ, J ν. µ=,...,q ν=,...,r Corollary 2.7. Uder the coditios of Theore 2.3, we obtai: p i w j (x i, y j ( x, ȳ ax p I k, J i w j (x i, y j l 2 D (I k, J l 0. (9 Proof. Subtractig I k I = {,..., } ad J l J = {,..., }, there is a stateet: p i w j (x i, y j fro every side of (9, we obtai that for every choice of p i w j (x i, y j ( x, ȳ p i w j (x i, y j 2 D (I k, J l 0. (20 Takig the ax of the right had side i (20 for I k I, I k 2 ad J l J, J l 2, the proof is akig through. Corollary 2.8. Uder the coditios of Theore 2.3, we obtai: p i w j (x i, y j ( x, ȳ ax p I k, J i w j (x i, y j l 2 D 2(I k, J l 0. (2

10 M. Adil Kha et al. / Filoat 30:3 (206, Proof. The proof is siilar to that of corollary 2.7 oly use D 2 (I k, J l istead of D (I k, J l. Now we give aother refieet of the Jese s iequality for the covex fuctio defied o the co-ordiates of the bidiesioal iterval i the plae: Theore 2.9. Let : [a, b] [c, d] R be a co-ordiate covex fuctio o [a, b] [c, d]. If x i [a, b], y j [c, d] k l such that x i+ = x i, y j+ = y j ad p i, w j > 0, i {, 2,..., }, j {, 2,..., }, with p i = ad w j =, for soe k ad l, 2 k ad 2 l, the we have ( x, ȳ k p r+ x i+r, ȳ + l x, w t+ y j+t (x i, ȳ + ( x, y j t=0 k p r+ x i+r, y j + l x i, w t+ y j+t (x i, y j, where x = x i, ȳ = y j. Proof. Sice yj : [a, b] R is covex, so by Jese s iequality, we have t=0 yj x i = yj x i k r= p r = yj k p r+ x i+r k yj p r+ x i+r k p r+ x i+r, y j, therefore, x i, y j k p r+ x i+r, y j. (22 O the other had, sice yj : [a, b] R is covex, so agai by Jese s iequality ad siple calculatios oe ca get k p r+ x i+r, y j ( x i, y j (23 the cobiatio of (22 ad (23 yields x i, y j k p r+ x i+r, y j ( x i, y j. (24 Siilarly, the covexity of xi : [c, d] R iplies the followig x i, y i l x i, w t+ y j+t t=0 ( x i, y j. (25

11 M. Adil Kha et al. / Filoat 30:3 (206, Multiplyig (24 ad (25 by ad respectively ad suig over j, i ad the addig the obtaied results, oe has the followig (x i, ȳ + ( x, y j k p r+ x i+r, y j + l x i, w t+ y j+t Furtherore by settig x i x ad y j ȳ i (24 ad (25 respectively we get x i, ȳ k p r+ x i+r, ȳ t=0 (x i, y j. (26 ( x i, ȳ (27 ad x, y i l x, w t+ y j+t t=0 ( x, y j. (28 Now addig the, we obtai ( x, ȳ k p r+ x i+r, ȳ + l x, w t+ y j+t t=0 (x i, ȳ + ( x, y j. Hece, we have the desired result. Ackowledgeets The authors would like to thak the referees for valuable suggestios ad coets which helped the authors to iprove this article substatially. Refereces [] M. Adil Kha, M. Awar, J. Jakšetić, J. Pečarić, O soe iproveets of the Jese iequality with soe applicatios, J. Iequal. Appl (2009, Article ID 32365, 5 pages. [2] M. Adil Kha, G.A. Kha, T. Ali, T. Batbold, A. Kılıça, Further refieet of Jese s type iequalities for the fuctio defied o the rectagle, Abstr. Appl. Aal. 203 ( [3] M. Adil Kha, G.A. Kha, T. Ali, A. Kılıça, O the refieet of Jeses iequality, Appl. Math. Coput. 262 ( [4] M.K. Bakula, J. Pečarić, O the Jese s iequality for covex fuctios o the co-ordiates i a rectagle fro the plae, Taiwaese J. Math. 0 ( [5] M.K. Bakula, J. Pečarić, J. Perić, O the coverse Jese s iequality, Appl. Math. Coput. 28 ( [6] I. Bretić, K.A. Kha, J. Pečarić, Refieet of Jese s iequality with applicatios to cyclic ixed syetric eas ad Cauchy eas, J. Math. Iequal. 9( ( [7] V. Čuljak, B. Ivaković, Oe refieet of Jese s discrete iequality ad applicatios, Math. Ieq. Appl. 2 ( [8] S.S. Dragoir, A iproveet of Jese s iequality, Bull. Math. Soc. Sci. Math. Rouaie 34(82 ( [9] S.S. Dragoir, Soe refieets of Ky Fa s iequality, J. Math. Aal. Appl. 63 ( [0] S.S. Dragoir, Soe refieets of Jese s iequality, J. Math. Aal. Appl. 68 ( [] S.S. Dragoir, A further iproveet of Jese s iequality, Takag J. Math. 25 ( [2] S.S. Dragoir, A ew iproveet of Jese s iequality, Idia J. Pure Appl. Math. 26 ( [3] S.S. Dragoir, J. Pečarić, L.E. Persso, Properties of soe fuctioals related to Jese s iequality, Acta Math. Hugarica 70 ( [4] S.S. Dragoir, O the Hadaard s iequality for covex fuctios o the co-ordiates i a rectagle fro the plae, Taiwaese J. Math. 5 ( [5] S.S. Dragoir, A ew refieet of Jese s iequality i liear spaces with applicatios, Math. Coput. Modellig 52 (

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