NEW OSTROWSKI-TYPE INEQUALITIES AND THEIR APPLICATIONS IN TWO COORDINATES
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1 At Mth Univ Comenine Vol LXXXV, (06, pp NEW OSTROWSKI-TYPE INEQUALITIES AND THEIR APPLICATIONS IN TWO COORDINATES G FARID Abstrt In this pper, new Ostrowski-type inequlities in two oordintes re estblished Also s n pplition of these inequlities we find bounds of nonnegtive differenes of Hdmrd inequlity in two oordintes Introdution nd preliminries An inequlity by Ostrowski [6 is onsidered by mny mthemtiins nd lot of rtiles reflet its motivtion (see, [, 5, 7,, 9, nd referenes there in it is stted in the following theorem Theorem Let f I R, where I is n intervl in R, be mpping differentible in I, the interior of I, nd, b I, < b If f (t M for ll t [, b, then we hve f(x [ f(tdt +b b (x ( + (b (b M for x [, b In [, Cheng gve the following Ostrowski-type inequlity Theorem Let I be n open intervl in R,, b I, < b f I R is differentible funtion suh tht there exist onstnts γ, Γ R with γ f (x Γ, x [, b Then we hve ( for ll x [, b f(x (x bf(b (x f( (b (x + (b x (Γ γ (b b f(tdt In [3, S S Drgomir gve Hdmrd inequlity for retngle in plne by defining onvex funtions on oordintes Reeived Mrh 7, 05; revised August, Mthemtis Subjet Clssifition Primry 6A5, 6D5 Key words nd phrses Ostrowski-type inequlities; Hdmrd inequlity; bounds
2 0 G FARID Definition 3 Let := [, b [, d R with < b nd < d A funtion f R lled onvex on oordintes if the prtil mppings f y [, b R, f y (u := f(u, y nd f x [, d R, f x (v := f(x, v re onvex, defined for ll y [, d nd x [, b Theorem Suppose tht f R is onvex on the oordintes on Then one hs the following inequlities ( + b f, + d [ b ( f x, + d dx + ( + b f b d, y dy (b (d [ b b f(x, ydxdy f(x, dx + b f(x, ddx + f(, ydy + d d [ f(, + f(, d + f(b, + f(b, d f(b, ydy P Cerone nd S S Drgomir [ found bounds of non-negtive differenes of lssil Hdmrd inequlity whih pper s upper nd lower bounds for midpoint nd trpezoidl qudrture rules N Ujević [ lso found new bounds nd mde omprison with P Cerone nd S S Drgomir results given in [ Motivted by suh bounds, we re interested in finding bounds of non-negtive differenes of Hdmrd inequlity in two oordintes by pplying Ostrowski-type inequlities in two oordintes In this pper, we give the versions of the bove inequlities ( nd ( in two oordintes As n pplition we pply these inequlities nd give bounds of non-negtive differenes of Hdmrd inequlity in two oordintes given in Theorem Ostrowski type inequlities in two oordintes First we give Ostrowski-type inequlity in two oordintes using Theorem Theorem Let f I J R, where I, J re open intervls in R, be mpping suh tht for, b I,, d J, < b, < d, the prtil mppings f y [, b R, f y (u := f(u, y nd f x [, d R, f x (v := f(x, v, defined for ll y [, d nd x [, b, re differentible nd f y(t M, t [, b, f x(t N, t [, d Then we hve f(x, + f(x, d d f(, y + f(b, y dx + dy ( b + ( f(x, ydxdy M + N (b (d d
3 NEW OSTROWSKI-TYPE INEQUALITIES IN TWO COORDINATES 09 Proof Applying Ostrowski inequlity for mpping f y t x = b, we hve f(b, y b Integrting over [, d, we hve ( f(b, ydy b f(t, ydt f(x, ydxdy (b M (b (d M Agin pplying Ostrowski inequlity for mpping f y t x = nd integrting over [, d, we hve (3 f(, ydy b Using ( nd (3, we n hve ( f(, y + f(b, y dy b f(x, ydxdy (b (d M f(x, ydxdy (b (d M Similrly, using inequlities getting fter pplying Ostrowski inequlity for mpping f x first t y =, then t y = d nd integrting over [, b we n hve (5 f(x, + f(x, d dx d Using ( nd (5 one n get ( f(x, ydxdy (b (d N Now in the following, we give version of Theorem in two oordintes Theorem Let f I J R, where I, J re open intervls in R, be mpping suh tht for, b I,, d J, < b, < d, the prtil mppings f y [, b R, f y (u := f(u, y nd f x [, d R, f x (v := f(x, v, defined for ll y [, d nd x [, b, re differentible with γ y f y(t Γ y, t [, b, γ x f x(t Γ x, t [, d Then we hve (6 f(x, + f(x, d dx + ( b + d f(, y + f(b, y dy f(x, ydxdy Γ x + Γ y (γ x + γ y (b (d
4 0 G FARID Proof Applying inequlity ( for mpping f y t x =, we hve f(, y + b (b f(b, y f(x, ydx (Γ y γ y b Integrting over [, d, we get f(, y + f(b, y dy f(x, ydxdy (7 b (b (d (Γ y γ y Now pplying inequlity ( for mpping f x t y =, we hve f(x, + d (d f(x, d f(x, ydx (Γ x γ x d Integrting over [, b, we hve f(x, + f(x, d dy f(x, ydxdy ( d (b (d (Γ x γ x Using inequlities (7 nd (, one n get (6 In the following results we give the bounds of non-negtive differene of Hdmrd inequlity in two oordintes given in Theorem Theorem 3 Under the ssumptions of Theorem, we hve [ b (f(x, + f(x, ddx + (f(, y + f(b, ydy b d b (9 f(x, ydxdy (b (d ((b M + (d N Proof From (, we hve d (f(, y + f(b, ydy (d (0 b f(x, ydxdy (b (d From (5, we hve (b ( (f(, y + f(b, ydx (b (d f(x, ydxdy Using inequlities (0 nd (, one n get (9 (b M (d N
5 NEW OSTROWSKI-TYPE INEQUALITIES IN TWO COORDINATES Theorem Under the ssumptions of Theorem, we hve [ b (f(x, + f(x, ddx + (f(, y + f(b, ydy b d b ( f(x, ydxdy (b (d ( (b (Γy γ y + (d (Γ x γ x Proof From (7, we hve d (f(, y + f(b, ydy (d (3 b f(x, ydxdy (b (d From (, we hve b (f(, y + f(b, ydx (b ( b f(x, ydxdy (b (d Using inequlities (3 nd (, one n get ( 6 (b (Γ y γ y (d (Γ x γ x Theorem 5 Under the ssumptions of Theorem, we hve ( f x, + d ( + b dx + f, y dy (5 ( b + f(x, ydxdy M + N (b (d d Proof Applying Ostrowski inequlity for mpping f y t x = +b, we hve ( + b f, y (b M f(t, ydt b Integrting over [, d, we hve ( + b (6 f, y dy b f(x, ydxdy (b (d M Similrly, pplying Ostrowski inequlity for mpping f x t y = +d, then integrting over [, b, we n hve ( f x, + d dy (b (d N (7 f(x, ydxdy d Using (6 nd (7 one n get (5 In the next result we give bounds of other non-negtive differene of Hdmrd inequlity in two oordintes given in Theorem
6 G FARID Theorem 6 Under the ssumptions of Theorem, we hve [ ( f x, + d ( + b dx + f, y dy ( b M(b + N(d f(x, ydxdy (b (d Proof From inequlity (6, we hve d ( + b f d, y (9 dy (b (d While from inequlity (7, we obtin b ( f x, + d (0 dy b (b (d Using (9 nd (0, one n get ( Aknowledgment f(x, ydxdy f(x, ydxdy Thnks to referee for suggestions (b M (d N Referenes Cheng X L, Improvement of some Ostrowski-Grüss type inequlites, Comput Mth Appl, (00, 09 Cerone P nd Drgomir S S, Midpoint-type rules from n inequlities point of view, Hndbook of Anlyti-Computtionl Methods in Applied Mthemtis, Editor: G Anstssiou, CRC Press, New York (000, Drgomir S S, On Hdmrd s inequlity for onvex funtions on the o-ordintes in retngle from the plne, Tiwnese J Mth (00, Ujević N, Some double integrl inequlities nd pplitions, At Mth Univ Comenine 7( (00, Liu W-J, Xue Q-L, nd Wng S-F, Severl new Perturbed Ostrowski-like type inequlities, J Inequl Pure nd Appl Mth (JIPAM, ( (007, rtile: 0 6 Ostrowski A, Über die Absolutbweihung einer dierentierbren Funktion von ihren Integrlmittelwert, Comment Mth Helv, 0 (93, Ozdemir M E, Kvurmi H, nd Set E, Ostrowski s type inequlities for (α, m-onvex funtions, Kyungpook Mth J, 50 (00, Qioling X, Jin Z, nd Wenjun L, A new generliztion of Ostrowski-type inequlity involving funtions of two independent vribles, Comput Mth Appl,60 (00, 9 9 Sriky M Z, On the Ostrowski type integrl inequlity, At Mth Univ Comeninee, Vol 79( (00, 9 3 G Frid, Deprtment of Mthemtis COMSATS, Institute of Informtion Tehnology, Attok Cmpus, Pkistn, e-mil: fridphdsms@hotmilom
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