On new inequalities of Hermite Hadamard Fejer type for harmonically convex functions via fractional integrals

Transkript

1 Kunt et l. SrinerPlus 6 5:635 DOI.86/s RESEARCH Oen Access On new ineulities o Hermite Hdmrd Fejer tye or hrmoniclly convex unctions vi rctionl interls Mehmet Kunt *, İmdt İşcn, Nzlı Yzıcı nd Uğur Gözütok *Corresondence: mkunt@ktu.e.tr Dertment o Mthemtics, Fculty o Sciences, Krdeniz Technicl University, Trzon, Turkey Full list o uthor inormtion is ville t the end o the rticle Astrct In this er, irstly, new Hermite Hdmrd tye ineulities or hrmoniclly convex unctions in rctionl interl orms re iven. Secondly, Hermite Hdmrd Fejer ineulities or hrmoniclly convex unctions in rctionl interl orms re uilt. Finlly, n interl identity nd some Hermite Hdmrd Fejer tye interl ineulities or hrmoniclly convex unctions in rctionl interl orms re otined. Some results resented here rovide extensions o others iven in erlier works. Keywords: Hermite Hdmrd ineulity, Hermite Hdmrd Fejer ineulity, Riemnn Liouville rctionl interls, Hrmoniclly convex unctions Mthemtics Suject Clssiiction: 6A5, 6A33, 6D Bckround Let : I R R e convex unction deined on the intervl I o rel numers nd, I. The ineulity + xdx + is well known in the literture s Hermite Hdmrd s ineulity Hdmrd 893; Hermite 883. The most well-known ineulities relted to the interl men o convex unction re the Hermite Hdmrd ineulities or their weihted versions, the so-clled Hermite Hdmrd-Fejér ineulities. Fejér 96 estlished the ollowin Fejér ineulity which is the weihted enerliztion o Hermite Hdmrd ineulity : Theorem Let :, R e convex unction. Then the ineulity + xdx xxdx + xdx holds, where :, R is nonnetive, interle nd symmetric to + /. 6 The Authors. This rticle is distriuted under the terms o the Cretive Commons Attriution 4. Interntionl License htt://cretivecommons.or/licenses/y/4./, which ermits unrestricted use, distriution, nd reroction in ny medium, rovided you ive rorite credit to the oriinl uthors nd the source, rovide link to the Cretive Commons license, nd indicte i chnes were mde.

2 Kunt et l. SrinerPlus 6 5:635 Pe o 9 For some results which enerlize, imrove nd extend the ineulities nd see Bomrdelli nd Vrošnec 869, İşcn 3, 4c, Minculete nd Mitroi, Srıky, Tsen et l.. We recll the ollowin ineulity nd secil unctions which re known s Bet nd hyereometric unction resectively: β x, y Γ xγ y Γ x + y t x t y dt, x, y >, F, ; c; z t t c zt dt, β, c c > >, z < see Kils et l. 6. Lemm Prudnikov et l. 98; Wn et l. 3 For < nd < we hve. The ollowin deinitions nd mthemticl reliminries o rctionl clculus theory re used urther in this er. Deinition Kils et l. 6 Let L,. The Riemnn Liouville interls J+ nd J o oder > with re deined y nd J + x Γ x x t tdt, x > J x t x tdt, x < Γ x resectively, where Γ is the Gmm unction deined y Γ e t t dt nd J+ x J x x. Becuse o the wide liction o Hermite Hdmrd tye ineulities nd rctionl interls, mny reserchers extend their studies to Hermite Hdmrd tye ineulities involvin rctionl interls not limited to inteer interls. Recently, more nd more Hermite Hdmrd ineulities involvin rctionl interls hve een otined or dierent clsses o unctions; see Dhmni, İşcn 3, 4, İşcn nd Wu 4, Mihi nd Ion 4, Srıky et l. 3, Wn et l., Wn et l. 3. İşcn 4 cn deined the so-clled hrmoniclly convex unctions nd estlished ollowin Hermite Hdmrd tye ineulity or them s ollows: Deinition Let I R\{} e rel intervl. A unction : I R is sid to e hrmoniclly convex, i xy t y + t x tx + ty 3

3 Kunt et l. SrinerPlus 6 5:635 Pe 3 o 9 or ll x, y I nd t,. I the ineulity in 3 is reversed, then is sid to e hrmoniclly concve. Theorem İşcn 4 Let : I R\{} R e hrmoniclly convex unction nd, I. I L, then the ollowin ineulities holds: x 4 + x dx +. Lti et l. 5 ve the ollowin deinition: Deinition 3 A unction :, R\{} R is sid to e hrmoniclly symmetric with resect to / + i x + x holds or ll x,. Chen nd Wu 4 resented Hermite Hdmrd Fejer tye ineulity or hrmoniclly convex unctions s ollows: Theorem 3 Let : I R\{} R e hrmoniclly convex unction nd, I. I L, nd :, R\{} R is nonnetive, interle nd hrmoniclly symmetric with resect to +, then x + x dx xx x dx + In this er, we ive new Hermite Hdmrd tye ineulities or hrmoniclly convex unctions in rctionl interl orms. We estlish new Hermite Hdmrd Fejer ineulities or hrmoniclly convex unctions in rctionl interl orms. We otin n interl identity nd some Hermite Hdmrd Fejer tye interl ineulities or hrmoniclly convex unctions in rctionl interl orms. Min results Throuhout this section, we write su t, or the continuous unction t, :, R. Lemm I :, R\{} R is interle nd hrmoniclly symmetric with resect to +, then / J + + / J + x x dx. J + + / +J + / 5

4 Kunt et l. SrinerPlus 6 5:635 Pe 4 o 9 with > nd hx /x, x,. Proo Since is hrmoniclly symmetric with resect to +, usin Deinition 3 we hve x, or ll x + x,. Hence, in the ollowin interl settin t + x nd dt dx ives J + / + Γ + This comletes the roo. Γ Γ Theorem 4 Let : I, R e unction such tht L,, where, I. I is hrmoniclly convex unction on,, then the ollowin ineulities or rctionl interls holds: + with > nd hx /x, x + + Γ + +,. t t x x x dt J + h / + +J + h / dx / + / x dx J + /. 6 Proo Since is hrmoniclly convex unction on,, we hve or ll t, t+t t+t + t+t + t+t + t+t t+t. 7 Multilyin oth sides o 7 y t, then intertin the resultin ineulity with resect to t over, we otin, t dt + t t + t t t + t + dt + t + t t dt t + t dt.

5 Kunt et l. SrinerPlus 6 5:635 Pe 5 o 9 Settin x t+t nd dx + dt ives nd the irst ineulity is roved. For the roo o the second ineulity in 6, we irst note tht, i is hrmoniclly convex unction, then, or ll t,, it yields Then multilyin oth sides o 8 y t nd intertin the resultin ineulity with resect to t over, we otin t + t, Γ J + t + t x dx + x x x dx x x dx x x dx h / + J + + h / +. t dt + t t + t + t dt + t + t dt 8 i.e. Γ J + + h / + J +. + h / The roo is comleted. Theorem 5 Let :, R e hrmoniclly convex unction with < nd L,. I :, R is nonnetive, interle nd hrmoniclly symmetric with resect to +, then the ollowin ineulities or rctionl interls holds: + J + J + + / + J with > nd hx /x, x / + J +,. + / / + J + + / + J + + / 9

6 Kunt et l. SrinerPlus 6 5:635 Pe 6 o 9 Proo Since is hrmoniclly convex unction on,, multilyin oth sides o 7 y t t+t, then intertin the resultin ineulity with resect to t over,, we otin t dt + t + t t + t + t t + t t dt t + t t + t + t t + t t + t Since is hrmoniclly symmetric with resect to +, usin Deinition 3 we hve x, or ll x + x,. Settin x t+t nd dx dt ives + Thereore, y Lemm we hve Γ J + + Γ J + nd the irst ineulity is roved. For the roo o the second ineulity in 9 we irst note tht i is hrmoniclly convex unction, then, multilyin oth sides o 8 y t t+t nd intertin the resultin ineulity with resect to t over, we otin, dt. + x dx x + x x dx + x + + x x x dx + x x dx + x + + x x x dx + x x x dx + + x x x dx. + / + J + / + J t + t + / + / dt

7 Kunt et l. SrinerPlus 6 5:635 Pe 7 o 9 t dt t + t t + t + t dt t + t t + t + t dt t + t i.e. Γ J + Γ / + J + + / + J + / + J + + /. The roo is comleted. Remrk In Theorem 5, i i we tke, then ineulity 9 ecomes ineulity 5 o Theorem 3. ii i we tke x, then ineulity 9 ecomes ineulity 6 o Theorem 4. iii i we tke nd x, then ineulity 9 ecomes ineulity 4 o Theorem. Lemm 3 Let : I, R e dierentile unction on I, the interior o I, such tht L,, where, I. I :, R is interle nd hrmoniclly symmetric with resect to +, then the ollowin eulity or rctionl interls holds: J + + J + / + J + Γ + / + J + + with > nd hx /x, x Proo It suices to note tht + t I + I I. t,. + / + / t s tdt h sds h tdt s h sds h t s sds h tdt s tdt sds h

8 Kunt et l. SrinerPlus 6 5:635 Pe 8 o 9 By intertion y rts nd Lemm we et t I s sds h t nd similrly Γ + Γ I t + + t t h tdt s sds t t h tdt + J + J + / J + / + J J + / s sds h t t t h tdt + s t sds t h tdt Multilyin oth sides y Γ we otin. This comletes the roo. + + / + / + + Γ + J + / + J + + / + Γ + J + / + J + + /. +J + + / Thus, we cn write + J + / + J I I I Γ + + / J. + / + J + + /

9 Kunt et l. SrinerPlus 6 5:635 Pe 9 o 9 Theorem 6 Let : I, R e dierentile unction on I, the interior o I, such tht L,, where, I nd <. I is hrmoniclly convex on,, :, R is continuous nd hrmoniclly symmetric with resect to +, then the ollowin ineulity or rctionl interls holds: where + J + / + J + + / J + / + J + + / C + C Γ + C ++ F, + ; + 3;, + ++ F, + ; + 3; + + F, + ; + 3; C + + F, + ; + ; +, F, + ; + 3; with < nd hx /x, x,. + Proo From Lemm 3 we hve + J + / + J + + / J + / + J + + / + t s s t dt ds h Γ + + t s s t dt ds h + t ds s t dt h Γ + ds t dt + t s h + t dt t t Γ. t + + dt t t

10 Kunt et l. SrinerPlus 6 5:635 Pe o 9 Settin t u+u nd dt ives + J + / + J + + / J + / + J + + / u u+u u+u Γ + + u. u+u u+u Since is hrmoniclly convex on,, we hve u + u. u + ut I we use 3 in, we hve + J + / + J + + / J + / + J + + / Γ + u u + u u+u + u u + u. u+u 3 4 Clcultin the ollowin interls y Lemm, we hve u + u + u + u u u + u u u u + u u u u u u + u u + u 4 uu u u u u + u u u u + u u u u + u u + vv v dv F, + ; + 3; + ++ F, + ; + 3; + C 5

11 Kunt et l. SrinerPlus 6 5:635 Pe o 9 nd similrly we et u u + u + u + u + u + u u + u + u u + u + u u u u + u u + u u u + u u u u u + u u u u + u u u + u u u + u u + u + u u 4 u + u u + u + u + v v dv vv v dv F, + ; + 3; + + F, + ; + ; F, + ; + 3; C. + 6 I we use 5 nd 6 in 4, we hve. This comletes the roo. Corollry In Theorem 6: I we tke we hve the ollowin Hermite Hdmrd Fejer ineulity or hrmoniclly convex unctions which is relted to the let-hnd side o 5: x + x dx xx x dx C + C, I we tke x we hve ollowin Hermite Hdmrd tye ineulity or hrmoniclly convex unctions in rctionl interl orms which is relted to the let-hnd side o 6: Γ + J + h / + + +J + h / C + C,

12 Kunt et l. SrinerPlus 6 5:635 Pe o 9 3 I we tke nd x we hve the ollowin Hermite Hdmrd tye ineulity or hrmoniclly convex unctions which is relted to the let-hnd side o 4: + x x dx C + C. Theorem 7 Let : I, R e dierentile unction on I, the interior o I, such tht L,, where, I. I,, is hrmoniclly convex on,, :, R is continuous nd hrmoniclly symmetric with resect to +, then the ollowin ineulity or rctionl interls holds: / + J / / J/+ / + J / / Γ + C C4 3 +C 5 +C C7 6 +C 8 + J/+ 7 where + C 3 + F, ; + ; +, + C 4 + F, ; + 3; +, C 5 C 3 C 4, C F, + ; + ;, C F, + ; + ; + + F, + ; + 3;, C 8 C 6 C 7, with > nd hx /x, x,. Proo Usin, ower men ineulity nd the hrmoniclly convexity o, it ollows tht

13 Kunt et l. SrinerPlus 6 5:635 Pe 3 o 9 + J + / + J + + / J + / + J + + / u u+u u+u Γ + + u u+u u+u Γ + u u+u u u+u u+u u + u+u u u+u u+u Γ + u u+u u u + u u+u u + u+u u u + u u+u Γ + u u+u u + u+u + u u u+u. u + u+u u u u+u + u + u+u 8 For the erin interls, we hve u u + u + + u u + u + v v + + F, ; + ; + C 3, + 9

14 Kunt et l. SrinerPlus 6 5:635 Pe 4 o 9 u + u + u + + u + u + u + v + v + + F, ; + 3; + C 4, + u u + u u C 3 C 4 C 5, u u + u + + u u + u u u + u u u + + F, + ; + ; C 6, u u + u u + u u u + u u u + u + + u u + u u + u + u u + u + u u u + u + u + + F, + ; + ; + + F, + ; + 3; C 7, 3 u + u + u u u + u C 6 C 7 C 8. u u + u u 4 I we use 9 4 in 8, we hve 7. This comletes the roo.

15 Kunt et l. SrinerPlus 6 5:635 Pe 5 o 9 Corollry In Theorem 7: I we tke we hve the ollowin Hermite Hdmrd Fejer ineulity or hrmoniclly convex unctions which is relted to the let-hnd side o 5: + x x dx C 3 +C 6 xx x dx C4 +C 5 C7 +C 8 I we tke x we hve the ollowin Hermite Hdmrd tye ineulity or hrmoniclly convex unctions in rctionl interl orms which is relted to the let-hnd side o 6: Γ + J + h / + + +J + h / C C4 3 +C 5 +C C7 6 +C 8, 3 I we tke nd x we hve the ollowin Hermite Hdmrd tye ineulity or hrmoniclly convex unctions which is relted to the let-hnd side o 4: x + x dx C C4 3 +C 5 +C C7 6 +C 8. We cn stte nother ineulity or > s ollows: Theorem 8 Let : I, R e dierentile unction on I, the interior o I, such tht L,, where, I. I, >, is hrmoniclly convex on,, :, R is continuous nd hrmoniclly symmetric with resect to +, then the ollowin ineulity or rctionl interls holds: + J /+ / + J / / J/+ / + J / / Γ + 5 C C 3 + 8,

16 Kunt et l. SrinerPlus 6 5:635 Pe 6 o 9 where C 9 C F with >, hx /x, x,; + ;, nd / + /. +, + + F, + ; + ;, Proo tht Usin, Hölder s ineulity nd the hrmoniclly convexity o, it ollows + J + / + J + + / J + / + J + + / u u+u u+u Γ + + u u+u u+u u u+u Γ + u+u u + u+u u+u u u+u Γ + u + u u + u+u u + u Γ + u + 3 u + u 8 u u + u 8 6

17 Kunt et l. SrinerPlus 6 5:635 Pe 7 o 9 For the erin interls, we hve u u + u + + u u + u + v v F C 9. dv +,; + ; + 7 Similrly, we hve u u + u + u u + u u u + u + u u + + F, + ; + ; C. 8 I we use 7 nd 8 in 6, we hve 5. This comletes the roo. Corollry 3 In Theorem 8: I we tke we hve the ollowin Hermite Hdmrd Fejer ineulity or hrmoniclly convex unctions which is relted to the let-hnd side o 5: x + x dx xx x dx C C I we tke x we hve ollowin Hermite Hdmrd tye ineulity or hrmoniclly convex unctions in rctionl interl orms which is relted to the let-hnd side o 6: + Γ + C C 3 + 8, J + h / + +J + h /,

18 Kunt et l. SrinerPlus 6 5:635 Pe 8 o 9 3 I we tke nd x we hve the ollowin Hermite Hdmrd tye ineulity or hrmoniclly convex unctions which is relted to the let-hnd side o 4: x + x dx C C Conclusion In this er, new Hermite Hdmrd tye ineulities or hrmoniclly convex unctions in rctionl interl orms re iven nd Hermite Hdmrd Fejer ineulities or hrmoniclly convex unctions in rctionl interl orms re uilt. Also, n interl identity nd some Hermite Hdmrd Fejer tye interl ineulities or hrmoniclly convex unctions in rctionl interl orms re otined.. Authors contriutions MK, İİ, NY, UG contriuted eully to the writin o this er. All uthors red nd roved the inl mnuscrit. Author detils Dertment o Mthemtics, Fculty o Sciences, Krdeniz Technicl University, Trzon, Turkey. Dertment o Mthemtics, Fculty o Sciences nd Arts, Giresun University, Giresun, Turkey. Acknowledements The uthors re very rteul to the reerees or helul comments nd vlule suestions. Also, Kunt nd İşcn re very rteul to their PhD suervisor Pro. Dr. Allh Çvuş. Cometin interests The uthors declre tht they hve no cometin interests. Received: 9 Ferury 6 Acceted: Aril 6 Reerences Bomrdelli M, Vrošnec S 869 Proerties o h-convex unctions relted to the Hermite Hdmrd Fejér ineulities. Comut Mth Al 589:877 Chen F, Wu S 4 Fejer nd Hermite Hdmrd tye ineulities or hrmoniclly convex unctions. J Al Mth 4, rticle id: Dhmni Z On Minkowski nd Hermite Hdmrd interl ineulities vi rctionl intertion. Ann Funct Anl :5 58 Fejér L 96 Uer die Fourierreihen, II, Mth. Nturwise. AnzUnr. Akd., Wiss, 4: in Hunrin Hdmrd J 893 Étude sur les roriétés des onctions entières et en rticulier d une onction considérée r Riemnn. J Mth Pures Al 58:7 5 Hermite Ch 883 Sur deux limites d une intérle déinie. Mthesis 3:8 83 İşcn İ 3 New estimtes on enerliztion o some interl ineulities or s-convex unctions nd their lictions. Int J Pure Al Mth 864: İşcn İ 3 Generliztion o dierent tye interl ineulities or s-convex unctions vi rctionl interls. Alicle Anlysis. doi:.8/ İşcn İ 4 On enerliztion o dierent tye interl ineulities or s-convex unctions vi rctionl interls. Mth Sci Al E-Notes :55 67 İşcn İ 4 Hermite Hdmrd tye ineulities or hrmoniclly convex unctions. Hcet J Mth Stt 436: İşcn İ 4c Some new enerl interl ineulities or h-convex nd h-concve unctions. Adv Pure Al Mth 5: 9. doi:.55/m-3-9 İşcn İ, Wu S 4 Hermite Hdmrd tye ineulities or hrmoniclly convex unctions vi rctionl interls. Al Mth Comut 38:37 44 Kils AA, Srivstv HM, Trujillo JJ 6 Theory nd lictions o rctionl dierentil eutions. Elsevier, Amsterdm Lti MA, Dromir SS, Momonit E 5 Some Fejer tye ineulities or hrmoniclly-convex unctions with lictions to secil mens. htt://rmi.or/ers/v8/v84 Mihi MV, Ion DA 4 Generliztion o some ineulities vi Riemnn Liouville rctionl clculus. Tmkn J Mth 45:7 5

19 Kunt et l. SrinerPlus 6 5:635 Pe 9 o 9 Minculete N, Mitroi F-C Fejér tye ineulities. Aust J Mth Anl Al 9: 8 Article Prudnikov AP, Brychkov YA, Mrichev OJ 98 Interl nd series, elementry Functions, vol. Nuk, Moscow Srıky MZ On new Hermite Hdmrd Fejér tye interl ineulities. Stud Univ Beş-Bolyi Mth 573: Srıky MZ, Set E, Yldız H, Bşk N 3 Hermite Hdmrd s ineulities or rctionl interls nd relted rctionl ineulities. Mth Comut Model 579:43 47 Tsen K-L, Yn G-S, Hsu K-C Some ineulities or dierentile mins nd lictions to Fejér ineulity nd weihted trezoidl ormul. Tiwn J Mth 54: Wn J, Li X, Fečkn M, Zhou Y Hermite-Hdmrd-tye ineulities or Riemnn Liouville rctionl interls vi two kinds o convexity. Al Anl 9:4 53. doi:.8/ Wn J, Zhu C 3 New enerlized Hermite Hdmrd tye ineulities nd lictions to secil mens. J Ineul Al 335: 5