On new Hermite-Hadamard-Fejer type inequalities for p-convex functions via fractional integrals

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1 CMMA, No., -5 7 Communiction in Mthemticl Modeling nd Applictions On new Hermite-Hdmrd-Fejer type ineulities or p-convex unctions vi rctionl integrls Mehmet Kunt nd Imdt Iscn Deprtment o Mthemtics, Fculty o Sciences, Krdeniz Technicl University, Trbzon, Turkey Deprtment o Mthemtics, Fculty o Sciences nd Arts, Giresun University, Giresun, Turkey Received: 5 September 6, Accepted: 3 December 6 Published online: 7 Jnury 7. Abstrct: In this pper, irstly, new Hermite-Hdmrd-Fejer type ineulities or p-convex unctions in rctionl integrl orms re built. Secondly, n integrl identity nd some Hermite-Hdmrd-Fejer type integrl ineulities or p-convex unctions in rctionl integrl orms re obtined. Finlly, some Hermite-Hdmrd nd Hermite-Hdmrd-Fejer ineulities or convex, hrmoniclly convex nd p-convex unctions re given. Mny results presented here or p-convex unctions provide extensions o others given in erlier works or convex, hrmoniclly convex nd p-convex unctions. Keywords: Hermite-Hdmrd ineulities, Hermite-Hdmrd-Fejer ineulities, rctionl integrl, convex unctions, hrmoniclly convex unctions, p-convex unctions. Introction Let : I R R be convex unction deined on the intervl I o rel numbers nd,b I with <b. The ineulity +b b xdx b is well known in the literture s Hermite-Hdmrd s ineulity 6,7. + b The most well-known ineulities relted to the integrl men o convex unction re the Hermite Hdmrd ineulities or its weighted versions, the so-clled Hermite-Hdmrd-Fejér ineulities. In 4, Fejér estblished the ollowing Fejér ineulity which is the weighted generliztion o Hermite-Hdmrd ineulity. Theorem. Let :, b R be convex unction. Then the ineulity +b b b wxdx xwxdx holds, where w :, b R is nonnegtive, integrble nd symmetric to + b/. + b b wxdx For some results which generlize, improve, nd extend the ineulities nd see,, 5, 9,,, 8, 9,,. Corresponding uthor e-mil: mkunt@ktu.e.tr c 7 BISKA Bilisim Technology

2 M. Kunt nd I. Iscn: On new Hermite-Hdmrd-Fejer type ineulities or p-convex unctions... We will now give deinitions o the right-hnd side nd let-hnd side Riemnn-Liouville rctionl integrls which re used throughout this pper. Deinition. 7. Let L,b. The right-hnd side nd let-hnd side Riemnn-Liouville rctionl integrls J+ α nd Jb α o order α > with b> re deined by J+ α x= x x t α tdt, x> nd Jb α Γα x= b t x α tdt, x<b Γα x respectively, where Γα is the Gmm unction deined by Γα= e t t α dt. Becuse o the wide ppliction o Hermite-Hdmrd type ineulities nd rctionl integrls, mny reserchers extend their studies to Hermite-Hdmrd type ineulities involving rctionl integrls not limited to integer integrls. Recently, more nd more Hermite-Hdmrd ineulities involving rctionl integrls hve been obtined or dierent clsses o unctions; see 3, 8, 4,, 3. In 9, İşcn gve the deinition o hrmoniclly convex unction nd estblished ollowing Hermite-Hdmrd type ineulity or hrmoniclly convex unctions s ollows. Deinition.Let I R\{} be rel intervl. A unction : I R is sid to be hrmoniclly convex, i xy tx+ ty t y+ t x 3 or ll x, y I nd t,. I the ineulity in 3 is reversed, then is sid to be hrmoniclly concve. Theorem. 9. Let : I R\{} R be hrmoniclly convex unction nd,b I with <b. I L,b then the ollowing ineulities holds: b b b x +b b x dx + b. 4 In, Chn nd Wu presented Hermite-Hdmrd-Fejér ineulity or hrmoniclly convex unctions s ollows: Theorem 3. Let : I R\{} R be hrmoniclly convex unction nd,b I with < b. I L,b nd w :,b R\{} R is nonnegtive, integrble nd hrmoniclly symmetric with respect to b +b, then b b +b wx b x dx xwx x dx + b b wx dx. 5 x In 4, Kunt et l. presented Hermite-Hdmrd nd Hermite-Hdmrd-Fejer ineulity or hrmoniclly convex unctions vi rctionl integrls s ollows: Theorem Let : I, Rbe unction such tht L,b where,b I with <b. I is hrmoniclly convex unction on, b, then the ollowing ineulities or rctionl integrls holds: b +b with α > nd gx= x, x b,. Γ α+ b α α J α +b b + g/+jα +b g/b b b + b 6 c 7 BISKA Bilisim Technology

3 CMMA, No., -5 7 / ntmsci.com/cmm 3 Theorem Let :,b R be hrmoniclly convex unction with < b nd L,b. I w :,b R is nonnegtive, integrble nd hrmoniclly symmetric with respect to b/ + b, then the ollowing ineulities or rctionl integrls holds: b Jα +b +w g/ b +b +J α +b w g/b b with α > nd gx= x, x b,. Jα +b w g/ b + +J α +b w g/b b + b Jα +b +w g/ b +J α 7 +b w g/b b In 4, Zhng nd Wn gve the deinition o p-convex unction on I R, in, İşcn gve dierent deinition o p-convex unction on I, s ollows: Deinition 3. Let I, be rel intervl nd p R\{}. A unction : I R is sid to be p-convex, i tx p + ty p /p t x+ t y 8 or ll x,y I nd t,. It cn be esily seen tht or p = nd p =, p-convexity reces to ordinry convexity nd hrmoniclly convexity o unctions deined on I,, respectively. In 5, Theorem 5, i we tke I,, p R\{} nd ht = t, then we hve the ollowing theorem. Theorem 6. Let : I, R be p-convex unction, p R\{}, nd,b I with <b. I L,b then the ollowing ineulities holds: p + b p /p p b b p p x x + b dx. 9 p In 5, Kunt nd İşcn presented Hermite-Hdmrd-Fejer ineulity or p-convex unctions s ollows. Theorem 7. Let : I, R be p-convex unction, p R\{},,b I with <b. I L,b nd w :,b R is nonnegtive, integrble nd p-symmetric with respect to p +b /p, p then the ollowing ineulities holds: p + b p /p b wx x p dx b xwx x p dx + b b wx dx. x p In 6, Kunt nd İşcn presented Hermite-Hdmrd ineulity or p-convex unctions vi rctionl integrls s ollows: Theorem 8. Let : I, R be p-convex unction, p R\{}, α > nd,b I with <b. I L,b, then the ollowing ineulities or rctionl integrls holds. i I p>, p + b p /p Γ α+ α b p p α J α + gbp +J α gp + b with gx=x /p, x p,b p, c 7 BISKA Bilisim Technology

4 4 M. Kunt nd I. Iscn: On new Hermite-Hdmrd-Fejer type ineulities or p-convex unctions... ii I p<, p + b p /p Γ α+ α p b p α J α + gp +J α gbp + b with gx=x /p, x b p, p. For some results relted to p-convex unctions nd its generliztions, we reer the reder to see 5,,,, 5, 6, 8, 9,4. In this pper, we built new Hermite-Hdmrd-Fejer type ineulities or p-convex unctions in rctionl integrl orms. We obtin n integrl identity nd some new Hermite-Hdmrd-Fejer type integrl ineulities or p-convex unctions in rctionl integrl orms. We give some new Hermite-Hdmrd nd Hermite-Hdmrd-Fejer ineulities or convex, hrmoniclly convex nd p-convex unctions. Min results Throughout this section, w = sup wt, or the continuous unction w :, b R. t,b Deinition 4. Let p R\{}. A unction w :, b, R is sid to be p-symmetric with respect to wx=w p + b p x p /p /p i holds or ll x,b. Lemm. Let p R\{}, α > nd w :,b, R is integrble, p-symmetric with respect to i I p>, J α + w gbp =J α w gp = with gx=x p, x p,b p, ii I p<, J α + w gp =J α w gbp = with gx=x p, x b p, p. J α + w gbp + J α w gp J α + w gp + J α w gbp /p, then Proo. i Let p >. Since w is p-symmetric with respect to p +b /p, p using Deinition 4 we hve w x /p = w p + b p x /p or ll x p,b p. Hence in the ollowing integrl setting t = p + b p x nd dt = dx gives J α + w gbp = Γ α = Γ α p b p x p α w b p x α w x /p dx= b p b p x α w p + b p x /p dx Γ α x /p dx=j α w gp. c 7 BISKA Bilisim Technology

5 CMMA, No., -5 7 / ntmsci.com/cmm 5 This completes the proo o i. ii The proo is similr with i. Theorem 9. Let : I, R be p-convex unction, p R\{}, α > nd,b I with <b. I L,b nd w :,b R is nonnegtive, integrble nd p-symmetric with respect to p +b /p, p then the ollowing ineulities or rctionl integrls holds. i I p>, p + b p /p J α + w gbp +J α w gp J α + w gbp +J α w gp + b J α p +b p + w gbp +J α w gp with gx=x /p, x p,b p, ii I p<, p + b p /p J α + w gp +J α w gbp J α + w gp +J α w gbp + b J α p +b p + w gp +J α w gbp 3 4 with gx=x /p, x b p, p. Proo. i Let p >. Since : I, R is p-convex unction, we hve x p +y p /p x,y I with t = in the ineulity8. x+ y or ll Choosing x=t p + tb p /p nd y=tb p + t p /p, we get p + b p /p t p + tb p /p + tb p + t p /p. 5 Multiplying both sides o5 by t α w t p + tb p /p nd integrting with respect to t over,, using Lemm -i, we get p + b p the let hnd side o3. /p J α + w gbp +J α w gp J α + w gbp +J α w gp For the proo o the second ineulity in 3 we irst note tht i is p-convex unction, then, or ll t,, it yields t p + tb p /p + tb p + t p /p + b. 6 c 7 BISKA Bilisim Technology

6 6 M. Kunt nd I. Iscn: On new Hermite-Hdmrd-Fejer type ineulities or p-convex unctions... Multiplying both sides o6 by t α w t p + tb p /p nd integrting with respect to t over,, using Lemm -i, we get J α + w gbp + J α w gp the right hnd side o3. This completes the proo o i. ii The proo is similr with i. Remrk.In Theorem 9, one cn see the ollowing. I one tkes p= nd α =, one hs, I one tkes p=, α = nd wx=, one hs, 3 I one tkes p =, one hs7, 4 I one tkes p= nd wx=, one hs6, 5 I one tkes p= nd α =, one hs5, 6 I one tkes p=, α = nd wx=, one hs4, 7 I one tkes α = nd wx=, one hs9, 8 I one tkes α =, one hs, 9 I one tkes wx=, one hs or p>, or p<. + b J α p +b p + w gbp +J α w gp Lemm. Let : I, R be dierentible unction on I nd,b I with < b, p R\{} nd α >. I L,b nd w :,b R is integrble, then the ollowing eulities or rctionl integrls holds. i I p>, p + b p /p Jα + w gbp Jα +J α + w gbp w gp +J α 7 w gp = t p ps p α w gsds g tdt Γ α b p b p t b p s α w gsds g tdt with gx=x /p, x p,b p, ii I p<, p + b p /p Jα + w gp Jα +J α + w gp w gbp +J α 8 w gbp = t b p b ps bp α w gsds g tdt Γ α p p t p s α w gsds g tdt with gx=x /p, x b p, p. Proo. i Let p >. It suices to note tht I = Γ α = Γ α Γ α p +b p p b p p b p t ps p α w gsds g tdt b p 9 t b p s α w gsds g tdt t α w gsds g tdt ps p b p b p s α w gsds g tdt = I I. t c 7 BISKA Bilisim Technology

7 CMMA, No., -5 7 / ntmsci.com/cmm 7 By integrtion by prts, we hve I = t p +b p Γ α gt α w gsds t p α w gtdt ps p p Γ α p p + b p /p p +b p = s p α w gsds t p α w gtdt Γ α p Γ α p p + b p /p = J α p +b p w gp J α w gp nd similrly I = b p Γ α gt t p + b p /p = Γ α p + b p /p = b p b p s α w gsds + Γ α b p b p b p s α w gsds+ Γ α J α + w gbp + J α + w gbp. A combintion o9, nd we hve7. This completes the proo o i. ii The proo is similr with i. b p b p t α w gtdt b p t α w gtdt Remrk.In Lemm, one cn see the ollowing. I one tkes p =, one hs, Lemm 4, I one tkes p = nd α =, one hs, Lemm., 3 I one tkes p=, α = nd wx=, one hs 3, Lemm., 4 I one tkes p=, one hs 4, Lemm 3, 5 I one tkes α = nd wx =, one hs 9, Lemm.7, 6 I one tkes α =, one hs 5, Lemm, 7 I one tkes wx =, one hs 6, Lemm. Theorem. Let : I, R be dierentible unction on I such tht L,b, where,b I nd <b. I is p-convex unction on,b or p R\{} nd α >, w :,b Ris continuous, then the ollowing ineulity or rctionl integrls holds. i I p>, p + b p /p Jα + w gbp Jα +J α + w gbp w gp +J α w gp w bp p α+ Γ α+ C α, p +C α, p b with gx=x /p, x p,b p, c 7 BISKA Bilisim Technology

8 8 M. Kunt nd I. Iscn: On new Hermite-Hdmrd-Fejer type ineulities or p-convex unctions... ii I p<, p + b p /p Jα + w gp Jα +J α + w gp w gbp +J α w gbp w p b p α+ Γ α+ C α, p C α, p b Proo. with gx=x /p, x b p, p, where C α, p= C α, p= + pu p + ub p u pu p + ub p i Let p >. Using Lemm -i, it ollows tht + /p /p + p + b p /p J α p +b p + w gbp +J α w gp p +bp t p ps p α w gsds g tdt Γ α + b p b p t b p s α w gsds g tdt t p ps p w gs ds α g t dt Γ α + b p b p t b p s w gs ds α g t dt w t α ds g t b p b p dt+ Γ α p ps p t w t p α Γ α+ p pt /p t /p bp dt+ u pu p + ub p + pu p + ub p /p, /p. J α + w gbp + J α b p s α ds g t dt b p t α pt /p t /p dt w gp. Setting t = u p + ub p nd dt = p b p gives p + b p /p Jα + w gbp Jα +J α + w gbp w gp +J α w gp w bp p α+ u p + ub p /p pu p + ub p /p Γ α+ + u p + ub p /p. pu p + ub p /p Since is p-convex unction on,b, we hve u p + ub p /p u + u b. 3 c 7 BISKA Bilisim Technology

9 CMMA, No., -5 7 / ntmsci.com/cmm 9 A combintion o nd3, we hve p + b p w bp p α+ Γ α+ = w bp p α+ Γ α+ = w bp p α+ Γ α+ /p J α + w gbp +J α w gp J α + w gbp + J α u pu p + ub p /p + u b + u pu p + ub p /p + u b + This completes the proo o i. ii The proo is similr with i. + pu p + ub p /p + u pu p + ub p /p u pu p + ub p /p + + pu p + ub p /p C α, p +C α, p b. Remrk. In Theorem, one cn see the ollowing. I one tkes p =, one hs, Theorem 6, I one tkes p=, α = nd wx=, one hs 3, Theorem., 3 I one tkes α = nd wx =, one hs 9, Teheorem 3.3, 4 I one tkes α =, one hs 5, Theorem 6, 5 I one tkes wx =, one hs 6, Theorem 5. Corollry. In Theorem, one cn see the ollowing. b w gp I one tkes p = nd wx =, one hs the ollowing Hermite-Hdmrd ineulity or convex unctions vi rctionl integrls. +b Γ α+ α b α J α +b + b+ Jα +b b + b, 4α+ I one tkes p = nd α =, one hs the ollowing Hermite-Hdmrd-Fejer ineulity or convex unctions. +b b b wxdx xwxdx w b + b, 8 3 I one tkes p =, one hs the ollowing Hermite-Hdmrd-Fejer ineulity or hrmoniclly convex unctions vi rctionl integrls. b +b w Γ α+ J α +b +w g + J α +b b w g J α +b b b b + w g b α+ C α, C α, b, b + J α +b b w g b 4 I one tkes p =, α = nd wx =, one hs the ollowing Hermite-Hdmrd ineulity or hrmoniclly convex unctions. b +b b b b x x dx b C, C, b, b c 7 BISKA Bilisim Technology

10 M. Kunt nd I. Iscn: On new Hermite-Hdmrd-Fejer type ineulities or p-convex unctions... 5 I one tkes p = nd α =, one hs the ollowing Hermite-Hdmrd-Fejer ineulity or hrmoniclly convex unctions. b b +b wx b x dx xwx x dx w b C, C, b, b 6 I one tkes p = nd wx =, one hs the ollowing Hermite-Hdmrd ineulity or hrmoniclly convex unctions vi rctionl integrls. b +b α b b Γ α+ b α α J α +b b b + g C α, C α, b. + J α +b b g b Theorem. Let : I, R be dierentible unction on I such tht L,b, where,b I nd <b. I,, is p-convex unction on,b or p R\{}, α >, w :,b R is continuous, then the ollowing ineulity or rctionl integrls holds. i I p>, p + b p /p Jα + w gbp Jα +J α + w gbp w gp +J α w gp w bp p α+ C 5 α, p C 6 α, p Γ α+ +C 7 α, p b +C 8 α, p C 9 α, p +C α, p b, with gx=x /p, x p,b p, ii I p<, p + b p /p Jα + w gp Jα +J α + w gp w gbp +J α w gbp w p b p α+ C 5 α, p Γ α+ C 6 α, p C 7 α, p b + C 8 α, p C 9 α, p C α, p b with gx=x /p, x b p, p, where C 5 α, p= C 7 α, p= C 9 α, p= pu p + ub p /p, C 6α, p= u pu p + ub p /p, C 8α, p= u pu p + ub p /p, C α, p= + pu p + ub p pu p + ub p + pu p + ub p /p, /p, /p. c 7 BISKA Bilisim Technology

11 CMMA, No., -5 7 / ntmsci.com/cmm Proo. i Let p>. Using, power men ineulity nd the p-convexity o it ollows tht p + b p /p Jα + w gbp Jα +J α + w gbp w gp +J α w gp w bp p α+ u p + ub p /p pu p + ub p /p Γ α+ + u p + ub p /p pu p + ub p /p w bp p α+ Γ α+ + pu p + ub p /p w bp p α+ Γ α+ + pu p + ub p /p pu p + ub p /p pu p + ub p pu p + ub p /p = w bp p α+ Γ α+ C 5 α, p u p + ub p /p u p + ub p /p pu p + ub p /p + /p + u pu p + ub p /p + pu p + ub p /p + pu p + ub p /p u pu p + ub p /p b b C 6 α, p +C 7 α, p b +C 8 α, p C 9 α, p +C α, p b. This completes the proo o i. ii The proo is similr with i. Remrk. In Theorem, one cn see the ollowing. I one tkes p =, one hs, Theorem 7, I one tkes α =, one hs 5, Theorem 7, 3 I one tkes wx =, one hs 6, Theorem 6. Corollry. In Theorem, one cn see the ollowing. I one tkes p =, α = nd wx =, one hs the ollowing Hermite-Hdmrd ineulity or convex unctions: +b b xdx b b C 5, C 6, +C 7, b +C 8, C 9, +C, b, I one tkes p = nd wx =, one hs the ollowing Hermite-Hdmrd ineulity or convex unctions vi rctionl integrls: +b Γ α+ α b α J α +b + b+jα +b b C α 5 α, C 6 α, +C 7 α, b +C 8 α, C 9 α, +C α, b, c 7 BISKA Bilisim Technology

12 M. Kunt nd I. Iscn: On new Hermite-Hdmrd-Fejer type ineulities or p-convex unctions... 3 I one tkes p = nd α =, one hs the ollowing Hermite-Hdmrd-Fejer ineulity or convex unctions: w b C 5, +b b b wxdx xwxdx C 6, +C 7, b +C 8, C 9, +C, b 4 I one tkes p =, one hs the ollowing Hermite-Hdmrd-Fejer ineulity or hrmoniclly convex unctions vi rctionl integrls: w Γ α+ b Jα +b +w g b +b +J α +b w g b b b α+ C 5 α, b C 6 α, C 7 α, b Jα +b + w g b +J α +b w g b b + C 8 α, C 9 α, C α, b, 5 I one tkes p =, α = nd wx =, one hs the ollowing Hermite-Hdmrd ineulity or hrmoniclly convex unctions: b b b x +b b x dx b C C 5, 6, b C 7, b + C 8, C 9, C, b, 6 I one tkes p = nd α =, one hs the ollowing Hermite-Hdmrd-Fejer ineulity or hrmoniclly convex unctions: b b wx b +b x dx xwx x dx b w C 5, b C 6, C 7, b + C 8, C 9, C, b, 7 I one tkes p = nd wx =, one hs the ollowing Hermite-Hdmrd ineulity or hrmoniclly convex unctions vi rctionl integrls. b α b b +b Γ α+ b α α b C 5 α, C 6 α, C 7 α, b J α +b b + g + J α +b b g b + C 8 α, C 9 α, C α, b, 8 I one tkes α = nd wx =, one hs the ollowing Hermite-Hdmrd ineulities or p-convex unctions. i I p>, p + b p /p p b x b p p dx x p, c 7 BISKA Bilisim Technology

13 CMMA, No., -5 7 / ntmsci.com/cmm 3 ii I p<, b p p C 5, p p b p C 5, p C 6, p +C 7, p b p + b p /p +C 8, p p b b p p x x p dx C 9, p +C, p b C 6, p C 7, p b + C 8, p C 9, p C, p b., Theorem. Let : I, R be dierentible unction on I such tht L,b, where,b I nd <b. I, >, is p-convex unction on,b or p R\{}, α >, + r =, w :,b R is continuous, then the ollowing ineulity or rctionl integrls holds. i I p>, p + b p /p Jα + w gbp Jα +J α + w gbp w gp +J α w gp w bp p α+ C α, p,r r + 3 b +Cα, p,r r 3 + b Γ α+ 8 8 where C α, p,r= pu p + ub p /p r, C α, p,r= pu p + ub p /p r, with gx=x /p, x p,b p, ii I p<, p + b p /p Jα + w gp Jα +J α + w gp w gbp +J α w gbp w p b p α+ C 3 α, p,r r + 3 b +C4α, p,r r 3 + b Γ α+ 8 8 where C 3 α, p,r= with gx=x /p, x b p, p. pu p + ub p /p r, C 4α, p,r= pu p + ub p /p r, c 7 BISKA Bilisim Technology

14 4 M. Kunt nd I. Iscn: On new Hermite-Hdmrd-Fejer type ineulities or p-convex unctions... Proo. i Let p>. Using, Hölder s ineulity nd the p-convexity o it ollows tht p + b p /p Jα + w gbp Jα +J α + w gbp w gp +J α w gp w bp p α+ u p + ub p /p pu p + ub p /p Γ α+ + u p + ub p /p pu p + ub p /p r w bp p α+ r pu p + ub p /p Γ α+ r + r pu p + ub p /p w bp p α+ Γ α+ = w bp p α+ Γ α+ = w bp p α+ Γ α+ + + u p + ub p /p u p + ub p /p r r pu p + ub p /p u + u b r r pu p + ub p /p u + u b r r +3 b pu p + ub p /p 8 r r 3 + b pu p + ub p /p b +Cα, p,r r C α, p,r r b. 8 This completes the proo o i. ii The proo is similr with i. Remrk. In Theorem, one cn see the ollowing. I one tkes p =, one hs, Theorem 8, I one tkes p=, α = nd wx=, one hs 3, Theorem.3, 3 I one tkes α =, one hs 5, Theorem 8, 4 I one tkes wx =, one hs 6, Theorem 7. Remrk.In Theorem, similrly the Corollry, i one tkes specil selections or p, α nd wx, one hs some ineulities or convex, hrmoniclly convex nd p-convex unctions. 3 Conclusion In Theorem 9, new Hermite-Hdmrd-Fejer type ineulities or p-convex unctions in rctionl integrl orms re built. In Lemm, n integrl identity nd in Theorem, Theorem nd Theorem, some new Hermite-Hdmrd-Fejer type integrl ineulities or p-convex unctions in rctionl integrl orms re obtined. In Corollry nd Corollry, some new Hermite-Hdmrd nd Hermite-Hdmrd-Fejer ineulities or convex, hrmoniclly convex nd p-convex unctions re given. Some results presented Remrk, Remrk nd Remrk, provide extensions o others given in erlier works or convex, hrmoniclly convex nd p-convex uctions. Competing interests The uthors declre tht they hve no competing interests. c 7 BISKA Bilisim Technology

15 CMMA, No., -5 7 / ntmsci.com/cmm 5 Authors contributions All uthors hve contributed to ll prts o the rticle. All uthors red nd pproved the inl mnuscript. Reerences M. Bombrdelli nd S. Vrošnec, Properties o h-convex unctions relted to the Hermite Hdmrd Fejér ineulities, Comp. Mth. Appl., 58 9, F. Chen nd S. Wu, Fejér nd Hermite-Hdmrd type ineulities or hrmoniclly convex unctions, J. Appl.Mth., volume 4, rticle id: Z. Dhmni, On Minkowski nd Hermite-Hdmrd integrl ineulities vi rctionl integrtion, Ann. Funct. Anl., L. Fejér, Uberdie Fourierreihen, II, Mth. Nturwise. Anz Ungr. Akd., Wiss, 4 96, , in Hungrin. 5 Z. B. Fng, R. Shi, On the p,h-convex unction nd some integrl ineulities, J. Ineul. Appl., , 6 pges. 6 J. Hdmrd, Étude sur les propriétés des onctions entières et en prticulier d une onction considérée pr Riemnn, J. Mth. Pures Appl., , Ch. Hermite, Sur deux limites d une intégrle déinie, Mthesis, 3 883, İ. İşcn, On generliztion o dierent type integrl ineulities or s-convex unctions vi rctionl integrls, Mth. Sci. Appl. E-Notes, 4, İ. İşcn, Hermite-Hdmrd type ineulities or hrmoniclly convex unctions, Hcet. J. Mth. Stt., , İ. İşcn, Ostrowski type ineulities or p-convex unctions, New Trends Mth. Sci., 43 6, 4-5. İ. İşcn, Hermite-Hdmrd type ineulities or p-convex unctions, Int. J. Anl. Appl., 6, İ. İşcn, Hermite-Hdmrd nd Simpson-like type ineulities or dierntible p-usi-convex unctions, doi:.34/rg , Avilble online t 3 U.S. Kırmcı, Ineulities or dierentible mppings nd pplictions to specil mens o rel numbers nd to midpoint ormul, Appl. Mth. Compt., 474, M. Kunt, İ. İşcn, N. Yzıcı, U. Gözütok, On new ineulities o Hermite-Hdmrd-Fejér type or hrmoniclly convex unctions vi rctionl integrls, Springerplus 5:635 6, M. Kunt, İ. İşcn, Hermite-Hdmrd-Fejer type ineulities or p-convex unctions, Arb J. Mth. Sci., 3 7, M. Kunt, İ. İşcn, Hermite-Hdmrd type ineulities or p-convex unctions vi rctionl integrls,moroccn J. Pure Appl. Anl., 3 7, A. A. Kilbs, H. M. Srivstv, J. J. Trujillo, Theory nd pplictions o rctionl dierentil eutions. Elsevier, Amsterdm 6. 8 M. V. Mihi, M. A. Noor, K. I. Noor, M. U. Awn, New estimtes or trpezoidl like ineulities vi dierentible p,h-convex unctions, doi:.34/rg , Avilble online t 9 M. A. Noor, K. I. Noor, M. V. Mihi, M. U. Awn, Hermite-Hdmrd ineulities or dierentible p-convex unctions using hypergeometric unctions, doi:.34/rg , Avilble online t M.Z. Srıky, On new Hermite Hdmrd Fejér type integrl ineulities, Stud. Univ. Bbeş-Bolyi Mth. 573, E. Set, İ. İşcn, M.Z. Srıky, M.E. Özdemir, On new ineulities o Hermite-Hdmrd-Fejer type or convex unctions vi rctionl integrls, Appl. Mth. Compt., K.-L. Tseng, G.-S. Yng nd K.-C. Hsu, Some ineulities or dierentible mppings nd pplictions to Fejér ineulity nd weighted trpezoidl ormul, Tiwnese journl o Mthemtics, 54, J. Wng, X. Li, M. Fečkn nd Y. Zhou, Hermite-Hdmrd-type ineulities or Riemnn-Liouville rctionl integrls vi two kinds o convexity, Appl. Anl., 9, doi:.8/ K. S. Zhng, J. P. Wn, p-convex unctions nd their properties, Pure Appl. Mth. 3 7, c 7 BISKA Bilisim Technology