SOME NEW HERMITE-HADAMARD TYPE INEQUALITIES FOR FUNCTIONS WHOSE HIGHER ORDER PARTIAL DERIVATIVES ARE CO-ORDINATED CONVEX

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1 FACTA UNIVERSITATIS (NIŠ) Ser. Mth. Inor. Vol. 7 No 3 (), SOME NEW HERMITE-HADAMARD TYPE INEQUALITIES FOR FUNCTIONS WHOSE HIGHER ORDER PARTIAL DERIVATIVES ARE CO-ORDINATED CONVEX Muhd Aer Lti nd Sbir Hussin Abstrt. In this pper we point out soe inequlities o Herite-Hdrd type or double integrls o untions whose prtil derivtives o higher order re o-ordinted onvex.. Introdution The ollowing deinition is well known in literture: A untion : I R, Ø I R, is sid to be onvex on I i the inequlity holds or ll x, y I nd λ, ]. (λx ( λ) y) λ (x) ( λ) (y), Mny iportnt inequlities hve been estblished or the lss o onvex untions but the ost ous is the Herite-Hdrd s inequlity. This double inequlity is stted s: ( ) b (.) () (b) (x) dx, b where : I R, Ø I R onvex untion,, b I with < b. The inequlities in (.) re in reversed order i onve untion. The inequlities (.) hve beoe n iportnt ornerstone in thetil nlysis nd optiiztion nd ny uses o these inequlities hve been disovered in vriety o settings. Moreover, ny inequlities o speil ens n be obtined or prtiulr hoie o the untion. Due to the rih geoetril signiine o Herite-Hdrd s inequlity (.), there is growing literture providing its new proos, extensions, reineents nd generliztions, see or exple, 3, 7,, 8, 9] nd the reerenes therein. Reeived July,.; Aepted Noveber 9,. Mthetis Subjet Clssiition. 6A33, 6A5, 6D7, 6D, 6D5 3

2 3 Muhd Aer Lti nd Sbir Hussin Let us onsider now bidiensionl intervl =:, b], d] in R with < b nd < d. A pping : R is sid to be onvex on i the inequlity (λx ( λ)z, λy ( λ)w) λ(x, y) ( λ)(z, w), holds or ll (x, y), (z, w) nd λ, ]. A odiition or onvex untions on, known s o-ordinted onvex untions, ws introdued by S. S. Drgoir, 5] s ollows: A untion : R is sid to be onvex on the o-ordintes on i the prtil ppings y :, b] R, y (u) = (u, y) nd x :, d] R, x (v) = (x, v) re onvex where deined or ll x, b], y, d]. A orl deinition or o-ordinted onvex untions y be stted s ollows: Deinition.. ] A untion : R is sid to be onvex on the o-ordintes on i the inequlity (tx ( t)y, su ( s)w) (x, u) t( s)(x, w) s( t)(y, u) ( t)( s)(y, w), holds or ll t, s, ] nd (x, u), (y, w). Clerly, every onvex pping : R is onvex on the o-ordintes but onverse y not be true, 5]. The ollowing Herite-Hdrd type inequlities or o-ordinted onvex untions on the retngle ro the plne R were estblished in ]: Theore.. ] Suppose tht : R is o-ordinted onvex on, then (.) ( b, d ) b b The bove inequlities re shrp. (b ) (d ) ( x, d ) dx d (x, ) (x, d)] dx d (x, y) dydx ( ) ] b, y dy (, y) (b, y)] dy (, ) (, d) (b, ) (b, d). In wht ollows is the interior o nd L ( ) is the spe o integrble untions over. The ollowing result will be very useul to estblish our one o the results in setion : ]

3 Inequlities o Herite-Hdrd Type or Double Integrls 33 Theore.. 9] Let : R be ontinuous pping suh tht the prtil derivtives kl (.,.), k =,,..., n, l =,,..., exist on nd re x k y l ontinuous on, then n (t, s) dsdt = k= l= () n X k (x) k= () n l= X k (x) Y l (y) kl (x, y) x k y l Y l (y) () n S (y, s) k (x, s) x k s ds K n (x, t) nl (t, y) t n y l dt K n (x, t) S (y, s) n (t, s) dsdt, where K n (x, t) := S (y, s) := { (t) n n!, t, x] (tb) n n!, t (x, b] { (s)!, s, y] (sd)!, s (y, d] nd X k (x) = (bx)k () k (x) k (k)! Y l (y) = (dy)l () l (y) l (l)!, or (x, y). In reent yers, ny uthors hve proved severl inequlities or o-ordinted onvex untions. These studies inlude, ong others, the works in ]-]-6], ]-7], ]. Alori et l. ]-6], proved severl Herite-Hdrd type inequlities or o-ordinted s-onvex untions. Drgoir, 5], proved the Herite- Hdrd type inequlities or o-ordinted onvex untions. Hwng et. l 6], lso proved soe Herite-Hdrd type inequlities or o-ordinted onvex untion o two vribles by onsidering soe ppings diretly ssoited to the Herite- Hdrd type inequlity or o-ordinted onvex ppings o two vribles. Lti et. l ]-3], proved soe inequlities o Herite-Hdrd type or dierentible o-ordinted onvex untion, produt o two o-ordinted onvex ppings nd or o-ordinted h-onvex ppings. Özdeir et. l ]-7], proved Hdrd s type inequlities or o-ordinted -onvex nd (α, )-onvex untions. By using the ollowing le: Le.., Le ] Let : R R be prtil dierentible pping on :=, b], d] in R with < b, < d. I L ( ), then the t s

4 3 Muhd Aer Lti nd Sbir Hussin ollowing equlity holds: (, ) (, d) (b, ) (b, d) (.3) b (x, y) dydx (b ) (d ) b (x, ) (x, d)] dx ] d (, y) dy (b, y)] dy b d = (b ) (d ) ( t) ( s) (t ( t) b, s ( s) d) dtds. t s Sriky, et. l ], proved the ollowing Herite-Hdrd type inequlities or dierentible o-ordinted onvex untions: Theore.3., Theore, Pge ] Let : R R be prtil dierentible pping on :=, b], d] in R with < b, < d. I t s is onvex on the o-ordintes on, then one hs the inequlities: (.) (, ) (, d) (b, ) (b, d) b (b ) (d ) (x, y) dydx A (b ) (d ) 6 t s (, ) t s (, d) t s (b, ) t s (b, d), where A = b (x, ) (x, d)] dx d (, y) dy (b, y)] dy Theore.., Theore 3, Pge 6-7] Let : R R be prtil dierentible pping on :=, b], d] in R with < b, < d. I t s q, q >, is onvex on the o-ordintes on, then one hs the inequlities: (.5) (, ) (, d) (b, ) (b, d) b (b ) (d ) (x, y) dydx A (b ) (d ) (p ) p t s (, ) q t s (, d) q t s (b, ) q t s (b, d) q q, ].

5 where A = b nd p q =. Inequlities o Herite-Hdrd Type or Double Integrls 35 (x, ) (x, d)] dx d (, y) dy (b, y)] dy ] Theore.5., Theore, Pge 8-9] Let : R R be prtil dierentible pping on :=, b], d] in R with < b, < d. I t s q, q, is onvex on the o-ordintes on, then one hs the inequlities: (.6) where (, ) (, d) (b, ) (b, d) b (b ) (d ) (x, y) dydx A (b ) (d ) 6 t s (, ) q t s (, d) q t s (b, ) q t s (b, d) q A = b (x, ) (x, d)] dx d (, y) dy (b, y)] dy We lso quote the ollowing result ro 3] to be used in the sequel o the pper: Theore.6. 3, Theore, pge 8] Let : R R be prtil dierentible pping on :=, b], d] with < b, < d. I q s t is onvex on the o-ordintes on nd q, then the ollowing inequlity holds: b ( b (.7) (x, y) dydx (b ) (d ), d ) d ( ) b (d ), y b ( dy x, d ) (b ) (d ) dx (b ) 6 q (, ) s t q (, d) s t q (b, ) s t q q (b, d) s t. q ],.

6 36 Muhd Aer Lti nd Sbir Hussin. Min Results In this setion we estblish new Herite-Hdrd type inequlities or double integrls o untions whose prtil derivtives o higher order re o-ordinted onvex untions. To ke the presenttion esier nd opt to understnd; we ke soe syboli representtion: A = b (x, ) (x, d)] dx d l= n k= (l ) (d ) l (l )! (k ) (b ) k (k )! (l ) (d ) l b (l )! l= n (k ) (b ) k d (k )! k= n k= l= (, y) (b, y)] dy l ] (, ) y l l (b, ) y l k ] (, ) x k k (, d) x k l (x, ) y l dx k (, y) x k dy (k ) (l ) (b ) k (d ) l (k )! (l )! B (n,) = n (, ) ; C (n,) = n (, d). D (n,) = n (b, ) ; E (n,) = n (b, d). It is obvious tht or = n = nd = n =, A = A. ] kl (, ) x kl. We quote the ollowing le ro 7], whih will help us estblish our in results: Le.. 7, Le.] Suppose : I R R,, b I with < b. I (n) exists on I nd (n) L (, b) or n, then we hve the identity: () (b) b n (x) dx = k= (b )n n! (k ) (b ) k (k) () (k )! t n (n t) (n) ( ( t) b)dt.

7 Inequlities o Herite-Hdrd Type or Double Integrls 37 Le.. Let : R < b; < d, be ontinuous pping suh tht n x n y exists on nd n x n y L ( ), or, n N,, n, then (.) (b ) n (d ) n!! t n s (n t) ( s) n (t ( t) b, s ( s) d) dtds A = (, ) (, d) (b, ) (b, d) (b ) (d ) (x, y) dydx. Proo. For n = =, the le oinides with Le.. Consider the se, or, n, then (.) (b ) n (d ) n!! t n s (n t) ( s) n (t ( t) b, s ( s) d) dtds (d ) = s (b ) n ( s) t n (n t)! n! n (t ( t) b, s ( s) d) dt ] ds An pplition o Le. with respet to the irst rguent yields: (.3) (b ) n (d ) n!! = n k= t n s (n t) ( s) n (t ( t) b, s ( s) d) dtds (d )! (d )! s ( s) (, s ( s) d) s ds s ( s) (b, s ( s) d) s ds (d )! (b ) s ( s) (x, s ( s) d) s dsdx (k ) (b ) k (k )! (d )! s ( s) k (, s ( s) d) x k s ds.

8 38 Muhd Aer Lti nd Sbir Hussin Now repeted pplition o Le. with respet to the seond rguent yields: (.) = (.5) = (d )! (, ) (, d) (d )! (b, ) (b, d) s ( s) (, s ( s) d) s ds (d ) (, y) dy l= s ( s) (b, s ( s) d) s ds (d ) (b, y) dy l= (k ) (d ) l (l )! (l ) (d ) l (l )! l (, ) y l. l (b, ) y l. (.6) nd (d )! (b ) = (b ) s ( s) (x, s ( s) d) s dsdx (x, ) dx (b ) (b ) (d ) b (x, y) dydx (x, d) dx (l ) (d ) l l= (l )! l (x, ) y l dx, (.7) (d )! n k= (k ) (b ) k (k )! s ( s) k (, s ( s) d) x k s ds = n k= n k= (k ) (b ) k (k )! (k ) (b ) k (k )! n (k ) (b ) k d (k )! k= n k= l= k (, ) x k k (, d) x k k (, y) x k dy (k ) (l ) (b ) k (d ) l kl (, ) (k )! (l )! x kl. Use (.)-(.7) in (.3) to get (.). This opletes the proo o the le.

9 Inequlities o Herite-Hdrd Type or Double Integrls 39 Theore.. Let : R < b; < d, be ontinuous pping suh tht n exists on nd n L ( ). I n is onvex on the o-ordintes on, or, n N,, n, then (.8) (, ) (, d) (b, ) (b, d) (b ) (d ) (x, y) dydx A (b )n (d ) ( n ) {( ) } B (n,) C (n,) (n )! ( )! Proo. Suppose, n. By Le., we hve: (.9) (.) (, ) (, d) (b, ) (b, d) (b ) (d ) (b )n (d ) n!! n {( ) D (n,) E (n,) }]. (x, y) dydx A By onvexity o n on the o-ordintes on (, ) (, d) (b, ) (b, d) t n s (n t) ( s) n (t ( t) b, s ( s) d) dtds (x, y) dydx A (b ) (d ) (b )n (d ) n (, ) n!! t n s (n t) ( s) dsdt n (, d) t n s ( s) (n t) ( s) dsdt n (b, ) t n s ( t) (n t) ( s) dsdt n (b, d) ( t n t n) (n t) ( s s ) ] ( s) dsdt

10 33 Muhd Aer Lti nd Sbir Hussin = (b )n (d ) ( n ) ( ) n (, ) n!! (n ) (n ) ( ) ( ) ( n ) n (, d) (n ) (n ) ( ) ( ) n ( ) n (b, ) (n ) (n ) ( ) ( ) n n ] (b, d) (n ) (n ) ( ) ( ). This opletes the proo o the theore. Theore.. Let : R < b; < d, be ontinuous pping suh tht n exists on nd n L ( ). I n q, q, is onvex on the o-ordintes on, or, n N,, n, then (.) (, ) (, d) (b, ) (b, d) b d (x, y) dydx A (b ) (d ) (b )n (d ) (n ) /q ( ) /q (n )! ( )! (n ) /q ( ) /q ( ) { ( n ) } B q (n,) ndq (n,) { (n ) C q (n,) neq (n,) Proo. Suppose, n. By Le. nd the power en inequlity, we hve (.) (, ) (, d) (b, ) (b, d) b d (x, y) dydx A (b ) (d ) (b )n (d ) { } /q t n s (n t) ( s) dsdt n!! { t n s (n t) ( s) }] q n q (t ( t) b, s ( s) d) /q dtds}..

11 Inequlities o Herite-Hdrd Type or Double Integrls 33 By the siilr rguents used to obtin (.8) nd the t t n s (n t) ( s) dsdt = (n ) ( ) (n ) ( ), we get (.). This opletes the proo o the theore. Theore.3. Let : R, < b; < d, be ontinuous pping suh tht n exist on nd n L ( ). I n q, q, is onvex on the o-ordintes on, or, n N,, n, then (.3) (b ) (d ) (b ) (d ) () n (d )! k= ()n (b ) n! l= (t, s) dsdt n k= l= (b )k (d ) l (k )! (l )! () k] (b ) k () k] () l] kl kl ( b, d x k y l k Q(s) k ( b, s) (k )! x k s ds () l] (d ) l l P (t) nl ( ) t, d (l )! t n y l dt (b ) n (d ) q B q (n,) Cq (n,) Dq (n,) Eq (n,), n/q (n )! ( )! ) where P (t) := (t ) n, t ], b (t b) n, t ( b, b] nd Q(s) := (s ), s ], d (s d), s ( d, d]. Proo. The proo ollows diretly ro Theore. by letting x b nd y

12 33 Muhd Aer Lti nd Sbir Hussin d, to obtin () n (d )! k= (.) (t, s) dsdt (b ) (d ) (b ) (d ) n () k] () l] (b ) k (d ) l kl (k )! (l )! k= l= () k] (b ) k ()n (b ) n! l= = k (k )! () l] (d ) l l (l )! () n (b ) (d )!n! kl ( b, ) d x k y l Q(s) k ( b, s) x k s ds P (t) nl ( ) t, d t n y l dt P (t)q(s) n (t, s) dsdt. An rguent prllel to tht o Theore. but with (.) in ple o Le. gives the desired result. We now derive results oprble to Theore. nd Theore. with onvity property insted o onvexity property. Theore.. Let : R < b; < d, be ontinuous pping suh tht n exists on nd n L ( ). I n q,q, is onve on the o-ordintes on, or, n N,, n, then (.5) (, ) (, d) (b, ) (b, d) (b ) (d ) (x, y) dydx A (n ) ( ) (b )n (d ) (n )! ( )! ( ) n (n )nb (n)(n), ( )d ()(). Proo. By the onvity o n q on the o-ordintes on nd the power en

13 Inequlities o Herite-Hdrd Type or Double Integrls 333 inequlity, the ollowing inequlity holds: n q (λx ( λ) y, v) λ n q (x, v) ( λ) n q (y, v) ( λ n (x, v) ( λ) n (y, v) q ), or ll x, y, b] nd λ, ] or soe ixed v, d]. Siilrly n (u, λz ( λ) w) λ n (u, z) ( λ) n (u, w), or ll z, w, d] nd λ, ] or soe ixed u, b], iplying n is onve on the o-ordintes on. By the Jensen s inequlity we hve (.6) t n (n t) n ] (t ( t) b, s ( s) d) dt ds ( ) s ( s) t n (n t) dt ( n ) tn (nt)(t(t)b)dt, s ( s) d tn (nt)dt t n s ds ( ) n (n )nb s (n)(n), s ( s) d ( s) ds ( ) (n ) ( ) n (n )nb (n)(n), ( )d ()() (n ) ( ). s ( s) = n n Applition o le. nd (.6), we get (.5). This opletes the proo o theore. Theore.5. Let : R < b; < d, be ontinuous pping suh tht n exist on nd n L ( ). I n q, q, is onve on the

14 33 Muhd Aer Lti nd Sbir Hussin o-ordintes on, or, n N,, n, then b (.7) (t, s) dsdt (b ) (d ) (b ) (d ) n () k] () l] (b ) k (d ) l kl (k )! (l )! kl x k y l k= l= () k] (b ) k () n (d )! k= ()n (b ) n! l= k (k )! () l] (d ) l l (l )! (b ) n (d ) n (n )! ( )! ( b, ) d Q(s) k ( b, s) x k s ds P (t) nl ( ) t, d t n y l dt n ( b, d x n y ). Proo. Siilr to proo o Theore. by using (.). detils or reder. Thereore we oit the Rerk.. On letting = n = in (.8), (.) nd (.3) respetively yield: (.8) (, ) (, d) (b, ) (b, d) (b ) (d ) (x, y) dydx A (b ) (d ) {B (,) C (,) D (,) E (,) }. (.9) (, ) (, d) (b, ) (b, d) (b ) (d ) 9 /q q b (x, y) dydx A (b ) (d ) B q (,) Cq (,) Dq (,) Eq (,) (.) (b ) (d ) (d ) (t, s) dsdt ( ) b, s ds ( b, d (b ) (d ) 9 6/q q ) b ( t, d ) dt (b ) B q (,) Cq (,) Dq (,) Eq (,). It y be noted tht the bounds in (.8), (.9) nd (.) re shrper thn the bounds o the inequlities proved in Theore.3, Theore.5 nd Theore.6 respetively.

15 Inequlities o Herite-Hdrd Type or Double Integrls Aknowledgent The uthors thnk to the nonyous reeree or his/her very useul nd onstrutive oents whih helped the uthors to iprove the inl version o the pper. R E F E R E N C E S. M. Alori nd M. Drus: Co-ordinted s-onvex untion in the irst sense with soe Hdrd-type inequlities, Int. J. Contep. Mth. Sienes, 3 (3) (8), M. Alori, M. Drus nd S. S. Drgoir: Inequlities o Herite- Hdrd s type or untions whose derivtives bsolute vlues re qusi-onvex, RGMIA Reserh Report Colletion, (suppl. ) (9). 3. S. S. Drgoir nd R. P. Agrwl: Two inequlities or dierentible ppings nd pplitions to speil ens o rel nubers nd to Trpezoidl orul, Appl. Mth. Lett. (5) (998) S.S. Drgoir: On Hdrd s inequlity or onvex untions on the oordintes in retngle ro the plne, Tiwnese Journl o Mthetis, (), S.S. Drgoir nd C.E.M. Pere: Seleted Topis on Herite- Hdrd Inequlities nd Applitions, RGMIA Monogrphs, Online: hdrd.htl]. 6. D. Y. Hwng, K. L. Tseng nd G. S. Yng: Soe Hdrd s inequlities or o-ordinted onvex untions in retngle ro the plne, Tiwnese Journl o Mthetis, (7), D. Y. Hwng: Soe inequlities or n-ties dierentible ppings nd pplitions, Kyungpook, Mth. J. 3(3), G. Hnn: Cubture rule ro generlized Tylor perspetive, PhD Thesis. 9. G. Hnn, S. S. Drgoir nd P. Cerone: A generl Ostrowski type inequlity or double integrls, Tkng J. Mth. Volue 33, Issue,.. U. S. Kiri: Inequlities or dierentible ppings nd pplitions to speil ens o rel nubers to idpoint orul, Appl. Mth. Coput. 7 () M. A. Lti nd M. Alori: Hdrd-type inequlities or produt two onvex untions on the o-ordinetes, Int. Mth. Foru, (7), 9, M. A. Lti nd M. Alori: On the Hdrd-type inequlities or h-onvex untions on the o-ordinetes, Int. J. o Mth. Anlysis, 3(33), 9, M. A. Lti nd S. S. Drgoir: On Soe New Inequlities or Dierentible Co-ordinted Convex Funtions, Journl o Inequlities nd Applitions, :8 doi:.86/9-x--8.. M.E. Özdeir, E. Set nd M.Z. Sriky: New soe Hdrd s type inequlities or oordinted -onvex nd ( α, )-onvex untions, RGMIA, Res. Rep. Coll., 3 (), Suppleent, Artile.

16 336 Muhd Aer Lti nd Sbir Hussin 5. M. E. Özdeir, H. Kvuri, A. O. Akdeir nd M. Avi: Inequlities or onvex nd s-onvex untions on =, b], d], Journl o Inequlities nd Applitions :, doi:.86/9-x M. E. Özdeir, M. A. Lti nd A. O. Akdeir: On soe Hdrd-type inequlities or produt o two s-onvex untions on the o-ordintes, Journl o Inequlities nd Applitions, :, doi:.86/9-x M. E. Özdeir, A. O. Akdeir nd M. Tun: On the Hdrd-type inequlities or o-ordinted onvex untions, rxiv:3.37v. 8. C. M. E. Pere nd J. E. Pečrić: Inequlities or dierentible ppings with pplitions to speil ens nd qudrture orul, Appl. Mth. Lett. 3 () J. E. Pečrić, F. Proshn nd Y. L. Tong: Convex Funtions, Prtil Ordering nd Sttistil Applitions, Adei Press, New York, 99.. M.Z. Sriky, E. Set, M.E. Özdeir nd S. S. Drgoir: New soe Hdrd s type inequlities or o-ordinted onvex untions, rxiv:5.7v th.ca]. Muhd Aer Lti College o Siene Deprtent o Mthetis University o Hil Hil, Sudi Arbi er lti@hotil.o Sbir Hussin Deprtent o Mthetis University o Engineering nd Tehnology Lhore, Pkistn sbirhus@gil.o