A GENERALIZATION OF HERMITE-HADAMARD S INEQUALITY

Transcription

1 Krgujevc Jourl o Mthemtics Volume 4() (7), Pges A GENERALIZATION OF HERMITE-HADAMARD S INEQUALITY MOHAMMAD W. ALOMARI Abstrct. I literture the Hermite-Hdmrd iequlity ws eligible or my resos, oe o the most surprisig d iterestig tht the Hermite-Hdmrd iequlity combie the midpoit d trpezoid ormule i iequlity. I this work, Hermite-Hdmrd like iequlity tht combies the composite trpezoid d composite midpoit ormule is proved. So tht, the clssicl Hermite-Hdmrd iequlity becomes specil cse o the preseted result. Some Ostrowski s type iequlities or covex uctios re proved s well.. Itroductio Let :, b R, be twice dieretible mppig such tht (x) exists o (, b) d = sup x (,b) (x) <. The the midpoit iequlity is kow s: + b (b )3 (x) dx (b ) 4, d, the trpezoid iequlity (x) dx (b ) () + (b) (b )3, lso hold. Thereore, the itegrl (x) dx c be pproximted i terms o the midpoit d the trpezoidl rules, respectively such s: (x) dx + b = (b ), Key words d phrses. Hermite-Hdmrd iequlity, Ostrowski iequlity, covex uctios. Mthemtics Subject Clssiictio. Primry: 6A5. Secodry: 6D5. Received: April 6, 6. Accepted: September 3, 6. 33

2 34 M. W. ALOMARI d (x) dx () + (b) = (b ), which re combied i useul d mous reltioship, kow s the Hermite- Hdmrd s iequlity. Tht is, + b (.) () + (b) (x) dx, b which hold or ll covex uctios deied o rel itervl, b. The rel begiig ws (lmost) i the lst twety ive yers, where, i 99 Drgomir 9 published his rticle bout (.). The mi result i 9 ws Theorem.. Let :, b is covex uctio oe c deie the ollowig mppig o, such s: b ( H (t) = tx + ( t) + b ) dx, (b ) the: () H is covex d mootoic o-decresig o, ; (b) oe hs the bouds or H d sup H (t) = t, i H (t) = t, b (x) dx = H (), (b ) + b = H (). Few yers ter 99, my uthors hve took ( rel) ttetio to the Hermite- Hdmrd iequlity d sequece o severl works uder vrious ssumptios or the uctio ivolved such s bouded vritio, covex, dieretible uctios whose -derivtive(s) belog to L p, b; ( p ), Lipschitz, mootoic, etc., hve bee published. For comprehesive list o results d excellet bibliogrphy we recommed the iterested to reer to 3, 4, 3. I 997, Yg d Hog 5, cotiued o Drgomir result (Theorem.) d they proved the ollowig theorem. Theorem.. Suppose tht :, b R, is covex d the mppig F :, R is deied by F (t) = ( + t b + t ) ( + t u + b + t ) u du, the:

3 A GENERALIZATION OF HERMITE-HADAMARD S INEQUALITY 35 () the mppig F is covex d mootoic odecresig o, ; (b) we hve the bouds b i F (t, s) = (x) dx = F (), t, (b ) sup F (t) = t, () + (b) = F (). For other closely relted results see, 5 8,. I terms o composite umericl itegrtio, we recll the composite midpoit rule, p. / (x) dx = h (x j ) + j= (b ) h (µ), 6 or some µ (, b), where C, b, is eve, h = b d x + j = + (j + )h, or ech j =,,..., + ; d, the composite trpezoid rule, p. 3 (x) dx = h () + (x j ) + (b) (b ) h (µ), or some µ (, b), where C, b, h = b d x j = + jh, or ech j =,,...,. The mi purpose o this work, is to combie the composite trpezoid d composite midpoit ormule i iequlity tht is similr to the clssicl Hermite-Hdmrd iequlity (.) or covex uctios deied o rel itervl, b. I this wy, we estblish covetiol geerliztio o (.) which is i tur most useul d hs very costructiol orm.. A Geerliztio o Hermite-Hdmrd s Iequlity Theorem.. Let :, b R be covex uctio o, b, the the double iequlity xk + x b k (.) h (t) dt h () + (x k ) + (b), holds, where x k = + k b b, k =,,,..., ; with h =, N. The costt i the let-hd side d i the right-hd side re the best possible or ll N. I is cocve the the iequlity is reversed. Not bee: ter revisio o this pper, the oymous reeree iormed us tht the iequlity (.) ws proved i more geerl cse i 4 (see lso 3, p. 3). We pprecite this remrk rom the reviewer.

4 36 M. W. ALOMARI Proo. Sice is covex o, b, the so is o ech subitervl, x j, j =,...,, the or ll t,, we hve (.) (t + ( t) x j ) t ( ) + ( t) (x j ). Itegrtig (.) with respect to t o, we get (.3) (t + ( t) x j ) dt () + (x j ). Substitutig u = t + ( t) x j, i the let hd side o (.3), we get (u) du x j ( ( ) + (x j )). Tkig the sum over j rom to, we get (.4) = (u) du (u) du x j mx {x j } j ( ( ) + (x j )) ( ( ) + (x j )) = h (x ) + (x ) + { ( ) + (x j )} + (x ) + (x ) j= = h () + (x j ) + (b). O the other hd, gi sice is covex o I x, the or t,, we hve (.5) ( txj + ( t) = + ( t) x ) j + t (tx j + ( t) ) + (( t) x j + t ).

5 A GENERALIZATION OF HERMITE-HADAMARD S INEQUALITY 37 Itegrtig iequlity (.5) with respect to t o, we get (.6) = (tx j + ( t) ) + (( t) x j + t ) dt (tx j + ( t) ) dt + (( t) x j + t ) dt. By puttig t = s i the secod itegrl o the right-hd side o (.6), we hve (.7) = (tx j + ( t) ) dt + (tx j + ( t) ) dt. (( t) x j + t ) dt Substitutig u = tx j + ( t), i the let hd side o (.7), d the tkig the sum over j rom to, we get (.8) xk + x k h (t) dt. From (.4) d (.8), we get the desired iequlity (.). To prove the shrpess let (.) hold with other costts C, C >, which gives (.9) xk + x b k C h (t) dt C h () + (x k ) + (b).

6 38 M. W. ALOMARI Let :, b R be the idetity mp (x) = x, the the right-hd side o (.9) reduces to b C h + x k + b ( = C h + + k b ) + b = C h + + b k + b = C b = C (b ) ( + b). + ( ) + b ( ) + b It ollows tht C, i.e., is the best possible costt i the right-hd side o (.). For the let-hd side, we hve b x k + x k C h { = C h + (k ) b }. = C b + b ( ) ( + ) = C b, which mes tht C, d thus is the best possible costt i the let-hd side o (.). Thus the proo o Theorem. is completely iished. Remrk.. I Theorem., i we tke =, the we reer to the origil Hermite- Hdmrd iequlity (.). I viewig o (.), ext we give direct shrp reiemets o Hermite-Hdmrd s type iequlities or covex uctios deied o rel itervl, b, ccordig to the umber o divisio (i our cse =,, 3, 4) i Theorem.. Corollry.. I Theorem., we hve () i =, the + b (b ) (t) dt (b ) () + (b) ;

7 A GENERALIZATION OF HERMITE-HADAMARD S INEQUALITY 39 (b) i =, the (c) i = 3, the (b ) 3 (b ) 6 (b ) (t) dt (b ) 4 (t) dt 3 + b + 3b () + + b + (b) ; 5 + b + b + 5b () + + b + b (b) ; (d) i = 4, the (b ) 7 + b 5 + 3b 3 + 5b + 7b (b ) (t) dt () b + b + 3b (b). Now, let :.b R be covex uctio o, b. Deie the mppigs H j, F j :, R, give by (.) H j (t) = ( tu + ( t) x ) j + x j du, u, x j, h d (.) F j (t) = h ( + t + t ) ( + t u + x j + t ) u du, where u, x j. Applyig Theorems. d., or :, x j R, j =,,...,. The the ollowig sttemets hold: () H j (t) d F j (t) re covex or ll t, d u, x j ; (b) H j (t) d F j (t) re mootoic odecresig or ll t, d u, x j. (c) we hve the ollowig bouds or H j (t) (.) h (u) du = H j (),

8 3 M. W. ALOMARI d xk + x k (.3) = H j (). (.4) (.5) d the ollowig bouds or F j (t) d ( ) + (x j ) = F j (), xj (u) du = F j (). h Hece, we my estblish two relted mppigs or the iequlity (.). Propositio.. Let be s i Theorem., deie the mppigs H, F :, R, give by H (t) = H j (t) d F (t) = F j (t), where H j (t) d F j (t) re deied i (.) d (.), respectively; the the ollowig sttemets hold: () H(t) d F (t) re covex or ll t, d u, b; (b) H(t) d F (t) re mootoic odecresig or ll t, d u, b; (c) We hve the ollowig bouds or H(t) d sup t, H (t) = h i H (t) = t, (u) du = H (), xk + x k = H (), d the ollowig bouds or F (t) sup F (t) = () + (x k ) + (b) = F (), t, d i t, F (t) = h (u) du = F (). Proo. Tkig the sum over j rom to, i (.) (.5) we get the required results, d we shll omit the detils. Remrk.. The iequlity (.) my writte i coveiet wy s ollows: xk + x k (x k ) () + (b) (t) dt (x k ), h

9 A GENERALIZATION OF HERMITE-HADAMARD S INEQUALITY 3 which is o Ostrowski s type. Some shrps Ostrowski s type iequlities or covex uctios deied o rel itervl, b, re proposed i the ext theorems. Theorem.. Let :, b R + be covex uctio o, b, the the iequlity (.6) (x) dx (b ) (y) h () + (x k ) + (b), holds or ll y, b. where, x k = + k b b, k =,,,..., ; with h =, N. The costt i the right-hd side is the best possible, i the sese tht it cot be replced by smller oe or ll N. I is cocve the the iequlity is reversed. Proo. Fix y, x j, j =,...,. Sice is covex o, b, the so is o ech subitervl, x j, i prticulr o, y, the or ll t,, we hve (.7) (t + ( t) y) t ( ) + ( t) (y), j =,...,. Itegrtig (.7) with respect to t o, we get (.8) (t + ( t) y) dt () + (y). Substitutig u = t + ( t) y, i the let hd side o (.8), we get (.9) y (u) du y ( ( ) + (y)). Now, we do similrly or the itervl y, x j, we thereore hve (.) (ty + ( t) x j ) t (y) + ( t) (x j ), j =,...,. Itegrtig (.) with respect to t o, we get (.) (ty + ( t) x j ) dt (y) + (x j). Substitutig u = ty + ( t) x j, i the let hd side o (.), we get (.) y (u) du x j y ( (y) + (x j )).

10 3 M. W. ALOMARI Addig the iequlities (.9) d (.), we get (.3) = y (u) du + (u) du y y x j y (u) du ( ( ) + (y)) + x j y ( (y) + (x j )) ( ) + x j y ( ) + (x j ) (y) ( ( ) + ( )) + h (y). Tkig the sum over j rom to, we get = = (u) du (u) du x j mx {x j } j { ( ) + (x j )} + h (y) ( ( ) + (x j )) + (b ) (y) = h (x ) + (x ) + { ( ) + (x j )} + (x ) + (x ) + (b ) (y) j= = h () + (x j ) + (b) + (b ) (y), which gives tht b (u) du (b ) (y) h () + (x j ) + (b), or ll y, x j, b or ll j =,,...,, which gives the desired result (.6). To prove the shrpess let (.6) hold with other costts C >, which gives b (.4) (x) dx (b ) (y) C h () + (x k ) + (b).

11 A GENERALIZATION OF HERMITE-HADAMARD S INEQUALITY 33 Let :, R be the idetity mp (x) = x, the the right-hd side o (.4) reduces to y C x k + = C ( ) + = C. Choose y =, it ollows tht C, i.e., right-hd side o (.6). is the best possible costt i the Theorem.3. Uder the ssumptios o Theorem., we hve (.5) (x) dx (b ) (y) h + mx j y x j + x j () + (x j ) + (b), or ll y, b. The costt i the right-hd side is the best possible or ll N. I is cocve the the iequlity is reversed. I prticulr, i = the b (x) dx (b ) (y) + y + b () + (b), or ll y, b. Proo. Repetig the steps o the proo o Theorem., thereore by (.3) (.6) (u) du y ( ) + x j y ( ) + h (y) { y xj mx, x } j y ( ( ) + ( )) + h (y) xj + y x j + x j ( ( ) + ( )) + h (y)

12 34 M. W. ALOMARI Tkig the sum over j rom to, we get (u) du xj = + y x j + x j { ( ) + (x j )} + h (y) xj mx + j y x j + x j ( ( ) + (x j )) + (b ) (y) h + mx j y x j + x j (x ) + (x ) + { ( ) + (x j )} + (x ) + (x ) + (b ) (y) j= h = + mx j y x j + x j () + (x j ) + (b) + (b ) (y), which gives tht (u) du (b ) (y) h + mx j y x j + x j () + (x j ) + (b) m or ll y, x j, b or ll j =,,...,, which gives the desired result (.5). The proo o shrpess goes likewise the proo o the shrpess o Theorem. d we shll omit the detils. Corollry.. Let α i, or ll i =,,,...,, be positive rel umbers such tht i= α i =, the uder the ssumptios o Theorem.3, we hve ( ) b (x) dx (b ) α i x i + h + mx j + i= i= α i x i + x j () + (x j ) + (b), or ll y, b. The costt i the right-hd side is the best possible. I is cocve the the iequlity is reversed. Theorem.4. Uder the ssumptios o Theorem.3, we hve b (.7) (u) du (x j ) b () + + (b),

13 A GENERALIZATION OF HERMITE-HADAMARD S INEQUALITY 35 or ll j =,,...,. The costt i the right-hd side is the best possible. I is cocve the the iequlity is reversed. Proo. Repetig the steps o the proo o Theorem.3, (.6) i we choose y = +x j, the we get (u) du x j ( ( ( ) + ( )) + h Tkig the sum over j rom to, we get x j (u) du { ( ) + (x j )} + h h { ( ) + (x j )} + h, which gives tht (u) du h ). h () + (x j ) + (b), or ll j =,,...,, which gives the desired result (.7). The proo o shrpess goes likewise the proo o the shrpess o Theorem. d we shll omit the detils. Theorem.5. Let I R be ope itervl d, b I, < b. Let : I R + be icresig covex uctio o, b, the the iequlity (b ) (.8) (t) dt (y) h, is vlid or ll y, b I. The costt i the right-hd side is the best possible, i the sese tht it cot be replced by greter oe. I is cocve the the iequlity is reversed. I prticulr, i = the (b ) (t) dt (y) b + b, Proo. Let y, x j be rbitrry poit such tht < y p y y +p < x j or ll j =,,..., with p >. It is well kow tht is covex o I i (y) p y+p y p (t) dt, or every subitervl y p, y + p, b I or some p >. But sice icreses o, b, we lso hve (y) y+p (t) dt (t) dt. p y p p

14 36 M. W. ALOMARI Choosig p h, (this choice is vilble sice it is true or every subitervl i I), thereore rom the lst iequlity we get (y) h y+p y p (t) dt h Agi by covexity we hve h Addig the lst two iequlities, we get h (y) + h (t) dt. (t) dt. (t) dt, or we write xj + x xj j h (t) dt h (y). Tkig the sum over j rom to, we get h (t) dt h (y) hece, (t) dt (b ) (y) h, holds by positivity o d this proves our ssertio. To prove the shrpess let (.8) holds with other costt C >, which gives (b ) (.9) (t) dt (y) C h. Let :, R + be the idetity mp (x) = x, the the right-hd side o (.9) reduces to y C j = C = C = C. j ( + ) Choose y =, it ollows tht C which mes tht C, i.e., 4 possible costt i the right-hd side o (.8). is the best

15 A GENERALIZATION OF HERMITE-HADAMARD S INEQUALITY 37 Corollry.3. Let α i, or ll i =,,,...,, be positive rel umbers such tht i= α i =, the uder the ssumptios o Theorem.5, we hve ( ) b (b ) (t) dt α i x i h. + i= The costt i the right-hd side is the best possible. I is cocve the the iequlity is reversed. I prticulr cse i =, the (b ) α + ( α) b (t) dt b + b, or ll α,. Ackowledgemets. I would like to thk the oymous reerees or their creul redig d I pprecite their vluble commets. Reereces M. Akkouchi, A result o the mppig H o S.S. Drgomir with pplictios Fct Uiversittis (NIS), Ser. Mth. Iorm. 7 (), 5. R. L. Burde d J. D. Fires, Numericl Alysis, 6th editio, Brooks/Cole Publishig, P. Ceroe d S. S. Drgomir, Midpoit Type Rules rom Iequlities Poit o View, Hdbook o Alytic-Computtiol Methods i Applied Mthemtics, New York, CRC Press,. 4 P. Ceroe d S. S. Drgomir, Trpezoidl-Type Rules rom Iequlities Poit o View, Hdbook o Alytic-Computtiol Methods i Applied Mthemtics, New York, CRC Press,. 5 S. S. Drgomir, Two reiemets o Hdmrd s iequlities, Coll. o Sci. Pp. o the Fc. o Sci. Krgujevc (99), S. S. Drgomir, J. E. Pečrić d J. Sdor, A ote o the Jese-Hdmrd s iequlity, Al. Numer. Theor. Approx. 9 (99), 8. 7 S. S. Drgomir, Some reiemets o Hdmrd s iequlity, Gz. Mt. Metod. (99), S. S. Drgomir, A mppig coected with Hdmrd s iequlities, A. Oster. Akd. Wiss. Mth.-Ntur. 3 (99), 7. 9 S. S. Drgomir, Two mppigs i coectio to Hdmrd s iequlities, J. Mth. Al. Appl. 67 (99), S. S. Drgomir d N. M. Ioescu, Some itgrl iequlities or dieretible covex uctios, Coll. o Sci. Pp. o the Fc. o Sci. Krgujevc 3 (99), 6. S. S. Drgomir, Some itegrl iequlities or dieretible covex uctios, Cotributios, Mcedoi Acd. o Sci. d Arts 3() (99), 3 7. S. S. Drgomir, O Hdmrd s iequlities or covex uctios, Mth. Blkic (N.S.) 6(4) (99), 5. 3 S. S. Drgomir d C. E. M. Perce, Selected topics o Hermite-Hdmrd iequlities d pplictios, RGMIA Moogrphs, Victori Uiversity,. 4 S. S. Drgomir, Some remrks o Hdmrd s iequlities or covex uctios, Extrct Mth. 9() (994),

16 38 M. W. ALOMARI 5 G. S. Yg d M. C. Hog, A ote o Hdmrd s iequlity, Tmkg J. Mth. 8() (997), Deprtmet o Mthemtics Fculty o Sciece d Iormtio Techology Irbid Ntiol Uiversity 6 Irbid, Jord E-mil ddress: mwomth@gmil.com