Avilble online t wwwtjnscom J Nonliner Sci Appl 9 6, 3 36 Reserch Article Properties nd integrl ineulities of Hdmrd- Simpson type for the generlized s, m-preinvex functions Ting-Song Du,b,, Ji-Gen Lio, Yu-Jio Li College of Science, Chin Three Gorges University, 443, Yichng, P R Chin b Hubei Province Key Lbortory of System Science in Metllurgicl Process Wuhn University of Science nd Technology, 43, Wuhn, P R Chin Communicted by Sh Wu Abstrct The uthors introduce the concepts of m-invex set, generlized s, m-preinvex function, nd explicitly s, m-preinvex function, provide some properties for the newly introduced functions, nd estblish new Hdmrd-Simpson type integrl ineulities for function of which the power of the bsolute of the first derivtive is generlized s, m-preinvex function By tking different vlues of the prmeters, Hdmrdtype nd Simpson-type integrl ineulities cn be deduced Furthermore, ineulities obtined in specil cse present refinement nd improvement of previously known results c 6 All rights reserved Keywords: Integrl ineulities of Hdmrd-Simpson type, Hölder s ineulity, s, m-preinvex function MSC: 6D5, 6A5, 6B Introduction nd Preliminries The following nottion is used throughout this pper We use I to denote n intervl on the rel line R =,, nd I to denote the interior of I For ny subset K R n, K is used to denote the interior of K R n is used to denote generic n-dimensionl vector spce nd R n denotes n n-dimensionl nonnegtive vector spce The nonnegtive rel numbers re denoted by R =, The set of integrble functions on the intervl, b] is denoted by L, b] Let us firstly recll some definitions of vrious convex functions Corresponding uthor Emil ddresses: tingsongdu@ctgueducn Ting-Song Du, JigenLio@63com Ji-Gen Lio, yujiolictgu@63com Yu-Jio Li Received 5--9
T-S Du, J-G Lio, Y-J Li, J Nonliner Sci Appl 9 6, 3 36 33 Definition 7] A function f : I R R is sid to be Godunov-Levin function if f is nonnegtive nd for ll x, y I, λ, we hve tht f λx λy fx λ fy λ Definition 6] For some s, m, ], function f :, b] R is sid to be s, m-convex in the second sense if for every x, y, b] nd λ, ] we hve tht f λx m λy λ s fx m λ s fy Definition 3 ] A set K R n is sid to be invex with respect to the mpping η : K K R n, if x tηy, x K for every x, y K nd t, ] Notice tht every convex set is invex with respect to the mpping ηy, x = y x, but the converse is not necessrily true For more detils plese see, 33] nd the references therein Definition 4 ] Let K R n be n invex set with respect to η : K K R n, for every x, y K, the η-pth P xν joining the points x nd ν = x ηy, x is defined by P xν = { z z = x tηy, x, t, ] Definition 5 ] The function f defined on the invex set K R n is sid to be preinvex with respect to η if for every x, y K nd t, ] we hve tht f x tηy, x tfx tfy The concept of preinvexity is more generl thn convexity since every convex function is preinvex with respect to the mpping ηy, x = y x, but the converse is not true Definition 6 3] Let K R be n invex set with respect to η A function f : K R is sid to be s-preinvex with respect to η, if for ll x, y K, t, ] nd some fixed s, ] we hve tht f x tηy, x t s fx t s fy The following ineulity is remrkble in the literture s Simpson type ineulity, which plys fundmentl nd importnt role in nlysis In prticulr, it is well pplied in numericl integrtion Theorem 7 5] Let f :, b] R be four-times continuously differentible mpping on, b with f 4 = sup x,b f 4 x < Then the following ineulity holds: f fb 3 f b ] b b fxdx f 4 b 4 Now it is time to recll some ineulities of Hdmrd type nd Simpson type for the kinds of convex functions mentioned bove tht hve been developed in recent decdes Theorem 6] Let f : I R R be differentible mpping on I such tht f L, b], where, b I with < b If f is s-convex on, b], for some fixed s, ], then f fb 4f 6 b ] b b s 46s 5 s 3 s 6 s b s s fxdx ] f f b
T-S Du, J-G Lio, Y-J Li, J Nonliner Sci Appl 9 6, 3 36 34 Theorem 9 4, ] Let f : I R R be differentible mpping on I, nd let, b I with < b If f x is convex on, b], then nd f fb b b f b b b fxdx b f f b fxdx 3 b f f b 4 Theorem 5] Let f :, b] R is differentible mpping whose derivtive is continuous on, b nd f = b f x dx <, then we hve the ineulity b fxdx b f fb 3 f b ] 3 f b 5 Theorem, 5] Let K R be n open invex subset with respect to η : K K R Suppose tht f : K R is differentible function If f is preinvex on K then for every, b K with ηb, we hve tht nd f f ηb, ηb, ηb, f ηb, ηb, ηb, fxdx fxdx ηb, f f b 6 ηb, f f b 7 Theorem ] Let A R be n open invex subset with respect to η : A A R Suppose tht f : A R is differentible function If >, r, s nd f is preinvex on A, then for every, b A with η, b, we hve tht b η, b f bη,b fxdx η, b b { η, b r f r 3 f b 4 r r r s 3 f s f b ] s s s Corollry 3 ] Under the conditions of Theorem, when r = s =, the following ineulity holds b η, b f bη,b fxdx η, b b η, b 4 4 f 3 4 f b 3 4 f ] 4 f b Currently, Hdmrd-type nd Simpson-type ineulities concerning different kinds of convex functions remin ttrctive topics for mny scholrs in the field of convex nlysis For further informtion bout this topic, the reder my refer to 3,, 9,, 6, 7,, 9,,, 3, 4, 7, 9, 3, 3] nd references cited therein In the recently published rticles] by Ltif et l, bsed on the differentible α, m-preinvex functions, they estblished Hdmrd-type integrl ineulities, nd in the pper 3] Qisr et l lso found some Simpson-type ineulity for differentible α, m-convex functions Motivted by this ide nd bsed on our previous works 4, 5, 3], in the present pper, the next section we introduce new concepts, to be ]
T-S Du, J-G Lio, Y-J Li, J Nonliner Sci Appl 9 6, 3 36 35 referred s the m-invex, the generlized s, m-preinvex function, nd the explicitly s, m-preinvex function respectively, nd then we give some interesting properties for the newly introduced functions Section 3 will derive n integrl identity with two prmeters for differentible mpping, then explore new Hdmrd- Simpson-type integrl ineulities for generlized s, m-preinvex functions Some ineulities obtined in specil cse present refinement nd improvement of previously known results New definitions nd properties As one cn see, the definitions of the s, m-convex, s-preinvex, Godunov-Levin functions hve similr forms This observtion leds us to generlize these vrieties of convexity Firstly, the so-clled m-invex, my be introduced s follows Definition A set K R n is sid to be m-invex with respect to the mpping η : K K, ] R n for some fixed m, ], if mx ληy, x, m K holds for ech x, y K nd ny λ, ] Exmple Let m = 4 nd X = π/,, π/] m cosy x, if x, π/], y, π/]; m cosy x, if x π/,, y π/, ; ηy, x, m = m cosx, if x, π/], y π/, ; m cosx, if x π/,, y, π/], then X is n m-invex set with respect to η for λ, ] nd m = 4 It is obvious tht X is not convex set Remrk 3 In Definition, under certin conditions, the mpping ηy, x, m could reduce to ηy, x For exmple, in the bove Exmple, when m =, then the m-invex set degenertes n invex set on X We next give new definitions, to be referred to s generlized s, m-preinvex function nd explicitly s, m-preinvex function respectively Definition 4 Let K R n be n open m-invex set with respect to η : K K, ] R n For f : K R nd some fixed s, m, ], if f mx ληy, x, m m λ s fx λ s fy is vlid for ll x, y K, λ, ], then we sy tht fx is generlized s, m-preinvex function with respect to η The function fx is sid to be strictly generlized s, m-preinvex function on K with respect to η, if strict ineulity holds on for ny x, y K nd x y Remrk 5 In Definition 4, it is worthwhile to note tht generlized s, m-preinvex function is n s, m- convex function on K with respect to ηy, x, m = y mx Definition 6 Let K R n be n open m-invex set with respect to η : K K, ] R n For f : K R nd some fixed s, m, ], if λ,, x, y K nd fx fy, we hve f mx ληy, x, m < m λ s fx λ s fy, then we sy tht fx is n explicitly s, m-preinvex function with respect to η
T-S Du, J-G Lio, Y-J Li, J Nonliner Sci Appl 9 6, 3 36 36 Exmple 7 Let fx = x, s =, nd y mx, if x, y ; y mx, if x, y ; ηy, x, m = mx y, if x, y ; mx y, if x, y Then fx is generlized, m-preinvex function with respect to η : R R, ] R nd some fixed m, ] However, it is obvious tht fx = x is not convex function on R By letting x =, y =, λ =, we hve fx = = fy nd f mx ληy, x, m = f m η,, m = m = m λs fx λ s fy Thus, f is not lso n explicitly s, m-preinvex function on R with respect to η for s = nd some fixed m, ] According to the bove definitions, we now derive some interesting properties of the generlized s, m- preinvex function nd the explicitly s, m-preinvex function s follows The proof of propositions, 9, nd re strightforwrd Proposition If K i, i I = {,,, n is fmily of m-invex sets in R n with respect to the sme η : R n R n, ] R for sme fixed m, ], then the intersection i I X i is n m-invex set Proposition 9 If f i : K R n R i =,,, n re generlized s, m-preinvex explicitly s, m- preinvex functions with respect to the sme η : K K, ] R for sme fixed s, m, ], then the function n f = i f i, i, i =,,, n i= is lso generlized s, m-preinvex explicitly s, m-preinvex functions on K with respect to the sme η for fixed s, m, ] Proposition If f i : K R n R i =,,, n re generlized s, m-preinvex explicitly s, m- preinvex functions nd with respect to η : K K, ] R for sme fixed s, m, ], then the function f = mx{f i, i =,,, n is lso generlized s, m-preinvex explicitly s, m-preinvex function on K with respect to the η for fixed s, m, ] In Proposition we prove tht combintion of generlized s, m-preinvex function with positively homogenous nd nondecresing function is generlized s, m-preinvex with respect to η on K for fixed s, m, ] Proposition Let K be nonempty m-invex set in R n with respect to η : K K, ] R n, f : K R be generlized s, m-preinvex function with respect to η for some fixed s, m, ], nd let g : W R W R be positively homogenous nd nondecresing function, where rngf W Then the composite function gf is generlized s, m-preinvex function with respect to η on K for fixed s, m, ]
T-S Du, J-G Lio, Y-J Li, J Nonliner Sci Appl 9 6, 3 36 37 Proof Since f is generlized s, m-preinvex function, then for ll x, y K f mx ληy, x, m m λ s fx λ s fy holds for ny λ, ] Since g is positively homogenous nd nondecresing function, then g f mx ληy, x, m g m λ s fx λ s fy = m λ s g fx λ s g fy, which follows tht gf is generlized s, m-preinvex function with respect to η on K for some fixed s, m, ] Proposition If g i : R n R i =,,, n re generlized s, m-preinvex functions with respect to η for sme fixed m, s, ], then the set M = {x R n : g i x, i =,,, n is n m-invex set Proof Since g i x, i =,,, n re generlized s, m-preinvex functions, then for ll x, y R n g i mx ληy, x, m m λ s g i y λ s g i x, i =,,, n holds for ny λ, ] When x, y M, we know g i x nd g i y, from the bove ineulity, it yields tht g i mx ληy, x, m, i =,,, n Tht is, mx ληy, x, m M Hence, M is n m-invex set Proposition 3 Let f : R R is generlized s, m-preinvex function with respect to η : R R, ] R for some fixed m, s, ] Assume tht f is monotone decresing, η is monotone incresing regrding m for fixed x, y R, nd m m m, m, ] If f is generlized s, m -preinvex function on R with respect to η, then f is generlized s, m -preinvex function on R with respect to η Proof Since f is generlized s, m -preinvex function, then for ll x, y R f m x ληy, x, m m λ s fx λ s fy Combining the conditions f is monotone decresing, η is monotone incresing regrding m for fixed x, y R, nd m m, it follows tht f m x ληy, x, m f m x ληy, x, m nd m λ s fx λ s fy m λ s fx λ s fy Following the bove two ineulities, we hve tht f m x ληy, x, m m λ s fx λ s fy Hence, f is lso generlized s, m -preinvex function on R with respect to η for fixed s, ], which completes the proof Proposition 4 Let K be nonempty m-invex set in R n with respect to η : K K, ] R n, nd f i : K Ri I = {,,, n be fmily of rel-vlued functions which re explicitly s, m-preinvex functions with respect to the sme η for sme fixed s, m, ] nd bounded from bove on K Then the function fx = sup{f i x, i I is lso n explicitly s, m-preinvex function on K with respect to the sme η for fixed s, m, ]
T-S Du, J-G Lio, Y-J Li, J Nonliner Sci Appl 9 6, 3 36 3 Proof Since ech f i xi I is n explicitly s, m-preinvex function with respect to the sme η for some fixed s, m, ], we hve for ech i I f i mx ληy, x, m < m λ s f i x λ s f i y, x, y K, λ, Therefore, for ech i I, f i mx ληy, x, m < m λ s sup i I f i x λ s sup f i y, x, y K, λ, i I Tking sup of the left-hnd side of the bove eution, we obtin sup f i mx ληy, x, m < m λ s sup f i x λ s sup f i y, x, y K, λ, i I i I i I Tht is, fx = sup{f i x, i I is lso n explicitly s, m-preinvex function on K with respect to the sme η for fixed s, m, ] Proposition 5 shows tht locl minimum of n explicitly s, m-preinvex function over n m-invex set is globl one under some conditions Proposition 5 Let K be nonempty m-invex set in R n with respect to η : K K, ] R n, nd f : K R be n explicitly s, m-preinvex function with respect to η for some fixed s, m, ] And let fixed s, m, ] stisfy m λ s λ s for λ, If x K is locl minimum to the problem of minimizing fx subject to x K, then x is globl one Proof Suppose tht x K is locl minimum to the problem of minimizing fx subject to x K Then there is n ε-neighborhood N ε x round x such tht f x fx, x K N ε x 3 If x is not globl minimum of fx on K, then there exists n x K such tht fx < f x By the explicit s, m-preinvexly of fx nd the condition m λ s λ s, f m x ληx, x, m < m λ s f x λ s fx < m λ s λ s ]f x < f x for ll < λ < For sufficiently smll λ >, it follows tht m x ληx, x, m K N ε x, which is contrdiction to 3 This completes the proof By Proposition 5, we cn conclude tht explicitly s, m-preinvex functions constitute n importnt clss of generlized convex functions in mthemticl progrmming The function in Exmple 7 is not n explicitly s, m-preinvex function with respect to η bsed on Proposition 5 3 Hdmrd-Simpson type integrl ineulities For estblishing our new integrl ineulities of Hdmrd-Simpson type for generlized s, m-preinvex function, we need the following key integrl identity, which will be used in the seuel
T-S Du, J-G Lio, Y-J Li, J Nonliner Sci Appl 9 6, 3 36 39 Lemm 3 Let K R be n open m-invex subset with respect to η : K K, ] R for some fixed m, ] nd let, b K, < b with m < m ηb,, m Assume tht f : K R is differentible function, f is integrble on m, m ηb,, m], nd k, t R, then for ech x m, m ηb,, m] we hve tht tfm kf m ηb,, m k tf m = ηb,, m λ tf m ληb,, m dλ Proof Set J = ηb,, m λ tf m ληb,, m dλ ηb,, m ηb,, m mηb,,m m λ kf ] m ληb,, m dλ fxdx λ kf ] m ληb,, m dλ Since, b K nd K is n m-invex set with respect to η, for every λ, ] nd some fixed m, ], we hve m ληb,, m K Integrting by prts yields { ] J = ηb,, m λ tf m ληb,, m ηb,, m f m ληb,, m dλ ] λ kf m ληb,, m f m ληb,, m dλ ηb,, m = t ηb,, m f m tfm f m ληb,, m dλ kf m ηb,, m k ηb,, m f m f m ληb,, m dλ ηb,, m = tfm kf m ηb,, m k tf m f m ληb,, m dλ Let x = m ληb,, m, then dx = ηb,, mdλ nd we hve J = tfm kf m ηb,, m k tf m which is reuired ηb,, m ηb,, m mηb,,m m 3 fxdx, Remrk 3 clerly, if m =, ηb,, = b nd pplying t = 6, k = 5 in Lemm 3, then we obtin 6 Lemm in 3] In wht follows, we estblish nother refinement of the Simpson s ineulity for generlized s, m- preinvex functions in the second sense Theorem 33 Let A R be n open m-invex subset with respect to η : A A, ] R for some fixed m, ] nd let, b A, < b with m < m ηb,, m Suppose tht f : A R is differentible function, f is generlized s, m-preinvex function on A for some fixed s, m, ], nd let k, t R, then for ech x m, m ηb,, m] the following ineulity holds: tfm kf m ηb,, m ] ηb,, m mν f ν f b, k tf m ηb,, m ηb,, m mηb,,m m fxdx 3
T-S Du, J-G Lio, Y-J Li, J Nonliner Sci Appl 9 6, 3 36 3 where nd t s k s k ts s 3 ] ts t v = s s s t s k s s s k t ] s ks k v = s s s Proof Since m ληb,, m A for every λ, ] nd some fixed m, ], by Lemm 3 nd the generlized s, m-preinvexity of f on A, we hve tfm kf ηb,, m mηb,,m m ηb,, m k tf m fxdx ηb,, m m ηb,, m λ t f ] dλ m ληb,, m λ k f dλ m ληb,, m ηb,, m m = ηb,, m { m λ t λ s f dλ λ t λ s f b dλ λ k λ s f dλ { m λ t λ s dλ λ t λ s dλ Using the fct tht λ t λ s dλ nd λ k λ s f b dλ ] λ k λ s dλ f b λ k λ s dλ] f λ k λ s dλ t s k s k ts s 3 ] ts t = s s s λ t λ s dλ λ k λ s dλ t s k s s s k t ] s ks k = s, s s the desired ineulity 3 is estblished Direct computtion yields the following corollries Corollry 34 Under the conditions of Theorem 33, if ηb,, m = b m, m =, t = 6, nd let k = 5, we hve 6 b b 6 f fb 4f ] b s 46s 5 s 3 s 6 s b s s fxdx ] f f b ; 33
T-S Du, J-G Lio, Y-J Li, J Nonliner Sci Appl 9 6, 3 36 3 If ηb,, m = b m, s = m =, nd let t = k =, we hve f fb b b fxdx b 3 Let m =, if ηb,, degenertes ηb,, s =, nd let t = k =, we hve f f ηb, ηb, ηb, fxdx f f b 34 ηb, f f b 35 Remrk 35 Ineulity 33 is the sme s ineulity of presented by Sriky in 6] Ineulity 34 is the sme s ineulity of 3 estblished by Drgomir in 4] Ineulity 35 is the sme s ineulity of 6 given by Brni in ] Thus, ineulity 3 is generliztion of these Simpson-type nd Hdmrd-type ineulities Corollry 36 The upper bound of the midpoint ineulity for the first derivtive is developed s follows: By putting fm = f m ηb,, m = f m ηb,,m in ineulity 3, we hve f ηb,, m mηb,,m m fxdx ηb,, m ηb,, m ] mν f ν f b, 36 m where v nd v re defined in Theorem 33 If ηb,, m = b m, m = s =, t = 6, nd let k = 5 in the bove ineulity 36, it yields tht 6 b f b fxdx 5b ] b f f b 37 7 3 Let m =, if ηb,, degenertes ηb,, s =, t = 6, nd let k = 5 in the bove ineulity 36, 6 we hve ηb, f ηb, fxdx 5 ηb, ηb, f f b 3 7 Remrk 37 It is noted tht the bove midpoint ineulity 37 is better thn the ineulity 4 presented by Kirmci in ]; Apprently, the result of ineulity 3 lso hs better result compred with ineulity 7 presented by Sriky in 5] We continue with Theorem 3 Let f be defined s in Theorem 33 with p =, p > If f is generlized s, m- preinvex function on A for some fixed s, m, ] nd let k, t R, then for ech x m, m ηb,, m] the following ineulity holds: tfm kf m ηb,, m { t p ] p tp ηb,, m p p s k ] p k p p m k tf m m sf ηb,, m s f ηb,, m mηb,,m m s f b ] s f b ] fxdx 39
T-S Du, J-G Lio, Y-J Li, J Nonliner Sci Appl 9 6, 3 36 3 Proof Since m ληb,, m A for every λ, ] nd some fixed m, ], by Lemm 3 nd the fmous Hölder s integrl ineulity, we hve tfm kf ηb,, m mηb,,m m ηb,, m k tf m fxdx ηb,, m m ηb,, m λ t f ] dλ m ληb,, m λ k f dλ m ληb,, m ηb,, m { λ t p dλ λ k p dλ p p f m ληb,, m f m ληb,, m ] dλ dλ ] Also, mking use of the generlized s, m-preinvexity of f, it follows tht tfm kf ηb,, m m ηb,, m k tf m ηb,, m ηb,, m { λ t p p dλ m λ s f λ s f b ] dλ λ k p dλ Direct clcultion yields tht p m λ s f λ s f b dλ ] mηb,,m m fxdx λ t p dλ = tp t p p nd λ k p dλ = k p k p p Similrly, we hve λ s dλ = λ s dλ = s s nd λ s dλ = λ s dλ = s s Therefore, combining the bove four eulities, this leds to the desired result The sttement in Theorem 3 is proved Corollry 39 Under the condition of Theorem 3, when s =, we hve tfm kf ηb,, m p p m ηb,, m k tf m { t p ] p 3m f tp f b 4 4 k p k p ] p m f 4 3 f b 4 ηb,, m ] ] ; ηb,, m mηb,,m m fxdx 3
T-S Du, J-G Lio, Y-J Li, J Nonliner Sci Appl 9 6, 3 36 33 Let m =, if ηb,, degenertes ηb,, k =, nd let t = in ineulity 3, we cn get f ηb, ηb, fxdx ηb, p ηb, 3 p 4 4 f 4 f b 4 f 3 ] 4 f b 3 Remrk 3 By substituting p = ineulity into ineulity 3 nd exchnging nd b, we cn deduce the In the following corollry, we hve the midpoint ineulity for powers in terms of the first derivtive Corollry 3 By substituting fm = f m ηb,, m = f m ηb,,m, t = 6, nd k = 5 6 into ineulity 39, we hve mηb,,m ηb,, m m ηb,, m p p p 6 3 { 3m f f b 4 4 fxdx f m p ] p ] m f 4 ηb,, m 3 f b 4 ] 3 In the following theorem, we obtin nother form of Simpson type ineulity for powers in term of the first derivtive Theorem 3 Let f be defined s in Theorem 33 If the mpping f for is generlized s, m- preinvex on A for some fixed s, m, ] nd let k, t R, then for ech x m, m ηb,, m] the following ineulity holds: tfm kf m ηb,, m k tf m { ηb,, m t t ] mξ f ξ f b k 3 k 5 mξ 3 f ξ 4 f b ], ηb,, m ηb,, m mηb,,m m fxdx where ξ = ξ = ξ 3 = ts t s ts 4t s 3 s s t s s ts 4t s, s s k s ks 4k s 3 s s s, s, 33
T-S Du, J-G Lio, Y-J Li, J Nonliner Sci Appl 9 6, 3 36 34 nd k s s ks 4k s ks k ξ 4 = s s s Proof Since m ληb,, m A for every λ, ] nd some fixed m, ], by Lemm 3 nd power-men integrl ineulity, it follows tht tfm kf ηb,, m mηb,,m m ηb,, m k tf m fxdx ηb,, m m ηb,, m λ t f ] dλ m ληb,, m λ k f dλ m ληb,, m ηb,, m { λ k dλ λ t dλ λ t f m ληb,, m λ k f m ληb,, m ] dλ dλ ] Using the generlized s, m-preinvexity of f, we hve tht tfm kf ηb,, m m ηb,, m k tf m ηb,, m m { ηb,, m λ t dλ λ t m λ s f λ s f b dλ λ k dλ By simple clcultions, we cn get nd λ k m λ s f λ s f b dλ ] mηb,,m ] fxdx λ t dλ = t t, λ k dλ = k 3 k 5, 34 ts t s ts 4t s 3 λ t λ s dλ = s s t s s ts 4t λ t λ s dλ = s s s k s ks 4k s 3 λ k λ s dλ = s s s, 35, 36 s, 37 k s s ks 4k s ks k λ k λ s dλ = s 3 s s Thus, our desired result cn be obtined by combining eulities 34-3, nd the proof is completed
T-S Du, J-G Lio, Y-J Li, J Nonliner Sci Appl 9 6, 3 36 35 Corollry 33 Let f be defined s in Theorem 3, if s =, t = 6, nd k = 5, the ineulity holds for 6 extended m-preinvex functions: 6 fm 4f ηb,, m 5 7 m ηb,, m f m ηb,, m ] 6m 96 f 9 96 f b mηb,,m fxdx ] ηb,, m m 9m 96 f 6 96 f b 39 In prticulr, let m =, if ηb,, degenertes ηb, in ineulity 39, the ineulity holds for convex function If f x Q, x I, we cn deduce tht b f fb f b ] b fxdx 5b 3 Q 3 36 Remrk 34 It is observed tht the ineulity 3 gives n improvement for the ineulity 3 with the integrl intervl length b Thus, Theorem 3 nd its conseuences generlize the min results in 5] Acknowledgements This work ws supported by the Ntionl Nturl Science foundtion of Chin under Grnt 396, the Hubei Province Key Lbortory of Systems Science in Metllurgicl Process of Chin under Grnt Z4, nd the Nturl Science Foundtion of Hubei Province, Chin under Grnts 3CFA3 References ] T Antczk, Men vlue in invexity nlysis, Nonliner Anl, 6 5, 473 44 3,, 4 ] A Brni, A G Ghznfri, S S Drgomir, Hermite-Hdmrd ineulity for functions whose derivtives bsolute vlues re preinvex, J Ineul Appl,, 9 pges, 35 3] F Chen, S Wu, Severl complementry ineulities to ineulities of Hermite-Hdmrd type for s-convex functions, J Nonliner Sci Appl, 9 6, 75 76 4] S S Drgomir, R P Agrwl, Two ineulities for differentible mppings nd pplictions to specil mens of rel numbers nd to trpezoidl formul, Appl Mth Lett, 99, 9 95 9, 35 5] S S Drgomir, R P Agrwl, P Cerone, On Simpson s ineulity nd pplictions, J Ineul Appl, 5, 533 579 7,, 34 6] N Eftekhri, Some remrks on s, m-convexity in the second sense, J Mth Ineul, 4, 49 495 7] E K Godunov, V I Levin, Ineulities for functions of brod clss tht contins convex, monotone nd some other forms of functions, Numer Mth Mth Phys, 66 95, 3 4 ] İ İşcn, Hermite-Hdmrd s ineulities for preinvex function vi frctionl integrls nd relted functionl ineulities, Americn J Mth Anl, 3, 33 3 9] İ İşcn, S Wu, Hermite-Hdmrd type ineulities for hrmoniclly convex functions vi frctionl integrls, Appl Mth Comput, 3 4, 37 44 ] U S Kirmci, Ineulities for differentible mppings nd pplictions to specil mens of rel numbers nd to midpoint formul, Appl Mth Comput, 47 4, 37 46 9, 37 ] M A Ltif, S S Drgomir, Some weighted integrl ineulities for differentible preinvex nd preusiinvex functions with pplictions, J Ineul Appl, 3 3, 9 pges ] M A Ltif, M Shoib, Hermite-Hdmrd type integrl ineulities for differentible m-preinvex nd α, m- preinvex functions, J Egyptin Mth Soc, 3 5, 36 4 3] J Y Li, On Hdmrd-type ineulities for s-preinvex functions, J Chonging Norm Univ Nturl Science Chin, 7, 5 6 4] Y J Li, T S Du, On Simpson type ineulities for functions whose derivtives re extended s, m-ga-convex functions, Pure Appl Mth Chin, 3 5, 47 497 5] Y J Li, T S Du, Some Simpson type integrl ineulities for functions whose third derivtives re α, m-gaconvex functions, J Egyptin Mth Soc, 4 6, 75 6] T Y Li, G H Hu, On the strictly G-preinvex function, J Ineul Appl, 4 4, 9 pges 7] M Mt lok, On some Hdmrd-type ineulities for h, h -preinvex functions on the co-ordintes, J Ineul Appl, 7 3, pges
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