RIEMANN-LIOUVILLE FRACTIONAL SIMPSON S INEQUALITIES THROUGH GENERALIZED (m, h 1, h 2 )-PREINVEXITY

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1 ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N RIEMANN-LIOUVILLE FRACTIONAL SIMPSON S INEQUALITIES THROUGH GENERALIZED m h h -PREINVEXITY Cheng Peng Chng Zhou Tingsong Du Deprtment of Mthemtics College of Science Chin Three Gorges University Yichng 443 Hubei P.R. Chin tingsongdu@ctgu.edu.cn Abstrct. The uthors first introduce the concept of generlized m h h -preinvex function. Second new Riemnn-Liouville frctionl integrl identity involving first order differentible function on n m-invex is found. By using this identity we present new Riemnn-Liouville frctionl Simpson s ineulities through generlized m h h - preinvexity. These ineulities cn be viewed s significnt generliztion of some previously known results. Keywords: Simpson s ineulity generlized m h h -preinvex functions Riemnn- Liouville frctionl integrls.. Introduction The following ineulity nmed Simpson s ineulity plys significnt role in nlysis. Theorem.. Let f : b] R be four times continuously differentible mpping on b nd f 4 := sup x b f 4 x <.Then the following ineulity holds:. f fb b f 3 88 f 4 b 4. ] b b fxdx The study of Simpson type integrl ineulities involving vrious kinds of convex functions hs been crried out by mny reserchers including Chun nd Qi 7] in the study of Simpson s ineulities for extended s-convex functions Du et l. ] in generliztion of Simpson type ineulity for extended. Corresponding uthor

2 CHENG PENG CHANG ZHOU TINGSONG DU 34 s m-convex functions Hussin nd Qisr 3] in generliztions of Simpson s ineulities through preinvexity nd Qisr et l. 8] in generliztions of Simpson s type ineulity vi m-convex functions. For more results nd recent development on the Simpson s ineulity see ] nd the references therein. In 3 Sriky et l. 34] considered the following interesting Hermite- Hdmrd-type ineulities contining Riemnn-Liouville frctionl integrls. Theorem. 34]. Let f : u v] R be positive function with u < v nd let f L u v]. Suppose f is convex function on u v] then the following ineulities for frctionl integrls hold:. u v f Γ v u J u fv J v fu] fu fv where the symbol Ju f nd J v f denote respectively the left-sided nd right-sided Riemnn-Liouville frctionl integrls of the order R defined by nd J u fx = Γ J v fx = Γ x u v x x t ftdt u < x t x ftdt x < v. Here Γ is the Gmm function nd its definition is Γ = e µ µ dµ. We observe tht for = the ineulity. cn be reduced to the following termed Hermite-Hdmrd ineulity u v f v fu fv.3 fxdx v u u where f : I R R is convex mpping on the intervl I of rel numbers nd u v I with u < v. For more recent results which generlize refine nd extend this clssic ineulity.3 One cn see contributions ] nd references therein. Recently Riemnn-Liouville frctionl Hermite-Hdmrd ineulities nd its extensively ppliction hve been ttched more nd more ttentions see ] however there re few works on the Simpson type frctionl integrls. Thus it is nturl to offer to study Simpson s ineulities involving Riemnn-Liouville frctionl integrl.

3 RIEMANN-LIOUVILLE FRACTIONAL SIMPSON S INEQUALITIES Stimulted by bove works in this pper we introduce clss of generlized m h h -preinvex functions nd derive new Riemnn-Liouville frctionl Simpson s ineulities involving the clss of functions whose first derivtives in bsolute vlues re generlized m h h -preinvex functions. Some results cn be regrded s generliztion of recent results tht ppered in Refs ].. Preliminries Definition. 43]. A set S R n is sid to be invex set with respect to the mpping η : S S R n if x tηy x S for every x y S nd t ]. The invex set S is lso termed n η-connected set. 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 tηy x m K holds for ech x y K nd ny t ]. We next explore new concept to be referred to s the generlized m h h - preinvex functions nd its vrint forms. Definition.3. Let K R be n open m-invex set with respect to η : K K ] R. A function f : K R h h : ] R if. fmx tηy x m mh tfx h tfy is vlid for ll x y K nd t ] with m ] then we sy tht fx is generlized m h h -preinvex function with respect to η. If the ineulity. reverses then f is sid to be m h h -preincve on K. Remrk.. In Definition.3 let h t = t s h t = t s nd s ] if. f mx tηy x m m t s fx t s fy is vlid for ll x y K nd t with m ] then we sy tht fx is generlized m s-godunov-levin-preinvex function with respect to η. We now discuss some specil cses of generlized m h h -preinvex function which ppers to be new ones. I If the mpping ηy x m with m = degenertes into ηy x h t = h t nd h t = ht then Definition.3 reduces to: Definition.4 ]. Let h : ] R be non-negtive function nd h. The function f on the invex set X is sid to be h-preinvex with respect to η if.3 f x tηy x h tfx htfy for ech x y X nd t ] where f >.

4 CHENG PENG CHANG ZHOU TINGSONG DU 348 II If the mpping ηy x m with m = degenertes into ηy x h t = t s nd h t = t s then Definition.3 reduces to: Definition.5 4]. Let S R n be n invex set with respect to η : S S R n. A function f : S R = is sid to be s-preinvex with respect to η nd s ] if for every x y S nd t ].4 f x tηy x t s fx t s fy. III If the mpping ηy x m with m = degenertes into ηy x h t = t s nd h t = t s then Definition.3 reduces to: Definition. 4]. A function f : K R is sid to be s-godunov-levin preinvex functions of second kind if.5 f x tηy x t s fx t s fy for ech x y K t nd s ]. IV If the mpping ηy x m with m = degenertes into ηy x h t = t nd h t = t then Definition.3 reduces to: Definition.7 4]. Let S R n be n invex set with respect to η : S S R n. A function f : S R is sid to be -preinvex with respect to η for ] if for every x y S nd t ]. f x tηy x t fx t fy. V If the mpping ηy x m with m = degenertes into ηy x h t = t nd h t = t then Definition.3 reduces to: Definition.8 43]. A function f defined on the invex set K R n is sid to be preinvex respecting η if.7 f x tηy x tfx tfy x y K t ]. VI If the mpping ηy x m = y mx with m = h t = h t nd h t = ht then Definition.3 reduces to: Definition.9 39]. Let h : J R R be positive function h. We sy tht f : I R R is h-convex if f is non-negtive nd for ll x y I nd t one hs.8 f tx ty h tfx htfy. VII If the mpping ηy x m = y mx with m = h t = t s nd h t = t s then Definition.3 reduces to:

5 RIEMANN-LIOUVILLE FRACTIONAL SIMPSON S INEQUALITIES Definition. 3]. Let s be rel number s ]. A function f : R is sid to be s-convex if.9 f tx ty t s fx t s fy. for ll x y nd t ]. VIII If the mpping ηy x m = y mx with m = h t = t s nd h t = t s then Definition.3 reduces to: Definition. ]. We sy tht the function f : K X is of s-godunov-levin type with s ] if. for ech x y X t. f tx ty t s fx t s fy IX If the mpping ηy x m = y mx with m = nd h t = h t = t t then Definition.3 reduces to: Definition. 37]. Let f : K R R be nonnegtive function we sy tht f : K R is tgs-convex function on K if the ineulity. f tx ty t tfx fy] holds for ll x y K nd t.we sy tht f is tgs-concve if f is tgs-convex. 3. Min results Let f : K R be differentible function throughout this section we will tke 3. K f ; η m b f m 4f m := Γ η b m J f mηbm Jm f m ηb m m ηb m ] f m ηb m ηb m ] where K R be n open m-invex subset with respect to η : K K ] R for some fixed m ] b K with < b > nd Γ is the Euler Gmm function. Before presenting our min results we clim the following integrl identity.

6 CHENG PENG CHANG ZHOU TINGSONG DU 35 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 ηb m >. Assume tht f : K R is first differentible function f is integrble on m mηb m] then the following identity for Riemnn-Louville frctionl integrl with > nd x m m ηb m] holds: 3. ηb m K f ; η m b = t 3 f 3 t m t f m t ] ηb m dt. ηb m Proof. By integrtion by prts we hve 3.3 f 3 t m t = ηb m 3 t t ηb m f tm = ηb m fm 3 f m Γ η b m J m f ηb m dt f tm t m t m ] ηb m ηbm m ηb m η m b m m = ηb m f m ] 3 f ηb m m m. ηb m ηb m ηb m ] dt u fudu Similrly we get 3.4 t f m t 3 t = ηb m f t m 3 ] t m ηb m ηb m dt ηb m

7 RIEMANN-LIOUVILLE FRACTIONAL SIMPSON S INEQUALITIES t ηb m f t m ηb m dt = ηb m η b m = ηb m f t m f m ηb m 3 f m mηbm m ηbm Γ η b m J mηbm f u m ηb m ] ηb m ηb m m ηb m ] 3 f ηb m m m. ηb m From 3.3 nd 3.4 we get 3.. This completes the proof. fudu Remrk 3.. If ηb m = b m with m = in Lemm 3. then the identity 3. reduces to the following identity f b 4f f b ] 3.5 Γ b J f = b 3 t t 3 ] b b Jb f t f t b ] dt f t b t which is proved by Mt lok in ]. Bsed on this identity he estblished some interesting ineulities for h-convex functions vi frctionl integrls. Remrk 3.. In Lemm 3. if the mpping ηb m with m = degenertes into ηb nd we choose = then 3. becomes f ] ηb 4f f ηb 3. ηb fxdx ηb ηb = f 4 3 t t ηb t f t ] ηb dt 3 which is proved by Wng et l. in 4]. Bsed on this identity they estblished some interesting Simpson type ineulities for s-preinvex functions.

8 CHENG PENG CHANG ZHOU TINGSONG DU If ηb = b in 3. it follows tht f 4f = b t 3 b f b ] b 3 t t f t b ] f t b t dt b fxdx which is proved by Sriky et l. in 33]. Bsed on this identity they estblished some interesting Simpson type ineulities for s-convex functions. We cn chieve the following conseuences by the bove frctionl integrl identity. Theorem 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 ηb m >. Suppose tht f : K R is first differentible function h h : ] R f is generlized m h h -preinvex on m m ηb m] then the following ineulity for Riemnn-Louville frctionl integrl with > nd x m m ηb m] holds: Kf ; η m b ηb m 3.8 m f h tdt f b h tdt. 3 Proof. From Lemm 3. f is generlized m h h -preinvex we get K f ; η m b ηb m 3 t f m t m ηb f m t ] ηb m dt ηb m 3 t t mh f t h f b ] 3.9 t mh f t h f b dt { ηb m ] = t t m h h f 3 t 3 t t h m f ηb m 3 ] t h f b dt ] h tdt f b h tdt dt

9 RIEMANN-LIOUVILLE FRACTIONAL SIMPSON S INEQUALITIES where we used the fct tht t 3 3 for ll t ]. The proof is completed. Corollry 3.. Under the ssumptions of Theorem 3. with h t = h t h t = ht if the mpping ηb m with m = degenertes into ηb then the ineulity 3.8 becomes the following frctionl ineulity for n h-preinvex function 3. f 4f ηb ] f ηb Γ η J ηb f b f f b htdt ηb 3 J f ηb ] ηb specilly for ηb = b we chieve ineulity for h-convex function f b 4f f b ] ] Γ b b 3. b J f J b f b f f b htdt 3 which is the sme s the ineulity estblished in T heorem]. Corollry 3.. In Theorem 3. when ηb m with m = reduces to ηb if s ] h t = t s h t = t s then the ineulity 3.8 becomes the following frctionl ineulity for s-preinvex function 3. f 4f ηb f Γ η J f b f f b ηb 3s ηb ] ηb J f ηb ] ηb specilly for ηb = b nd = then from the proof of Theorem 3. it follows tht the following ineulity for s-convex function holds f b 4f f b ] b fxdx b 3.3 b s 4s 5 s 3 s f s f b s s which is one of the ineulities given in 33 T heorem7].

10 CHENG PENG CHANG ZHOU TINGSONG DU 354 Corollry 3.3. In Theorem 3. if h t = t s h t = t s nd s we get frctionl ineulity for generlized m s-godunov-levin-preinvex function 3.4 Kf ; η m b ηb m m f f b ] 3 s specilly for = nd ηb m = b m with m = then from the proof of Theorem 3. it follows tht the following ineulity for s-godunov-levin functions holds 3.5 f 4f 3 b f b ] b b fxdx b s 4 s 5 s 3 s s s s f f b ]. Corollry 3.4. In Theorem 3. when ηb m with m = degenertes into ηb if h t = t h t = t for ] then the ineulity 3.8 becomes the following frctionl ineulity for -preinvex function 3. f 4f Γ η b ηb 3 ηb J f f f b ] f ηb ηb J f ηb ] ηb specilly for ηb = b nd = then from the proof of Theorem 3. it follows tht the following ineulity holds 3.7 f 4f 5b 7 b f f b. f b ] b This is one of the ineulities given in 3 Theorem 5]. b fxdx Corollry 3.5. In Theorem 3. if = h t = h t = t t nd ηb m = b m with m = then from the proof of Theorem 3. it follows tht the following ineulity for tgs-convex function holds 3.8 f b 4f f b ] b b f f b. 777 b fxdx

11 RIEMANN-LIOUVILLE FRACTIONAL SIMPSON S INEQUALITIES Theorem 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 ηb m >. Assume tht f : K R is first differentible function h h : ] R f is m h h -preinvex on m m ηb m] p = nd > then the following ineulity for Riemnn-Louville frctionl integrl with > nd x m m ηb m] holds Kf ; η m b 3.9 { ηb m t t m f h dt f b h t t ] m f h dt f b h dt. Proof. From Lemm 3. nd the Hölder ineulity we hve Kf ; η m b 3. { ηb m t p p 3 dt f m t ηb m dt 3 t p p dt f m t Becuse f is generlized m h h -preinvex we get 3. nd 3. f m t mf ηb m dt h t f m t mf dt f b ηb m dt h t dt f b ηb m dt t h dt t h dt. ] dt. Using the fct tht t 3 3 for ll t ] nd using the lst two ineulities in 3. we obtin 3.9. This completes the proof of the theorem. Corollry 3.. Under the circumstnce of Theorem 3. with h t = h t h t = ht if the mpping ηb m with m = degenertes into ηb then

12 CHENG PENG CHANG ZHOU TINGSONG DU 35 the ineulity 3.9 becomes the following ineulity for n h-preinvex function f ] ηb 4f f ηb ] Γ b b b J f J b f 3.3 ηb t t ] { f h dt f b h dt t t ] f h dt f b h dt specilly for = ηb = b then from the proof of Theorem 3. it follows tht the following ineulity for h-convex function holds f b 4f f b ] b fxdx b { b p p t f h dt 3p 3.4 t ] f b h dt t t ] f h dt f b h dt. Corollry 3.7. In Theorem 3. when ηb m with m = reduces to ηb if s ] h t = t s nd h t = t s then the ineulity 3.8 becomes the following ineulity for s-preinvex function 3.5 f 4f ηb J f Γ η b ηb s f s f ηb ] ηb J f ηb { f s f b f b s ] s ] ] ηb specilly for = then from the proof of Theorem 3. it follows tht the following ineulity holds 3. f 4f ] ηb f ηb ηb ηb fxdx

13 RIEMANN-LIOUVILLE FRACTIONAL SIMPSON S INEQUALITIES ηb 3 3 s s { f s f b ] ] s f f b ] which is the ineulities given in 4 Theorem 3.]. Specilly for ηb = b then from the proof of Theorem 3. it follows tht the following ineulity for s-convex function holds 3.7 f b 4f f b ] b fxdx b { b p s p f f b 3p s s f s f b s. s This is one of the ineulities given in 33 Theorem 9]. Corollry 3.8. In Theorem 3. if h t = t s h t = t s nd s we get frctionl ineulity for generlized m s-godunov-levin-preinvex function Kf ; η m b 3.8 s ηb m m f s f b { m f s ] s ] f b s specilly for = ηb m with m = reduces to ηb then from the proof of Theorem 3. it follows tht the following ineulity holds f ηb 4f ηb s s { s f f b ] ] f ηb ηb fxdx ηb ] f s f b ] furthermore if ηb = b then from the proof of Theorem 3. it follows tht the following ineulity for s-godunov-levin function holds 3.3 f 4f b f b ] b b fxdx

14 CHENG PENG CHANG ZHOU TINGSONG DU 358 { b p s p f f b 3p s s s f b f s. s Corollry 3.9. In Theorem 3. when ηb m with m = reduces to ηb if h t = t h t = t for ] then the ineulity 3.8 becomes the following frctionl ineulity for -preinvex function 3.3 f 4f Γ η b ηb ηb J f ] f ηb ηb J f ηb ] ηb { ] f f b ] f f b. Furthermore if ηb = b with = then from the proof of Theorem 3. it follows tht the following ineulity holds 3.3 f b 4f b p f b ] b p 3p 3 4 f 4 f b b 4 f 3 4 f b ] fxdx which is given by Sriky et l. in 3 T heorem]. Corollry 3.. In Theorem 3. if h t = t h t = t ηb m = b m with m = then the following frctionl ineulity holds 3.33 f b 4f f b ] Γ b b J f b 4 f 3 4 f b J b f ] b 3 4 f ] 4 f b.

15 RIEMANN-LIOUVILLE FRACTIONAL SIMPSON S INEQUALITIES Corollry 3.. In Theorem 3. if h t = h t = t t nd ηb m = b m with m = we obtin frctionl ineulity for tgs-convex function f b 4f f b ] ] Γ b b 3.34 b J f J b f b 3 f f b furthermore if = then from the proof of Theorem 3. it follows tht the following ineulity holds f b 4f f b ] b fxdx b 3.35 b p p f f b. 3p Theorem 3.3. 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 ηb m >. Suppose tht f : K R is first differentible function h h : ] R f is m h h -preinvex on m m ηb m] nd then the following ineulity for Riemnn-Louville frctionl integrl with > nd x m m ηb m] holds K f ; η m b ηb m ] 3 { t t ] 3.3 m f h dt f b h dt m f h t dt f b t h dt Proof. From Lemm 3. nd power men ineulity we hve Kf ; η m b 3.37 ηb m t 3 dt t 3 f m t 3 t f m t ηb m dt ηb m dt ]. ].

16 CHENG PENG CHANG ZHOU TINGSONG DU 3 By the generlized m h h -preinvex of f nd using the fct tht t 3 = 3 t 3 for ll t ] we get 3.38 nd 3.39 t 3 f m t ηb m dt mf t h dt f b 3 3 t f m t ηb m dt mf t h dt f b 3 By simple computtion 3.4 t h dt t h dt. t 3 dt = ] 3 3. Using the lst three ineulities in 3.37 we obtin 3.3. This completes the proof of the theorem. Corollry 3.. Under the ssumptions of Theorem 3.3 with h t = h t h t = ht if the mpping ηb m with m = degenertes into ηb then the ineulity 3.3 becomes the following ineulity for n h-preinvex function 3.4 f 4f ηb f ηb ] Γ η J f b ηb ] 3 { t t f h dt f b h ηb J f ηb t t f h dt f b h ] dt dt ] ] ηb specilly for ηb = b nd = we chieve ineulity for h-convex function f b 4f f b ] b 3.4 fxdx b

17 RIEMANN-LIOUVILLE FRACTIONAL SIMPSON S INEQUALITIES... 3 b 5 { t f h t f h dt f b t dt f b h t ] h dt. ] dt Corollry 3.3. In Theorem 3.3 when ηb m with m = reduces to ηb if s ] h t = t s h t = t s then the ineulity 3.3 becomes the following frctionl ineulity for s-preinvex function 3.43 f 4f ηb Γ η J f b ηb 3 { f s f s f b ] f ηb ηb J ] f b s s ] ] ηb s f ] ηb specilly for ηb = b = then from the proof of Theorem 3.3 it follows tht the following ineulity for s-convex function holds 3.44 f 4f b 5 3 b f b ] b fxdx b { 5 s s 4 s s 73 s 3 s f s s s 3 s 3 s s s f s 3 s 3 s s s f b 5s s 4 s s 73 s 3 s f b s s which is one of the ineulities proved in 33 Theorem ]. Corollry 3.4. In Theorem 3.3 if h t = t s h t = t s nd s we get frctionl ineulity for generlized m s-godunov-levin-preinvex

18 CHENG PENG CHANG ZHOU TINGSONG DU 3 function 3.45 Kf ; η m b ηb m 3 { m f ] s m f s f b f b s s ] ] s specilly for = nd ηb m = b m with m = then from the proof of Theorem 3.3 it follows tht the following ineulity for s-godunov-levin functions holds 3.4 f b 4f f b ] b fxdx b { b s s 4 s 7 s3 s 3 s f b s s s3 s 3 s s s f s3 s 3 s s s f b 5 s s 4 s 7 s3 s 3 s f s s. Corollry 3.5. In Theorem 3.3 when ηb m with m = reduces to ηb if h t = t h t = t for ] then the ineulity 3.3 becomes the following frctionl ineulity for -preinvex function 3.47 f 4f Γ η b ηb 3 ηb J f { ] f ηb ηb J f ηb ] f f b ] ] f f b ] ηb

19 RIEMANN-LIOUVILLE FRACTIONAL SIMPSON S INEQUALITIES specilly for ηb = b = then from the proof of Theorem 3.3 it follows tht the following ineulity holds f b 4f f b ] b fxdx b 3.48 { 9 f 5b f b f 9 f b which is confirmed by Sriky et l. in 3 T heorem7]. Corollry 3.. If = h t = h t = t t nd ηb m = b m with m = then from the proof of Theorem 3.3 it follows tht the following ineulity for tgs-convex function holds 3.49 f 4f 5b 3 b f b ] b f f b. 8 b fxdx Corollry 3.7. In Theorem 3.3 if h t = t h t = t nd ηb m = b m with m = then the following frctionl ineulity holds f b 4f f b ] ] Γ b b b J f J b f 3.5 b ] 3 { 4 f 3 4 f b 3 4 f ] 4 f b. Acknowledgements This work ws supported by the Ntionl Nturl Science Foundtion of Chin No References ] G.A. Anstssiou Generlised frctionl Hermite-Hdmrd ineulities involving m-convexity nd s m convexity Fct Univ. Ser. Mth. Inform ] M. U. Awn M.A. Noor M.V. Mihi K.I. Noor Frctionl Hermite- Hdmrd ineulities for differentible s-godunov-levin functions Filomt

20 CHENG PENG CHANG ZHOU TINGSONG DU 34 3] W.W. Breckner Stetigkeitsussgen für eine klsse verllgemeinerter konvexer funktionen in topologischen lineren rumen Pupl. Inst. Mth ] F.X. Chen A note on Hermite-Hdmrd ineulities for products of convex functions vi Riemnn-Liouville frctionl integrls Itl. J. Pure Appl. Mth ] F.X. Chen Extensions of the Hermite-hdmrd ineulity for convex functions vi frctionl integrls J. Mth. Ineul ] H. Chen U.N. Ktugmpol Hermite-Hdmrd nd Hermite-Hdmrd- Fejér type ineulities for generlized frctionl integrls J. Mth. Anl. Appl ] L. Chun F. Qi Ineulities of Simpson type for functions whose third derivtives re extended s-convex functions nd pplictions to mens J. Comput. Anl. Appl ] S.S. Drgomir M.I. Bhtti M. Ibl M. Muddssr Some new Hermite- Hdmrd s type frctionl integrl ineulities J. Comput. Anl. Appl ] S.S. Drgomir Ineulities of Hermite-Hdmrd type for h-convex functions on liner spces Proyecciones Antofgst ] T.S. Du J.G. Lio Y.J. Li Properties nd integrl ineulities of Hdmrd-Simpson type for the generlized s m-preinvex functions J. Nonliner Sci. Appl ] T.S. Du Y.J. Li Z.Q. Yng A generliztion of Simpson s ineulity vi differentible mpping using extended s m-convex functions Appl. Mth. Comput ] E.K. Godunov V.I. Levin Ineulities for functions of brod clss tht contins convex monotone nd some other forms of functionsrussin Numericl mthemtics nd mthemticl physicsrussin ] S. Hussin S. Qisr Generliztions of Simpson s type ineulities through preinvexity nd preusiinvexity Punjb Univ. J. Mth. Lhore ] S. Hussin S. Qisr More results on Simpson s type ineulity through convexity for twice differentible continuous mppings SpringerPlus 5 Article Number 77 9 pges.

21 RIEMANN-LIOUVILLE FRACTIONAL SIMPSON S INEQUALITIES ] M. Ibl S. Qisr M. Muddssr A short note on integrl ineulity of type Hermite-Hdmrd through convexity J. Comput. Anl. Appl ] İ. İşcn S.H. Wu Hermite-Hdmrd type ineulities for hrmoniclly convex functions vi frctionl integrls Appl. Mth. Comput ] M.A. Ltif S.S. Drgomir Generliztion of Hermite-Hdmrd type ineulities for n-times differentible functions which re s-preinvex in the second sense with pplictions Hcet. J. Mth. Stt ] Y.J. Li T.S. Du A generliztion of Simpson type ineulity vi differentible functions using extended s m ϕ -preinvex functions J. Comput. Anl. Appl ] Y.J. Li T.S. Du Some Simpson type integrl ineulities for functions whose third derivtives re m-ga-convex functions J. Egyptin Mth. Soc ] Y.J. Li T.S. Du B. Yu Some new integrl ineulities of Hdmrd- Simpson type for extended s m-preinvex functions Itl. J. Pure Appl. Mth ] M. Mt lok Ineulities for h-preinvex functions Appl. Mth. Comput ] M. Mt lok Some ineulities of Simpson type for h-convex functions vi frctionl integrls Abstr. Appl. Anl. 5 5 Article ID pges. 3] M.A. Noor K.I. Noor M.V. Mihi M.U. Awn Frctionl Hermite- Hdmrd ineulities for some clsses of differentible preinvex functions Politehn. Univ. Buchrest Sci. Bull. Ser. A Appl. Mth. Phys ] M.A. Noor K.I. Noor M.U. Awn S. Khn Hermite-Hdmrd ineulities for s-godunov-levin preinvex functions J. Adv. Mth. Stud ] M.A. Noor K.I. Noor M.U. Awn S. Costche Some integrl ineulities for hrmoniclly h-convex functions Politehn. Univ. Buchrest Sci. Bull. Ser. A Appl. Mth. Phys ] M. E. Özdemir M. Gürbüz Ç. Yildiz Ineulities for mppings whose second derivtives re usi-convex or h-convex functions Miskolc Mth. Notes

22 CHENG PENG CHANG ZHOU TINGSONG DU 3 7] M.E. Özdemir M.A. Ardic Simpson type ineulities for first order differentible preinvex nd preusiinvex functions rxiv preprint rxiv: ] S. Qisr C. He S. Hussin A generliztions of Simpson s type ineulity for differentible functions using m-convex functions nd pplictions J. Ineul. Appl. 3 3 Article Number 58 3 pges. 9] S. Qisr C.J. He On new ineulities of Hermite-Hdmrd type for generlized convex functions Itl. J. Pure Appl. Mth ] F. Qi B.Y. Xi Some integrl ineulities of Simpson type for GA-ε-convex functions Georgin Mth. J ] A. Sglm H. Yildirim M.Z. Sriky Some new ineulities of Hermite- Hdmrd s type Kyungpook Mth. J ] M.Z. Sriky E. Set M.E. Özdemir On new ineulities of Simpson s type for convex functions Comput. Mth. Appl. RGMIA Res. Rep. Coll. 3 no. rticle 8 pges. 33] M.Z. Sriky E. Set M.E. Özdemir On new ineulities of Simpson s type for s-convex functions Comput. Mth. Appl ] M.Z. Sriky E. Set H. Yldiz N. Bşk Hermite-Hdmrd s ineulities for frctionl integrls nd relted frctionl ineulities Mth. Comput. Modelling ] E. Set İ. İşcn M.Z. Sriky M.E. Özdemir On new ineulities of Hermite-Hdmrd-Fejér type for convex functions vi frctionl integrls Appl. Mth. Comput ] Y. Shung Y. Wng F. Qi Integrl ineulities of Simpson s type for m-convex functions J. Nonliner Sci. Appl ] M. Tunç E. Göv Ü. Şnl On tgs-convex function nd their ineulities Fct Univ. Ser. Mth. Inform ] M. Tunç Ç. Yildiz A. Ekinci On some ineulities of Simpson s type vi h-convex functions Hcet. J. Mth. Stt ] S. Vrošnec On h-convexity J. Mth. Anl. Appl ] Y. Wng S.H. Wng F. Qi Simpson type integrl ineulities in which the power of the bsolute vlue of the first derivtive of the integrnd is s-preinvex Fct Univ. Ser. Mth. Inform

23 RIEMANN-LIOUVILLE FRACTIONAL SIMPSON S INEQUALITIES ] Y. Wng M.M. Zheng F. Qi Integrl ineulities of Hermite-Hdmrd type for functions whose derivtives re -preinvex J. Ineul. Appl. 4 4 Article Number 97 pges. 4] J.R. Wng X.Z. Li C. Zhu Refinements of Hermite-Hdmrd type ineulities involving frctionl integrls Bull. Belg. Mth. Soc. Simon Stevin ] T. Weir B. Mond Pre-invex functions in multiple objective optimiztion J. Mth. Anl. Appl ] S.H. Wu B. Sroysng J.S. Xie Y.M. Chu Prmetrized ineulity of Hermite-Hdmrd type for functions whose third derivtive bsolute vlues re usi-convex SpringerPlus 4 5 Article Number 83 9 pges. 45] B.Y. Xi F. Qi Integrl ineulities of Simpson type for logrithmiclly convex functions Adv. Stud. Contemp. Mth. Kyungshng ] R.Y. Xing Refinements of Hermite-Hdmrd type ineulities for convex functions vi frctionl integrls J. Appl. Mth. & Informtics ] Z.Q. Yng Y.J. Li T.S. Du A generliztion of Simpson type ineulity vi differentible functions using s m-convex functions Itl. J. Pure Appl. Mth ] S. Zheng T.S. Du S.S. Zho L.Z. Chen New Hermite-Hdmrd ineulities for twice differentible ϕ-mt-preinvex functions J. Nonliner Sci. Appl Accepted:.4.7