INTEGRAL INEQUALITIES FOR DIFFERENTIABLE RELATIVE HARMONIC PREINVEX FUNCTIONS (SURVEY)

Transcription

1 TWMS J. Pure Appl. Mth., V.7, N., 6, pp.3-9 INTEGRAL INEQUALITIES FOR DIFFERENTIABLE RELATIVE HARMONIC PREINVEX FUNCTIONS SURVEY M.A. NOOR, K.I. NOOR, S. IFTIKHAR Abstrct. In this pper, we consider nd investigte the reltive hrmonic preinvex unctions, which uniies severl new known clsses o hrmonic preinvex unctions. We derive severl new integrl ineulities such s Hermite-Hdmrd, Simpson s, trpezoidl or the reltive hrmonic preinvex unctions. Since the reltive hrmonic preinvex unctions include, convex unction, hrmonic convex unctions, preinvex unctions, reltive hrmonic convex nd reltive preinvex unctions s specil cses, results obtined in this pper continue to hold or these problems. Severl open problems hve been suggested or urther reserch in these res. Keywords: hrmonic convex unctions, preinvex unctions, hrmonic preinvex unctions, h- convex unctions, midpoint ineulity, trpezoidl ineulity, Simpson s ineulity, Hermite- Hdmrd type ineulity. AMS Subject Clssiiction: 6D5, 6D, 9C3.. Introduction It is well known tht ineulities hve plyed undmentl role in the development o lmost ll the ields o pure nd pplied sciences nd re continuing to do so. Ineulities present very ctive nd scinting ield o reserch. In recent yers, wide clss o integrl ineulities re being derived vi dierent concepts o convexity. These integrl ineulities re useul in physics, where upper nd lower bounds or nturl phenomen described by integrls such s mechnicl work virtul work re reuired. Integrl ineulities re closely relted the convex unctions nd their vrint orms. As result o interction between dierent brnches o mthemticl nd engineering sciences, convex unctions hve been extended nd generlized in severl directions rom dierent point o views. The ides nd techniues o convex unctions re being used in vriety o diverse res o sciences nd proved to be productive nd innovtive. These cts hve inspired nd motivted the reserchers to generlize nd extend the concept o convexity in vrious directions. The development o convexity theory cn be viewed s the simultneous pursuit o two dierent lines o reserch. It revels the undmentl cts on the ulittive behvior o the solution to importnt clsses o problems; on the other hnd, it lso helps us to develop us highly eicient nd powerul new numericl techniues to solve complicted nd complex problems. In ct, convexity theory provides us sound bsis or computing the pproximte nd nlyticl solutions o lrge number o seemingly unrelted problems in generl nd uniied rmework. For exmple, the vritionl ineulities, which cn be regrded s nturl extension o vritionl principles, re relted to the simple ct tht the minimum o dierentible convex unction on convex set in ny normed spce cn be Mthemtics Deprtment, COMSATS Institute o Inormtion Technology, Islmbd, Pkistn e-mil: noormslm@gmil.com, khlidnoor@hotmil.com, sbh.itikhr@gmil.com Mnuscript received Jnury 6. 3

2 4 TWMS J. PURE APPL. MATH., V.7, N., 6 chrcterized by vritionl ineulity. However, it remrkble nd mzing tht vritionl ineulities llow mny diversiied pplictions in ever brnch o pure nd pplied sciences. On other hnd, unction is convex unction, i nd only i, it stisies the Hermite-Hdmrd type ineulity. To be more precise, unction is convex unction on the intervl I = [, b], i nd only i, stisies the ineulity + b b b x dx + b, x [, b], which is clled the Hermite-Hdmrd ineulity or convex unctions, see [3,, 5]. For novel pplictions o the Hermite-Hdmrd ineulity, see Khttri [5]. Convexity theory is n eective nd powerul techniue or studying wide clss o problems which rise in vrious brnches o pure nd pplied sciences. Convex unctions hve been generlized nd extended in severl directions using interesting nd novel ides. Severl new clsses o convex unctions nd convex sets hve been introduced nd investigted. Vrious new ineulities relted to these new clsses o convex unctions hve been derived by reserchers, see, or exmple, [4, 6-6, 8-, 5-9, 3-33, 35-5, 56] nd the reerence therein. Motivted by the ongoing reserch in this ield, Vrosnec [54] introduced clss o convex unctions with respect to n rbitrry non-negtive unction h. This clss o convex unction is commonly known s h-convex unction. She hs shown tht this clss contins severl previously known clsses o convex unctions s specil cses. Hnson [] introduced nd investigted nother clss o generlized convex unctions, which is clled invex unctions. Ben-Isrel nd Mond [5] introduced the concepts o invex sets nd preinvex unctions. They shown tht the dierentible preinvex unctions re invex unctions, but the converse my not true. These preinvex unctions re not convex unctions, but they enjoy some nice properties, which convex unctions hve. Noor [7] proved tht the minimum o the dierentible preinvex unctions on the invex set cn be chrcterized by clss o vritionl ineulities, which is clled the vritionl-like-ineulity or pre-vritionl ineulity. It is known tht the invex unctions nd preinvex unctions re euivlent under some suitble conditions, see [35, 36]. For recent pplictions o the preinvex unctions in dierent brnches o pure nd pplied sciences, see [4, 9,,, 36, 3, 49, 35, 45]. In 7, Noor [3] hs shown tht unction is preinvex unction, i nd only i, it stisies the ineulity o the type, + ηb, ηb, +ηb, xdx + b,, b [, + ηb, ], which is clled the Hermite-Hdmrd-Noor type ineulity. We remrk tht i ηb, = b, then both ineulities nd re the sme. Severl integrl ineulities or vrious types o preinvex unctions hve been obtined in recent yers, see [47, 48, 49] nd the reerences therein. Noor et. l. [49] introduced clss o preinvex unctions with respect to n rbitrry unction h. This clss o h-preinvex unctions is known s reltive preinvx unctions. They hve shown tht this clss contins severl previously known clsses o preinvex unctions nd convex unctions s specil cses. It is worth mentioning tht the weighted rithmetic men is used to deine the convex set. Relted to the rithmetic men, we hve hrmonic men, which hs pplictions in electricl circuit theory nd other brnches o sciences. It is known tht tht the totl resistnce o set o prllel resisters is obtined by dding up the reciprocl o the individul resistnce vlue nd then considering the reciprocl o their totl. For exmple, i r nd r re the resistnce

3 M.A. NOOR et l: INTEGRAL INEQUALITIES FOR DIFFERENTIABLE... 5 o two prllel resisters, then the totl resistnce R = r + = r r, r r + r which is hl o the hrmonic men. Noor [34] hs used hrmonic men to suggest some itertive methods or solving nonliner eutions. Anderson et l. [] nd Iscn [4]. hve considered nd studied some other properties o the hrmonic convex unctions. In prticulr, it hs been shown by Isn[4] tht unction is hrmonic convex, i nd only i, it stisies the ineulity o the type b + b b b b x x dx + b,, b [, b], 3 which is clled the Hermite-Hdmrd ineulity or hrmonic convex unction. Noor nd Noor [37] hve shown tht the optimlity conditions o the dierentible hrmonic convex unctions on the hrmonic convex set cn be expressed by clss o vritionl ineulities, which is clled the hrmonic vritionl ineulity. This shows tht hrmonic convex unctions hve similr properties, which convex unctions hve. This llows us to use the nlogue results o the convex unctions to suggest similr numericl methods or the hrmonic convex unctions. This is itsel n interesting problem. Noor et l. [38] introduced the clss o reltive hrmonic unctions with respect to n rbitrry nonnegtive unction h. This clss is more generl nd contins severl known clsses o hrmonic convex unctions s specil cses. For the chrcteriztion nd other spects o the reltive hrmonic convex unctions, see the reerences. We would like to emphsize tht reltive preinvex unctions nd reltive hrmonic convex unctions re two dierent extensions nd generliztions o the reltive convex unctions. These clsses re uite dierent in nture nd hve dierent pplictions. It is nturl to uniy these two clsses o reltive convex unctions. Motivted nd inspired by the ongoing reserch ctivities in this ield, Noor et l. [47, 48] introduced nd investigted new clss o convex unctions with respective to n rbitrry nonnegtive unction, which is clled reltive hrmonic preinvex unctions. It is shown tht this new clss uniies severl new clsses o hrmonic preinvex nd hrmonic convex unctions such s Breckner type o s-hrmonic preinvex unctions, Godunov- Levin type o s-hrmonic preinvex unctions nd hrmonic P -preinvex unctions. We remrk tht the reltive hrmonic prienvex unctions theory is uite brod, we shll content ourselves to give the lvor o the ides nd techniues involved. The techniues used to estblish the results re beutiul blend o ides o pure nd pplied mthemticl sciences. In this pper, we hve presented the results regrding the derivtion o integrl ineulities such s Hermite-Hdmrd, trpezoidl, Simpson s or the reltive hrmonic preinvex unctions. We hve included severl new results which we nd our coworkers hve recently obtined such s severl Hermite-Hdmrd, trpezoidl nd Simpson s type ineulities or the reltive hrmonic preinvex unctions. The rmework chosen should be seen s model setting or generl results or other clsses o convex unctions. It is true tht ech o these res o pplictions reuires specil considertion o peculirities o the clss o convex unctions t hnd nd the ineulities tht model it. However, mny o the concepts nd techniues, we hve discussed re undmentl to ll o these pplictions. Generl nd uniied rmework re importnt nd signiicnt scientiic vlue, both s mens o summriztion existing nd to provide ides nd tools or explining reltionship nd perorming investigtion. In this pper, we ocus bsiclly on presenting the stte-rt-o generlizing convexity nd invexity by mens o the hrmonic mens. We would like to mention tht the results obtined nd discussed in this pper my motivte nd bring lrge number o novel, potentil pplictions, extensions, generliztions

4 6 TWMS J. PURE APPL. MATH., V.7, N., 6 nd interesting topics in these dynmicl ields. We hve given only the bsic nd undmentl concepts in this dynmic ield.. Preliminries nd bsic results In this section, we recll the bsic concepts nd results in the convex nlysis. For more detils, see [5, 5] nd the reerences therein. Let K be set in the inite dimensions Eucliden spce R n, whose inner product nd norm re denoted by.,. nd., respectively. Deinition.. A set K in R n is sid to be convex set, i nd only i, tu + tv K, u, v K, t [, ]. Deinition.. A unction on the convex set K is sid to be convex unction, i nd only i, tu + tv tu + tv, u, v K, t [, ]. I t =, the deinition. reduces to; u + v u + v, u, v K, which is clled Jensen convex unction. It is known tht unction is convex unction on the intervl I = [, b], i nd only i, it stisies the Hermite-Hdmrd ineulity. For the dierentible convex unction, we hve the ollowing celebrted result. Theorem.. Let K be nonempty convex set in R n. nd let be dierentible convex unction on the set K. Then u K is the minimum o, i nd only i u K stisies the ineulity u, v u, v K. 4 The ineulity o the type 4 is known s vritionl ineulity, which ws introduced nd investigted by Stmpcchi [53] in 964. This shows tht the vritionl ineulities re connected with the theory o convex unctions. Vritionl ineulities cn be regrded s nturl extension o the vritionl principles, the origin o which cn be trced bck to Euler, Newton, Bernoulli brothers nd Lgrnge. Vritionl ineulities provides us uniied nd generl rmework to study wide clss o unrelted problems, which rise in pure, pplied, regionl, economics, trnsporttion, structurl nlysis, gme theory nd engineering sciences, see [3, 6, 3, 8, 9, 34, 36, 37, 3, 35, 45, 53]. We now recll the concept o reltive convex unctions with respective to n rbitrry nonnegtive unction h, which is due to [54]. Deinition.3. [54]. Let h : [, ] J R be non-negtive unction. A unction : I = [, b] R R is reltive convex unction with respect to n rbitrry unction h, i tx + ty h tx + hty, x, y I, t [, ]. It is cler tht, i ht = t, then the reltive convex unctions re exctly the convex unctions. For dierent, pproprite nd suitble choice o the rbitrry unction h, one cn obtin severl clsses o convex unctions, which re being investigted by the reserchers. We now consider n other importnt clss o convex unctions, which is known s preinvex unctions.

5 M.A. NOOR et l: INTEGRAL INEQUALITIES FOR DIFFERENTIABLE... 7 Deinition.4. []. A set K η R is sid to be invex set with respect to the biunction η,, i nd only i, x + tηy, x I, x, y K η, t [, ]. The invex set I is lso clled η-connected set. Note tht, i ηb, = b, then invex set becomes the convex set. Clelry, every convex set is n invex set, but the converse is not true. Deinition.5. [55]. Let K η be n invex set in R n. Then unction : K η R R is sid to be preinvex unction with respect to the biunction η,, i nd only i, x + tηy, x tx + ty, x, y K η, t [, ]. Noor[3] hs shown tht unction is preinvex unction, i nd only i, it stisies the Hermite-Hdmrd-Noor ineulity. For dierentible preinvex unctions, we hve the ollowing result, which is minly due to Noor [7]. Theorem.. Let K η be nonempty inves set in R n. nd let be dierentible preinvex unction on the set K η. Then u K η is the minimum o, i nd only i u K η stisies the ineulity u, ηv, u, v K η. 5 Ineulity o type 5 is clled the vritionl-like ineulity. For the pplictions nd other properties o the vritionl-like ineulities, see [7, 3, 36]. We now consider concepts o the hrmonic convex set nd hrmonic convex unctions. For more inormtion, see Anderson et l.[] nd Iscn [4]. Deinition.6. [4]. A set K h R \ {} is sid to hrmonic convex set, i nd only i, uv v + tu v K h, u, v K h, t [, ]. Deinition.7. [4]. A unction : K h R \ {} R is sid to be hrmonic convex unction, i nd only i, xy tx + ty, x, y K h, t [, ]. tx + ty Noor nd Noor [37] hve shown tht the minimum o dierentible hrmonic convex unctions on the hrmonic convex set cn be chrcterized by clss o vritionl ineulities, which is clled hrmonic vritionl ineulities. In this direction, we hve the ollowing result. Theorem.3. Let K h be nonempty hrmonic convex set in R n. nd let be dierentible hrmonic convex unction on the set K h. Then u K h is minimum o, i nd only i, u K h stisies the ineulity uv u, u v, v K h. 6 The ineulity o the type 6 is clled the hrmonic vritionl ineulity, see Noor nd Noor [34, 37]. Noor et l.[38] nd Mihi et l.[3] introduced the reltive hrmonic convex unctions with respect to n rbitrry nonnegtive unction,

6 8 TWMS J. PURE APPL. MATH., V.7, N., 6 Deinition.8. Let h : [, ] J R be non-negtive unction. unction on the hrmonic convex set K h is sid to reltive hrmonic convex unctions with respect to the nonnegtive unction h, i nd only i, xy tx + ty h tx + hty, x, y K h, t [, ]. I ht = t, then the reltive hrmonic convex unctions become hrmonic convex unctions. For pproprite choice nd suitble choice o the unction h, one cn obtin severl new clsses o hrmonic convex unctions. This shows tht the reltive hrmonic convex unctions re uite generl nd uniying ones. From the bove discussion, it ollows tht the reltive preinvex unctions nd the reltive hrmonic convex unctions re two dierent generliztion o the reltive convex unctions. Noor et l. [47, 48] introduced new clss o reltive convex unctions, which includes ll these clsses o convex unctions s specil cses. Deinition.9. [48]. A set I = [, + ηb, ] R \ {} is sid to be hrmonic invex set with respect to the biunction η,, i xx + ηy, x I, x, y I, t [, ]. x + tηy, x I ηy, x = y x, then hrmonic invex set reduces to hrmonic set. Clerly, every hrmonic set is invex set but the converse is not true. We now introduce the concept o the reltive hrmonic preinvex unctions, which re minly due to Noor et l [47, 48]. Deinition.. Let h : [, ] J R be non-negtive unction. A unction : I = [, + ηb, ] R \ {} R is reltive hrmonic preinvex unction with respect to n rbitrry nonnegtive unction h nd n rbitrry biunction η,, i xx + ηy, x h tx + hty, x, y I, t [, ]. 7 x + tηy, x Note tht or t =, we hve Jensen type reltive hrmonic preinvex unction. xx + ηy, x h [x + y], x, y I. x + ηy, x We now discuss some specil cses o Deinition., which pper to be new ones. I. I ht = t in 7, then Deinition. reduce to the deinition o hrmonic preinvex unctions. Deinition.. [48]. A unction : I R \ {} R is sid to be hrmonic preinvex unction with respect to η,, i xx + ηy, x tx + ty, x, y I, t [, ]. x + tηy, x II. I ht = t s in 7, then Deinition. reduces to the deinition o Breckner type o s-hrmonic preinvex unctions. Deinition.. A unction : I R \ {} R is sid to be s-hrmonic preinvex unction with respect to η, nd s, ], i xx + ηy, x t s x + t s y, x, y I, t [, ]. x + tηy, x

7 M.A. NOOR et l: INTEGRAL INEQUALITIES FOR DIFFERENTIABLE... 9 III. I ht = t s in 7, then Deinition. reduces to the deinition o Godunov-Levin type o s-hrmonic preinvex unctions. Deinition.3. A unction : I R \ {} R is sid to be Godunov-Levin type o s- hrmonic preinvex unction, where s, with respect to η,, i xx + ηy, x x + tηy, x t s x + y, x, y I, t,. ts It is obvious tht or s =, s-hrmonic Godunov-Levin preinvex unctions reduces to hrmonic P -preinvex unctions. I s =, s-hrmonic Godunov-Levin preinvex unctions reduces to hrmonic Godunov-Levin preinvex unctions. V. I ht = in 7, then Deinition. reduces to the deinition o hrmonic P -preinvex unctions. Deinition.4. A unction : I R \ {} R is sid to be hrmonic P-preinvex unction with respect to η,, i xx + ηy, x x + y, x, y I, t [, ]. x + tηy, x Deinition.5. [5]. Two unctions, g re sid to be similrly ordered is g-monotone, i nd only i, x y, gx gy, x, y R n. Noor et l [47, 48] hve proved tht the pproduct o two reltive hrmonic preinvex unctions is gin reltive preinvex unction. Lemm.. Let nd g be two similrly ordered reltive hrmonic preinvex unctions. ht + h t, then the product g is gin reltive hrmonic preinvex unction. I We need the ollowing ssumption bout the biunction η.,., which is due to Mohn nd Neogy[4]. This ssumption hve been used to prove the existence o solution o vritionl-like ineulities. We use to prove the let hnd o the Hermire-Hdmrd type ineulities; Condition C : Let I R be n invex set with respect to biunction η, : I I R. For ny x, y I nd ny t [, ], we hve ηy, y + tηx, y = tηx, y ηx, y + tηx, y = tηx, y. Note tht or every x, y I, t, t [, ] nd rom condition C, we hve ηy + t ηx, y, y + t ηx, y = t t ηx, y. It is remrked tht this condition is utomticlly stisied or the convex unctions. We now derive Hermite-Hdmrd ineulities or reltive hrmonic preinvex unctions. Theorem.4. Let : I = [, + ηb, ] R \ {} R be reltive hrmonic preinvex unction, where, + ηb, I with < + ηb,. I L[, + ηb, ] nd condition C holds, then + ηb, + ηb, +ηb, x h + ηb, ηb, x dx [ + b] ht. 8 Proo. Let be reltive hrmonic preinvex unction with t = in the ineulity 7. Then xx + ηy, x h [x + y], x, y I, t [, ]. x + ηy, x

8 TWMS J. PURE APPL. MATH., V.7, N., 6 Let Then, using condition C, we hve + ηb, h + ηb, x = y = + ηb, + tηb,, + ηb, + tηb,. [ ] + ηb, + ηb, +. + tηb, + tηb, Integrting both sides o the bove ineulity with respect to t over [, ], we hve + ηb, + ηb, +ηb, x h + ηb, ηb, x dx + ηb, = + tηb, which is the reuired result. [ h t + ht + ηb, ] = [ + + ηb, ] ht [ + b] ht, Now we discuss some specil cses o Theorem.4, which pper to be new ones. I. I ht = t, then Theorem.4 reduces to the ollowing result. Corollry.. [48]. Let : I = [, + ηb, ] R \ {} R be hrmonic preinvex unction. I L[, + ηb, ], then + ηb, + ηb, +ηb, x + ηb, ηb, x dx + b. II. I ht = t s, then Theorem.4 reduces to the ollowing result. Corollry.. Let : I = [, + ηb, ] R \ {} R be s-hrmonic preinvex unction. I L[, + ηb, ], then + ηb, s + ηb, +ηb, x + ηb, ηb, x dx + b. s + III. I ht = t s, then Theorem.4 reduces to the ollowing result. Corollry.3. Let : I = [, + ηb, ] R \ {} R be s-hrmonic Godunov-Levin preinvex unction. I L[, + ηb, ], then + ηb, s+ + ηb, +ηb, x + ηb, ηb, x dx + b. s IV. I ht =, then Theorem.4 reduces to the ollowing result. Corollry.4. Let : I = [, + ηb, ] R \ {} R be hrmonic P-preinvex unction. I L[, + ηb, ], then + ηb, + ηb, +ηb, x + ηb, ηb, x dx + b.

9 M.A. NOOR et l: INTEGRAL INEQUALITIES FOR DIFFERENTIABLE Min results We need the ollowing result, which plys n importnt role in the derivtion o the min results. Lemm 3.. Let : I = [, + ηb, ] R \ {} R be dierentible unction on the interior I o o I. I L[, + ηb, ] nd λ [, ], then where Proo. Let Now = I = = Υ, + ηb, ; λ + ηb, ηb, + [ t λ + ηb, + tηb, + tηb, t + λ + ηb, + tηb, + tηb, Υ, + ηb, ; λ + ηb, + + ηb, = λ + λ + ηb, + ηb, ηb, + ηb, ηb, + +ηb, x x dx. ], t λ + ηb, + tηb, + tηb, t + λ + ηb, + tηb, + tηb, + ηb, ηb, + ηb, ηb, + = I + I. + ηb, ηb, I = = + ηb, t λ + tηb, = λ + ηb, + ηb, Similrly, one cn show tht I = = λ + ηb, ηb, λ + ηb, + t λ + ηb, + tηb, + tηb, t + λ + ηb, + tηb, + tηb, ] t λ + ηb, + tηb, + tηb, + λ + ηb, + tηb, + ηb, + tηb, t + λ + ηb, + tηb, + tηb, + ηb, + ηb,. + ηb, + tηb,.

10 TWMS J. PURE APPL. MATH., V.7, N., 6 Thus + ηb, + + ηb, I + I = λ + λ + ηb, + ηb, ηb, +ηb, x x dx, which is the reuired result. Theorem 3.. Let : I = [, + ηb, ] R \ {} R be dierentible unction on the interior I o o I. I L[, + ηb, ] nd is reltive hrmonic preinvex unction on I or nd λ [, ], then Υ, + ηb, ; λ [ + ηb, ηb, κ, b; λ [κ, b; λ, h +κ 3, b; λ, h b ] + κ 4, b; λ [κ 5, b; λ, h ] +κ 6, b; λ, h b ], where one cn evlute these integrls using ny mthemticl sotwre i.e, mple. κ, b; λ = t λ, 9 + tηb, κ, b; λ, h = h t t λ, + tηb, κ 3, b; λ, h = ht t λ, + tηb, κ 4, b; λ = t + λ, + tηb, κ 5, b; λ, h = h t t + λ, 3 + tηb, κ 6, b; λ, h = ht t + λ. 4 + tηb,

11 M.A. NOOR et l: INTEGRAL INEQUALITIES FOR DIFFERENTIABLE... 3 Proo. Using Lemm 3. nd the power men ineulity, we hve Υ, + ηb, ; λ [ + ηb, ηb, + t λ + ηb, + tηb, + tηb, t + λ + ηb, + tηb, + tηb, + ηb, ηb, + [ ] t λ + tηb, t λ + ηb, + tηb, + tηb, t + λ + tηb, t + λ + ηb, + tηb, + tηb, + ηb, ηb, + [ ] t λ + tηb, t λ + tηb, [ h t + ht b ] t + λ + tηb, t + λ + tηb, [ h t + ht b ] + ηb, ηb, ] [ κ, b; λ [κ, b; λ, h + κ 3, b; λ, h b ] +κ 4, b; λ [κ 5, b; λ, h + κ 6, b; λ, h b ] ], which is the reuired result. I =, then, Theorem 3. reduces to the ollowing result, which ppers to be new one. Corollry 3.. Let : I = [, + ηb, ] R \ {} R be dierentible unction on the interior I o o I. I L[, + ηb, ] nd is reltive hrmonic preinvex unction on I nd λ [, ], then Υ, + ηb, ; λ + ηb, ηb, [ [κ, b; λ, h + κ 3, b; λ, h] +[κ 5, b; λ, h + κ 6, b; λ, h] b ], where κ, b; λ, h, κ 3, b; λ, h, κ 5, b; λ, h, κ 6, b; λ, h re given by,, 3 nd4 respectively.

12 4 TWMS J. PURE APPL. MATH., V.7, N., 6 Theorem 3.. Let : I = [, + ηb, ] R \ {} R be dierentible unction on the interior I o o I. I L[, + ηb, ] nd is reltive hrmonic preinvex unction on I or p, >, p + = nd λ [, ], then Υ, + ηb, ; λ [ + ηb, ηb, + +ηb, κ 7, b; p, λ +ηb, p ht +ηb, +κ 8, b; p, λ +ηb, + b ] p ht, 5 where κ 7, b; p, λ = κ 8, b; p, λ = t λ p, 6 + tηb, t + λ p. 7 + tηb, Proo. Using Lemm 3. nd the Holder s integrl ineulity, we hve Υ, + ηb, ; λ [ + ηb, ηb, t λ + ηb, + tηb, + tηb, t + λ + + ηb, + tηb, ] + tηb, + ηb, ηb, + = [ t + λ p + tηb, t λ p + tηb, p [ p + ηb, + tηb, t λ p + ηb, ηb, + tηb, +ηb, + ηb, +ηb, x ηb, x dx t + λ p + + tηb, p + ηb, ηb, p + ηb, + tηb, +ηb, +ηb, +ηb, ] x x ]. 8 Using the reltive hrmonic preinvexity o, we obtin the ollowing ineulities rom ineulity 8 +ηb, + ηb, +ηb, x [ ηb, x dx + + ηb, ] ht, 9 + ηb, nd + ηb, ηb, +ηb, +ηb, +ηb, x dx x [ + ηb, ] + b ht. + ηb,

13 M.A. NOOR et l: INTEGRAL INEQUALITIES FOR DIFFERENTIABLE... 5 A combintion o 8- gives the reuired ineulity 5. Theorem 3.3. Let : I = [, + ηb, ] R \ {} R be dierentible unction on the interior I o o I. I L[, + ηb, ] nd is reltive hrmonic preinvex unction on I or p, >, p + = nd λ [, ], then Υ, + ηb, ; λ + ηb, ηb, λ p+ + λ p+ p [ κ9, b;, h p + where + κ, b;, h b + κ, b;, h + κ, b;, h b ], κ 9, b;, h = h t, + tηb, κ, b;, h = ht, + tηb, κ, b;, h = h t, 3 + tηb, κ, b;, h = ht. 4 + tηb, Proo. Using Lemm 3. nd the Holder s integrl ineulity, we hve Υ, + ηb, ; λ [ + ηb, ηb, t λ +ηb, + tηb, [ + tηb, ] +ηb, ] + t + λ + tηb, [ + tηb, ] = [ +ηb, + tηb, [ + tηb, ] + ηb, ηb, p t λ p + t + λ p p +ηb, + tηb, ] [ + tηb, ] [ + ηb, ηb, t λ p p +ηb, + tηb, [ + tηb, ] p + t + λ p +ηb, ] + tηb, [ + tηb, ]

14 6 TWMS J. PURE APPL. MATH., V.7, N., 6 = + ηb, ηb, + t + λ p + ηb, ηb, [ t λ p p h t + ht b [ + tηb, ] p h t + ht b ] [ + tηb, ] λ p+ + λ p+ p + p [ κ9, b;, h +κ, b;, h b + κ, b;, h + κ, b;, h b ], which is the reuired result. Remrk 3.. For ht = t, ht = t s, ht = t s nd ht =, the clss o reltive hrmonic preinvex unctions reduces to the clss o hrmonic preinvex unctions, s-hrmonic convex unctions, s-hrmonic Godunov-Levin unctions nd hrmonic P -preinvex unctions respectively. This shows tht the clss o reltive hrmonic preinvex unctions is uite generl nd uniying one. Conseuently, results obtined in this pper continue to hold or ll these new clsses o hrmonic preinvex unctions. With the suitble nd pproprite choice vlue o λ, one cn obtin integrl ineulities or midpoint, trpezoidl, Simpson s rule nd three point trpezoidl rule, respectively. We leve this to the interested reders. Conclusion nd uture reserch In this pper, we hve presented the stte-o-the rt in convexity theory nd severl spects o reltive hrmonic preinvex unctions. These new concepts re very recent ones nd oer gret opportunities or uture reserch. It is expected tht the interply mong ll these clsses o convex unctions will certinly led to some innovtive, interesting nd signiicnt results. In this pper, our min im hve been to describe the bsic ides nd techniues, which hve used to derive the integrl ineulities, the oundtion, we hve lid is uite brod nd lexible. Quntum clculus is brnch o mthemtics nd hs mny pplictions in physics. Noor et l. [4, 4] hs estblished some untum estimtes or the convex unctions. These untum Hermite-Hdmrd ineulities nd their vrint orms re useul or untum mechnics where upper nd lower bounds or nturl phenomen described by integrls re reuently reuired. In spite o their importnce, little reserch hs been crried out in this direction. To the best o knowledge, no untum estimtes hve been derived or hrmonic convex unctions. This ield s new one nd eorts re needed to develop sound bsis or pplictions. The study o these spects o the integrl ineulities is ruitul nd growing ield o intellectul endevor. 4. Acknowledgment The uthors would like to thnk Dr. S. M. Junid Zidi, Rector, COMSATS Institute o Inormtion Technology, Pkistn, or providing excellent reserch nd cdemic environments. We re lso grteul to Pro. Dr. Fikret Aliev or his kind invittion to write this expository pper. The uthor would like to express their sincere grtitude to the reerees or their vluble nd constructive comments. Reerences [] Alomri, M., Drus, M., Drgomir, S.S., 9, New ineulities o Simpson s type or s-convex unctions with pplictions, RGMIA Res. Rep. Coll, 4.

15 M.A. NOOR et l: INTEGRAL INEQUALITIES FOR DIFFERENTIABLE... 7 [] Anderson, G.D., Vmnmurthy, M.K., Vuorinen, M., 7, Generlized convexity nd ineulities, Journl o Mthemticl Anlysis nd Applictions, 335, pp [3] Biocchi, C., Cpelo, A., 984, Vritionl nd Qusi-vritionl Ineulities, J. Wiley nd Sons, New York. [4] Brni, A., Ghznri, A.G., Drgomir, S.S.,, Hermite-Hdmrd ineulity or unctions whose derivtives bsolute vlues re preinvex, Journl o Ineulities nd Applictions, : 47. [5] Ben-Isrel, A., Mond, B., 986, Wht is invexity? Journl o Austrl. Mthemticl Society, Ser. B, 8, pp.-9. [6] Breckner, W.W., 978, Stetigkeitsussgen ur eine Klss verllgemeinerter Konvex unktionen in topologischen lineren Rumen Publictions Institutiue o Mthemtics, 3, pp.3-. [7] Drgomir, S.S., Agrwl, R.P., Cerone, P.,, On Simpson ineulity nd pplictions, Journl o Ineulities nd Applictons, 5, pp [8] Drgomir, S.S., Brnett, N.S., 998, An Ostrowski type ineulity or mppings whose second derivtive re bounded nd pplictions, RGMIA Res. Rep. Coll,, Art 9. [9] Drgomir, S.S., Peccric, J., Persson, L. E., 995, Some ineulities o Hdmrd type, Soochow Journl o Mthemtics,, pp [] Godunov, E.K., Levin, V.I., 985, Nervenstv dlj unkcii sirokogo klss soderzscego vypuklye monotonnye i nekotorye drugie vidy unkii. Vycislitel. Mt. i. Fiz. Mezvuzov. Sb. Nuk. MGPI Moskv, pp in Russin. [] Hdmrd, J., 983, Etude sur les proprietes des onctions entieres e.t en prticulier dune onction consideree pr Riemnn, J. Mth. Pures Appl., 58, pp.7-5. [] Hnson, M.A., 98, On suiciency o the Kuhn-Tucker conditions, Journl o Mthemticl Anlysis nd Applictions, 8, pp [3] Hermite, C., 883, Sur deux limites d une intgrle dinie, Mthesis, 3, pp. 8. [4] Iscn, I., 4, Hermite-Hdmrd nd Simpson like ineulities or dierentible hrmoniclly convex unctions, Journl o Mthemtics, 4: Article ID 34635, p. [5] Khttri, S.K.,, Three proos o ineulity e < + n n+.5, Americn Mthemticl Monthly, 7, pp [6] Kinderlehrer, D., Stmpcchi, G., 98, An Introduction to Vritionl Ineulities nd Applictions, Acdemic Press, London. [7] Kirmci, U.S., 4, Ineulities or dierentible mppings nd pplictions to specil mens o rel numbers to midpoint ormul, Applied Mthemtics nd Computtion, 47, pp [8] Kirmci, U.S., Bkul, M.K., Ozdemir, M.E., Pecric, J., 7, Hdmrd type ineulities or s-convex unctions, Applied Mthemtics nd Applictins, 93, pp [9] Lti, M.A., 3, Some ineulities or dierentible preusiinvex unctions with pplictions, Konurlp Journl o Mthemtics,, pp. 79. [] Lti, M.A., Drgomir, S.S., 3, Some Hermite-Hdmrd type ineulities or unctions whose prtil derivtives in bsloute vlue re preinvex on the co-oordintes, Fct Universittis NIS Ser. Mthemtics- Inormtics, 83, pp [] Lti, M.A., Drgomir, S.S., Momonit, E., 4, Some Weighted Integrl Ineulities or Dierentible Preinvex nd Preusiinvex Functions, RGMIA. [] Lions, J.L., Stmpcchi, G., 967, Vritionl ineulities, Communiction on Pure nd Applied Mthemtics,, pp [3] Mihi, M.V., Noor, M.A., Noor, K.I., Awn, M.U., 5, Some integrl ineulities or hrmoniclly h- convex unctions involving hypergeometric unctions, Applied Mthemtics nd Computtion, 5, pp DOI:.6/j.mc [4] Mohn S.R., Neogy, S.K., 995, On invex sets nd preinvex unctions, Journl o Mthemticl Anlysis nd Applictions, 89, pp [5] Niculescu, C.P., Persson, L.E., 6, Convex Functions nd Their Applictions, Springer-Verlg, New York. [6] Noor, M.A., 975, On Vritionl Ineulities, Ph.D. Thesis, Brunl University, London, United Kingdom. [7] Noor, M.A., 994, Vritionl-like ineulities, Optimiztion, 3, pp [8] Noor, M.A.,, New pproximtion schemes or generl vritionl ineulities, Journl o Mthemticl Anlysis nd Applictions, 5, pp.7-9. [9] Noor, M.A.,, Some deveoplment in generl vritionl ineulities, Applied Mthemtics nd Computtion, 5, pp

16 8 TWMS J. PURE APPL. MATH., V.7, N., 6 [3] Noor, M.A., 4, Fundmentl o mixed usi vritionl ineulities, Interntionl Journl o Pure nd Applied Mthemtics, 5, pp [3] Noor, M.A., 5, Invex euilibrium problems, Journl o Mthemticl Anlysis nd Applictions, 3, pp [3] Noor, M.A., 7, Hermite-Hdmrd integrl ineulities or log-preinvex unctions, Journl o Mthemticl Anlysis nd Approximtion Theory,, pp.6-3. [33] Noor, M.A., 9, Hdmrd integrl ineulities or product o two preinvex unctions, Nonliner Anlysis Forum, 4, pp [34] Noor, M.A., 8-6, Numericl nd Convex Anlysis, Lecture Notes, COMSATS Institute o Inormtion Technology, Islmbd, Pkistn. [35] Noor, M.A., Noor, K.I., 6, Generlized preinvex unctions nd their properties, Journl o Applied Mthemtics nd Stochstics Anlysis, 6, pp.3, doi:.55/jamsa/6/736 [36] Noor, M.A., Noor, K.I., 6, Some chrcteriztions o strongly preinvex unctions, Journl o Mthemticl Anlysis dn Applictions, 36, pp [37] Noor, M.A., Noor, K.I., 6, Hrmonic vritionl ineulities, Applied mthemtics dn Inormtion Science, 5. In Press. [38] Noor, M.A., Noor, K.I., Awn, M.U., Costche, S.,5, Some integrl ineulities or hrmoniclly h- convex unctions, University POLITEHNICA o Buchrest Scientiic BulletinA series: Applied Mthemtics nd Physics, 77, pp.5-6. [39] Noor, M.A., Noor, K.I., Awn, M.U., 5, Integrl ineulities or coordinted hrmoniclly convex unctions, Complex Vribles nd Elliptic Eutions, 66, pp [4] Noor, M.A. Noor, K.I., Awn, M.U., 5, Some untum estimtes or Hermite-Hdmrd ineulities, Applied Mthemtics nd Computtion, 5, pp [4] Noor, M.A., Noor, K.I., Awn, M.U., 5, Some untum integrl ineulities vi preinvex unctions, Applied Mthemtics nd Computtion, 69, pp [4] Noor, M.A., Noor, K.I., Awn, M.U., 4, Integrl ineulities or hrmoniclly s-godunov-levin unctions. FACTA Universittis NIS-Mthemtics-Inormtics, 94, pp [43] Noor, M.A., Noor, K.I., Awn, M.U., 5, Frctionl Ostroswki ineulities or s, m-godunov-levin Functions, FACTANIS, Mthemtics-Inormtics, 34, pp [44] Noor, M.A., Noor, K.I., Awn, M.U., 4, Some chrcteriztions o hrmoniclly log-convex unctions, Proceeding o Jngjeon Mthemticl Society, 7, pp.5-6. [45] Noor, M.A., Noor, K.I., Awn, M.U., 4, Hermite-Hdmrd ineulities or s-godunov-levin preinvex unctions, Journl o Advnced Mthemticl Studies, 7, pp.-9. [46] Noor, M.A., Noor, K.I., Itikhr, S., 5, Nonconvex unctions nd integrl ineulities, Punjb University Journl o Mthemtics, 47, pp.9-7. [47] Noor, M.A., Noor, K.I., Itikhr, S., 6, Frctionl Ostrowski ineulities or hrmonic h-preinvex unctions, FACTA Universittis NIS, Mthemtics-Inormtics, 3. [48] Noor, M.A., Noor, K.I., Itikhr, S., 6, Hermite-Hdmrd ineulities or hrmonic preinvex unctions, Sussure, 6, pp [49] Noor, M.A., Noor, K.I., Awn, M.U., Li, J., 4, On Hermite-Hdmrd ineulities or h-preinvex unctions, Filomt 87, pp [5] Pecric, J., Proschn, F., Tong, Y.L., 99, J. Pecric, F. Proschn, nd Y. L. Tong, Convex Functions, Prtil Orderings, nd Sttisticl Applictions, Acdemic Press, New York. [5] Sriky, M.Z., Sglm, A., Yildirim, H., 8, On some Hermite-type ineulities or h-convex unctions, Journl o Mthemticl Ineulities, 3, pp [5] Shi, H.N., Zhng, J., 3, Some new judgement theorems o Schur geometric nd Schur hrmonic convexities or clss o symmetric unctions, Journl o Ineulities nd Applictions, 3: 57. [53] Stmpcchi, G., 964, Formes bilineires coercivities sur les ensembles convexes, Comptes Rendus de lacdemie des Sciences, Pris, 58, pp [54] Vrosnec, S., 7, On h-convexity, Journl o Mthemticl Anlysis nd Applictions, 36, pp [55] Weir, T., Mond, B., 988, Preinvex unctions in multiobjective optimiztion, Journl o Mthemticl Anlysis nd Applictions, 36, pp [56] Yng, X.M., Li, D.,, On properties o preinvex unctions, Journl o Mthemticl Anlysis nd Applictions, 56, pp.9-4.

17 M.A. NOOR et l: INTEGRAL INEQUALITIES FOR DIFFERENTIABLE... 9 Muhmmd Aslm Noor, or photogrph nd biogrphy, see TWMS J. Pure Appl. Mth., V.4, N., 3, p.68 Khlid Inyt Noor obtined her Ph.D. rom Wles University UK. She is Proessor. She hs vst experience o teching nd reserch t university levels in vrious countries including Irn, Pkistn, Sudi Arbi, Cnd nd United Arb Emirtes. Her ield o interest nd speciliztion is Complex nlysis, Geometric unction theory, Functionl nd Convex nlysis. She hs published more thn 45 reserch rticles in reputed interntionl journls o mthemticl nd engineering sciences. Sbh Itikhr is currently Ph.D scholr t COMSATS Institute o Inormtion nd Technology, Islmbd, Pkistn. Her ield o interest is convex nlysis nd mthemticl ineulities.