TJMM 9 (7), No., 35-4 ON CO-ORDINATED OSTROWSKI AND HADAMARD S TYPE INEQUALITIES FOR CONVEX FUNCTIONS II MUHAMMAD MUDDASSAR, NASIR SIDDIQUI, AND MUHAMMAD IQBAL Abstrt. In this rtile, we estblish vrious ineulities or some dierentible mppings tht re linked with the illustrious Hermite-Hdmrd nd Ostrowski integrl ineulity or onvex untions o severl-vribles on the o-ordintes.the generlized integrl ineulities ontribute some better estimtes thn some lredy presented.. Introdution In 938, A. Ostrowski proved shrp estimte o dierentible untion, whose irst derivtive is bounded by its integrl men s ollows: Theorem. [ Let : [, b R be dierentible untion on (, b) with bounded derivtive, tht is, <, then (x) [ ( ) (t)dt b x b 4 (b ) (b ), () or ll x [, b. The onstnt 4 is the best possible one. Ineulity () hs lot o pplitions in dierent ields o mthemtis suh s probbility nd numeril nlysis et. Due to tht reson () mke the use o ttention mong mthemtiins nd reserhers. Ineulity () ws improved, generlized nd extended in dierent diretions by using dierent tehniues [, 3, 5, 7. Drgomir [6 deined onvex untion on the o-ordintes s ollows: Deinition. Let us onsider the bi-dimensionl intervl = [, b [, d in R with < b, < d. A untion : [, b R, y (u) = (u, y) nd x : [, d R, x (v) = (x, v) re onvex whose deined or ll y [, d nd x [, b. Rell tht the untion : R is onvex on i (λx ( λ)z, λy ( λ)w) λ(x, y) ( λ)(z, w), holds or ll (x, y), (z, w) nd λ [,. Clerly, every onvex (onve) untion : R is onvex (onve) on the oordintes but onverse my not be true [6. By using this deinition mny reserhers ormulted Ostrowski type ineulities nd in prtiulr gve some shrp estimtes or the let nd right Hdmrd ineulity nd some relted results [, 4, 6, 8, 9, 3. The min im o this pper is to estblish some new Ostrowski type ineulities or o-ordinted onvex untions nd s n pplitions we hve derived let Hdmrd type Mthemtis Subjet Clssiition. Primry 6B5, 6D, 6D5; Seondry 6A48, 6A5, 5A4. Key words nd phrses. o-ordinted onvex untion; Ostrowski ineulity; Hermite- Hdmrd ineulity. 35
36 M. MUDDASSAR, N. SIDDIQUI, AND M. IQBAL ineulities. Among those some re shrper thn the exiting one s nd some generlized the exiting one s. This pper is orgnized in the ollowing wy. Ater this Introdution in Setion, we ormulted our min results nd some relted pplitions.. Min Results Lemm. Let : R R be prtil dierentible untion on := [, b [, d with < b nd < d. I L( ), then where, (x, y) A = b I µ,ν = (x µ) (y ν) b (u, y)du d (u, v)dudv A = d (t )(s ) (x, v)dv, I µ,ν, (tµ ( t)x, sν ( s)y)dtds with I µ,ν is positive when (µ, ν) = (, ) nd (b, d) otherwise negtive. Proo. Consider I, = (x ) (y ) (t )(s ) (t ( t)x, s ( s)y)dtds = (x ) (y ) ( ) (s ) (t ) (t ( t)x, s ( s)y)dt ds [ (x )(y ) = ( s) (x, s ( s)y)ds s (t ( t)x, s ( s)y)dsdt = Similrly [ (x )(y ) (x, y) I,d = (x ) (y d) = (x )(y d) [ (x, y) ( s) s (x, s ( s)y)ds (t ( t)x, y)dt (t ( t)x, s ( s)y)dsdt. (t )(s ) (t ( t)x, sd ( s)y)dtds (x, sd ( s)y)ds (t ( t)x, y)dt (t ( t)x, sd ( s)y)dsdt,
OSTROWSKI AND HADAMARD S TYPE INEQUALITIES ON CO-ORDINATES 37 I b, = (x b) (y ) = nd (x b)(y ) I b,d = (x b) (y d) = (x b)(y d) [ (x, y) [ (x, y) whih ompletes the proo. (t )(s ) (tb ( t)x, s ( s)y)dtds (x, s ( s)y)ds (tb ( t)x, y)dt (tb ( t)x, s ( s)y)dsdt, (t )(s ) (tb ( t)x, sd ( s)y)dtds (x, sd ( s)y)ds (tb ( t)x, y)dt (tb ( t)x, sd ( s)y)dsdt, Theorem. Let : R R be prtil dierentible untion on := [, b [, d with < b nd < d. I is onvex untion on the o-ordintes on, then (x, y) (u, v)dudv A (x µ) (y ν) 36 [ (µ, ν) (µ, y) (x, ν) 4 (x, y) Proo. By lemm, we hve (x, y) (u, v)dudv A I µ,ν () I µ,ν (x µ) (y ν) Sine : R is oordinted onvex on dtds I µ,ν (x µ) (y ν) { ( s)( t) t (µ, sν ( s)y) ( t) } (x, sν ( s)y) dtds (x µ) (y ν) { ( s)( t) st (µ, ν) ( s)t (µ, y) ( t)s (x, ν) } (x, y) dtds, whih ompletes the proo.
38 M. MUDDASSAR, N. SIDDIQUI, AND M. IQBAL Corollry. By setting x = b nd y = d ( ) b, d b (b )(d ) 36 6 (b )(d ) 64 in () we hve d (u, v)dudv A { (µ, ν) ( µ, d ) ( ) b, ν 4 ( b, d )} (µ, ν) (3) Remrk. It my be noted tht (3) gives better pproximtion to [9, Theorem. Theorem 3. Let : R R be prtil dierentible untion on := [, b [, d with < b nd < d. I, > with p =, is onvex untion on the oordintes on, then (x, y) (u, v)dudv A / (x µ) (y ν) (p ) /p ( (µ, ν) (µ, y) Proo. By lemm, we hve (x, y) I µ,ν (x µ) (y ν) By Höder s ineulity I µ,ν (x µ) (y ν) ( (x, ν) (u, v)dudv A / (x, y) ) (4) I µ,ν dtds ( t) p ( s) p dsdt) /p ( ) / dtds By oordinted onvexity o on, we hve (x µ) (y ν) ( { I µ,ν st (p ) /p (µ, ν) ( s)t (µ, y) ( t)s (x, ν) } / (x, y) dtds), whih ompletes the proo.
OSTROWSKI AND HADAMARD S TYPE INEQUALITIES ON CO-ORDINATES 39 Remrk. By setting (t, s) M or (t, s) [, b [, d, Theorem 3 redues to [, Theorem 4. The ollowing theorem gives tighter estimte thn tht o previous theorem 3. Theorem 4. Let : R R be prtil dierentible untion on := [, b [, d with < b nd < d. I,, is onvex untion on the o-ordintes on, then (x, y) (u, v)dudv A 3 / (x µ) (y ν) 4 ( (µ, ν) (µ, y) Proo. By lemm, we hve (x, y) I µ,ν (x µ) (y ν) By Höder s ineulity: (x, ν) (u, v)dudv A 4 / (x, y) ) (5) I µ,ν dtds I µ,ν (x µ) (y ν) ( /p dsdt) ( By oordinted onvexity o on, we hve ) / dtds I µ,ν (x µ) (y ν) ( { st /p (µ, ν) ( s)t (µ, y) ( t)s (x, ν) } / (x, y) dtds), whih omplete the proo. Remrk 3. By setting (t, s) M or (t, s) [, b [, d, Theorem 4 redues to [, Theorem 3. Theorem 5. Let : R R be prtil dierentible untion on := [, b [, d with < b nd < d. I, > with /p / = is onve untion on the
4 M. MUDDASSAR, N. SIDDIQUI, AND M. IQBAL o-ordintes on, then (x, y) (x µ) (y ν) (p ) /p Proo. By lemm, we hve (x, y) I µ,ν (x µ) (y ν) By Höder s ineulity: I µ,ν (x µ) (y ν) ( (u, v)dudv A ( µ x, ν y ) (6) (u, v)dudv A I µ,ν dtds ( t) p ( s) p dsdt) /p ( By oordinted onvity o on, we hve dtds [ ( tµ ( t)x, ν y ) dt ( ) µ x, sν ( s)y ( µ x, ν y ), ds ) / dtds whih ompletes the proo. Theorem 6. Let : R R be prtil dierentible mpping on := [, b [, d with < b nd < d. I,, is onve untion on the o-ordintes on, then (x, y) (u, v)dudv A (x µ) (y ν) ( µ x 4, ν y ) (7) 3 3
OSTROWSKI AND HADAMARD S TYPE INEQUALITIES ON CO-ORDINATES 4 Proo. By the onvity o on the o-ordintes on nd power-men ineulity, the ollowing ineulity holds: (tµ ( t)x, v) t (µ, v) ( t) (x, v) ( t (µ, v) ( t) ) (x, v) or ll µ, x [, b, t [, nd ixed v [, d (tµ ( t)x, v) t (µ, v) ( t) (x, v). Similrly (u, sν ( s)y) s (u, ν) ( s) (u, y), or ll ν, y [, d, s [, nd ixed u [, b, showing tht is onve on o-ordintes on. Now by lemm (x, y) (u, v)dudv A I µ,ν. I µ,ν (x µ) (y ν) dtds. By oordinted onvity o on, nd Jensen s integrl ineulity, we hve dtds [ = ( t) ( s) ds ds [ ( ( t) ( s)ds) ( ) ( s)(sν ( s)y)ds tµ ( t)x, ( s)ds dt = ( t) ( tµ ( t)x, ν y ) dt 3 ( ( t)dt) ( ) ( t)(tµ ( t)x)dt, ν y ( t)dt 3 = ( µ x 4, ν y ) dt, 3 3 whih ompletes the proo. Remrk 4. By setting x = b, y = d, Theorem 6 redues to [9, Theorem 5. Authors ontributions: All uthors ontributed eully in this rtile. They red nd pproved the inl mnusript. Aknowledgments: This reserh pper is mde possible through the help nd support rom HEC, Pkistn nd Univ. o the Engg. & Teh. Txil, Pkistn. We grteully
4 M. MUDDASSAR, N. SIDDIQUI, AND M. IQBAL knowledge the time nd expertise devoted to reviewing ppers by the dvisory editors, the members o the editoril bord, nd the reerees. Finlly, We would like to express our tribute to Pro. Dr. Muhmmd Ibl Bhtti (Chirmn, Dept. o Mth., UET, Txil), Our dotorl Supervisor, who died young in his work. As tlented nd outstnding young proessor, you sriied your lie to the ountry when your reer ws thriving. We will never orget how muh you red bout our study nd dily lie. We will miss you orever! Reerenes [ Alomri, M. nd Drus, M., Some Ostrowski type ineulities or onvex untions with pplitions, RGMIA 3 (). Artile No. 3. [ Alomri, M. nd Drus, M., The Hdmrd s ineulity or s-onvex untions o -vribles, Int. Jour. Mth. Anl. (3), (8), 69 638. [3 Alomri, M., Drus, M., Drgomir, S.S., Cerone, P., Ostroski type ineulities or untions whose derivtives re s-onvex in seond sense, Appl. Mth. Lett. 3 (), 7 76. [4 Bkul M.K., nd Pečrić, J., On the Jensen s ineulity or onvex untions on the o-ordintes in retngle rom the plne, Tiwnese Journl o Mthemtis 5 (6), 7 9. [5 Cerone, P., Drgomir, S.S., Roumeliots, J., An ineulity o Ostrowski-Grüss type or twie dierentible mppings nd pplitions in numeril integrtion, Kyungpook Mth. J. 39 (999), 33 34. [6 Drgomir, S.S., On Hdmrd ineulity or onvex untions on o-ordintes in retngle rom the plne, Tiwnese Journl o Mthemtis, 5 (), 775 788. [7 Drgomir, S.S., Cerone, P., Roumelities, J., A new generliztion o Ostrowski integrl ineulity or mpping whose derivtives re bounded nd pplitions in numeril integrtion nd or speil mens, Appl. Mth. Lett. 3 (), 9 5. [8 Lti, M.A. nd Alomri, M., Hdmrd-type ineulities or produt two onvex untions on the o-ordintes, Int. Mth. Forum 4 (47), (9), 37 338. [9 Lti, M.A. nd Drgomir, S.S., On some new ineulities or dierentible o-ordinted onvex untions, Jour. Ine. Appl. :8, doi:.86/9-4x--8. [ Lti, M.A., Hussin, S. nd Drgomir, S.S., New Ostrowski type ineulities o o-ordinted onvex untions, ON-LINE:http://rgmi.org/ppers/v4/v449.pd. [ Mitrinović, D.S., Pečrić, J.E. nd Fink, A.M., Ineulities Involving Funtions nd their Integrls nd Derivtives, Kluwer Ademi Publishers, Dortreht, 99. [ Ostrowski, A., Über die Absolutbweihung einer dierentiebren unktin von ihren intergrlmittelwert, Comment. Mth. Helv. (938), 6 7. [3 Özdemir, M.E., Kvurmi, H., Akdemir, A.O. nd Avi, M., Ineulities or onvex nd s-onvex untions on = [, b [, d, Jour. Ine. Appl.,. Deprtment o Mthemtis Govt. College o Siene, Lhore - Pkistn E-mil ddress: mlik.muddssr@gmil.om Deprtment o Mthemtis University o Engineering nd Tehnology, Txil - Pkistn E-mil ddress: nsir.siddiui@uettxil.edu.pk Deprtment o Mthemtis Govt. Islmi College, Civil Lines Lhore- Pkistn E-mil ddress: mibl.bki@gmil.om