Hermite-Hadamard-Fejér type inequalities for harmonically convex functions via fractional integrals

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1 NTMSCI 4, No. 3, New Trends in Mthemticl Sciences Hermite-Hdmrd-Fejér type ineulities or hrmoniclly conve unctions vi rctionl interls Imdt Iscn, Mehmet Kunt nd Nzli Yzici 3 Deprtment o Mthemtics, Fculty o Sciences nd Arts, Giresun University,, Giresun, Turkey,3 Deprtment o Mthemtics, Fculty o Sciences, Krdeniz Technicl University, 6, Trzon, Turkey Received: 3 June 6, Accepted: July 6 Pulished online: 3 Auust 6. Astrct: In this pper, irstly, Hermite-Hdmrd-Fejér type ineulity or hrmoniclly conve unctions in rctionl interl orms hve een estlished. Secondly, n interl identity nd some Hermite-Hdmrd-Fejér type interl ineulities or hrmoniclly conve unctions in rctionl interl orms hve een otined. The some results presented here would provide etensions o those iven in erlier works. Keywords: Hermite-Hdmrd ineulity, Hermite-Hdmrd-Fejér ineulity, Riemnn-Liouville rctionl interl, hrmoniclly conve unction. Introction Let : I R R e conve unction deined on the intervl I o rel numers nd, I with <. The ineulity d is well known in the literture s Hermite-Hdmrd s ineulity 5. The most well-known ineulities relted to the interl men o conve unction re the Hermite Hdmrd ineulities or its weihted versions, the so-clled Hermite-Hdmrd-Fejér ineulities. In 4, Fejér estlished the ollowin Fejér ineulity which is the weihted enerliztion o Hermite-Hdmrd ineulity. Theorem. Let :, R e conve unction. Then, the ineulity d d holds, where :, R is nonnetive, interle nd symmetric to /. d For some results which enerlize, improve nd etend ineulities nd see, 6, 7, 5, 7. We recll the ollowin ineulity nd specil unctions which re known s Bet nd hypereometric unction Correspondin uthor e-mil: mkunt@ktu.e.tr c 6 BISKA Bilisim Technoloy

2 4 I. Iscn, M. Kunt nd N. Yzici: Hermite-Hdmrd-Fejer type ineulities... respectively Γ Γ y β,y Γ y t t y dt,,y>, F,;c;z Lemm. For < nd < we hve see 4,9. t t c zt dt, c>>, z <see. β,c. We will now ive deinitions o the riht-hnd side nd let-hnd side Riemnn-Liouville rctionl interls which re used throuhout this pper. Deinition. Let L,. The riht-hnd side nd let-hnd side Riemnn-Liouville rctionl interls J nd J o order > with > re deined y J Γ J Γ t tdt, > nd t tdt, <, respectively, where Γ is the Gmm unction deined y Γ e t t dt see. Becuse o the wide ppliction o Hermite-Hdmrd type ineulities nd rctionl interls, mny reserchers etend their studies to Hermite-Hdmrd type 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 3,, 9, 6,, 9. In, İşcn ve deinition o hrmoniclly conve unctions nd estlished ollowin Hermite-Hdmrd type ineulity or hrmoniclly conve unctions s ollows. Deinition. Let I R\{} e rel intervl. A unction : I R is sid to e hrmoniclly conve, i y t ty t y t 3 or ll,y I nd t,. I the ineulity in 3 is reversed, then is sid to e hrmoniclly concve. Theorem. Let : I R\{} R e hrmoniclly conve unction nd, I with <. I L, then the ollowin ineulities holds: d. 4 see. In, İşcn nd Wu presented Hermite-Hdmrd s ineulities or hrmoniclly conve unctions in rctionl interl orms s ollows. c 6 BISKA Bilisim Technoloy

3 NTMSCI 4, No. 3, / 4 Theorem 3. Let : I, R e unction such tht L,, where, I with <. I is hrmoniclly conve unction on,, then the ollowin ineulities or rctionl interls holds: with > nd h/. In 3 Lti et. l. ve the ollowin deinition. Γ J/ h/ J / h/ Deinition 3. A unction :, R\{} R is sid to e hrmoniclly symmetric with respect to / i holds or ll,. In Chn nd Wu presented Hermite-Hdmrd-Fejér ineulity or hrmoniclly conve unctions s ollows. Theorem 4. Let : I R\{} R e hrmoniclly conve unction nd, I with <. I L, nd :, R\{} R is nonnetive, interle nd hrmoniclly symmetric with respect to, then d 5 d 6 d. In this pper, we irstly presented Hermite-Hdmrd-Fejér ineulity or hrmoniclly conve unction in rctionl interl orms which is the weihted enerliztion o Hermite-Hdmrd ineulity or hrmoniclly conve unctions 5. Secondly, we otined some new ineulities connected with the riht-hnd side o Hermite-Hdmrd-Fejér type interl ineulity or hrmoniclly conve unction in rctionl interls. Min results Throuhout this section, let sup t, or the continuous unction :, R. t, Lemm. I :, R\{} R is interle nd hrmoniclly symmetric with respect to / with <, then J / h/ J / h/ J / h/j / h/ with > nd h/,,. Proo. Since is hrmoniclly symmetric with respect to /, rom Deinition 3 we hve, or ll,. Settin t nd dt d ives J/ h/ Γ t This completes the proo. Γ t dt Γ dj/ h/. d c 6 BISKA Bilisim Technoloy

4 4 I. Iscn, M. Kunt nd N. Yzici: Hermite-Hdmrd-Fejer type ineulities... Theorem 5. Let :, R e hrmoniclly conve unction with < nd L,. I :, R is nonnetive, interle nd hrmoniclly symmetric with respect to /, then the ollowin ineulities or rctionl interls holds: J/ h/ J J/ h/ / h/ J/ h/ with > nd h/,,. Proo. Since is hrmoniclly conve unction on,, we write tt tt tt or ll t,. Multiplyin oth sides o y t to t over,, we hve tt t dt t t t t t t t t t t t t tt dt J / h/ J/ h/ tt tt 7. then intertin the resultin ineulity with respect t t t dt t t t t Since is hrmoniclly symmetric with respect to /, rom Deinition 3, we hve or ll,, tt. Settin, nd d dt ives { { { Usin Lemm, we hve d d d d d }. dt. } d } d J/ Γ h/ J J/ h/ Γ / h/ J/ h/. This ineulity ives the let hnd side o 7. On the other hnd, since is hrmoniclly conve unction, then, or ll t,, we hve t t t t. 9 c 6 BISKA Bilisim Technoloy

5 NTMSCI 4, No. 3, / 43 Then multiplyin oth sides o 9 y t,, we hve t t t t tt t t t t nd intertin the resultin ineulity with respect to t over dt t dt. t t t t It mens tht J Γ / h/ J/ J/ h/ Γ h/ J/ h/. This ineulity ives the riht hnd side o 7. The proo is completed. Remrk. In Theorem 5, one cn see the ollowin. i I one tkes, then ineulity 7 ecomes ineulity 6 o Theorem 4. ii I one tkes, then ineulity 7 ecomes ineulity 5 o Theorem 3. iii I one tkes nd, then ineulity 7 ecomes ineulity 4 o Theorem. Lemm 3. Let : I, R e dierentile unction on I such tht L,, where, I nd <. I :, Ris interle nd hrmoniclly symmetric with respect to /, then the ollowin eulity or rctionl interls holds J/ h/j / h/ t Γ s hsds t dt J/ h/j / h/ s hsds h tdt with > nd h/,,. Proo. It suices to note tht t s hsds I t s hsds h tdt t s hsds h tdt t I I. s hsds h tdt By intertion y prts nd usin Lemm, we hve t I hsds s ht hsds s Γ J/ h/j / h/ Γ t ht htdt t ht htdt J / h/j / h/ J / h/. c 6 BISKA Bilisim Technoloy

6 44 I. Iscn, M. Kunt nd N. Yzici: Hermite-Hdmrd-Fejer type ineulities... Similrly we hve I t s hsds ht s hsds Γ J/ h/j / h/ Γ A comintion o, nd 3 ives t ht htdt t ht htdt J / h/j / h/ J / h/. 3 { J/ I I I Γ h/ } J J/ h/ / h/ J/ h/. 4 Multiplyin oth sides o 4 yγ we hve. This completes the proo. Remrk. In Lemm 3, i one tkes, then eulity ecomes eulity in, Lemm 3. Theorem 6. Let : I, R e dierentile unction on I such tht L,, where, I nd <. I is hrmoniclly conve on,, :, R is continuous nd hrmoniclly symmetric with respect to /, then the ollowin ineulity or rctionl interls holds. J / h/j / h/ J/ h/j / h/ C C 5 Γ where F,; 3; C F, ; 3;, F, ; 3; F,; 3; C F, ; 3; F, ; ; F, ; 3; with < nd h/,,. Proo. From Lemm 3 we hve J / h/j / h/ J/ h/j / h/ t Γ s hsds s hsds h t t dt. 6 c 6 BISKA Bilisim Technoloy

7 NTMSCI 4, No. 3, / 45 Since is hrmoniclly symmetric with respect to /, usin Deinition 3 we hve or ll,,. t s hsds s hsds t s t hsds s hsds t t s hsds t I we use7 in 6, we hve Settin t uu t t t s hs ds, t, s hs ds t, t,. 7 J/ h/j / h/ J/ h/j / h/ t t s hs ds h t dt Γ t s hs ds t h t dt t s ds Γ t t t dt t s ds t t t dt., nd dt ives J/ h/j / h/ J/ h/j / h/ Γ Since is hrmoniclly conve on,, we hve u u uu uu u u uu uu. uu u u. 9 c 6 BISKA Bilisim Technoloy

8 46 I. Iscn, M. Kunt nd N. Yzici: Hermite-Hdmrd-Fejer type ineulities... I we use9 in, we hve J / h/j / h/ J/ h/j / h/ Γ Usin Lemm, we hve nd u u uu u u u uu u u u uu u u u uu u u u uu u u uu u uu u u Clcultin ollowin interls, we hve u uu u u uu u u u u u uu u uu uu uu uu uu u uu uu u u u uu uu u u u u u uu u u uu u u u uu u uu u u u. u u uu uu u u uu vv v dv uu u u. uu u u uu u F,; 3; F, ; 3; C 3 F, ; 3; c 6 BISKA Bilisim Technoloy

9 NTMSCI 4, No. 3, / 47 nd uu uu uu uu vv v u uu u uu dv u u u v v uu u u dv F,; 3; F, ; 3; F, ; ; F, ; 3; C. 4 I we use,, 3 nd4 in, we hve5. This completes the proo. Corollry. In Theorem 6, one hs the ollowin. I one tkes, one hs the ollowin Hermite-Hdmrd-Fejér ineulity or hrmoniclly conve unctions which is relted the riht-hnd side o 6: d d C C, I one tkes, one hs the ollowin Hermite-Hdmrd type ineulity or hrmoniclly conve unction in rctionl interl orms which is relted the riht-hnd side o5: { } Γ J / h/ J/ h/ C C, 3 I one tkes nd, one hs the ollowin Hermite-Hdmrd type ineulity or hrmoniclly conve unction which is relted the riht-hnd side o 4: d C C. Theorem 7. Let : I, R e dierentile unction on I such tht L,, where, I nd <. I,, is hrmoniclly conve on,, :, R is continuous nd hrmoniclly symmetric with respect to /, then the ollowin ineulity or rctionl interls holds: J / h/j / h/ J/ h/j / h/ Γ C 3 C 4 C 5 C 6 C 7 C 5 c 6 BISKA Bilisim Technoloy

10 4 I. Iscn, M. Kunt nd N. Yzici: Hermite-Hdmrd-Fejer type ineulities... where C 3 C 4 F, ; 3;, F, ; 3;, C 5 C 3 C 4, C 6 F,; ; F, ; ; C3 F,; 3; C 7, F, ; 3; C4 C C 6 C 7, with < nd h/,,., Proo. By usin power men ineulity nd the hrmoniclly conveity o in, we hve J / h/j / h/ J/ h/j / h/ Γ Γ u u uu Γ u u uu uu u u uu uu u u uu u u uu uu u u uu Γ u u uu u u uu u u uu uu u u uu u u u u uu u u uu u u u u u u u uu u uu u u uu u u u uu u 6 c 6 BISKA Bilisim Technoloy

11 NTMSCI 4, No. 3, / 49 Clcultin ollowin interls y Lemm, we hve u u uu u uu v v u u u dv F, ; ; C3, 7 u u uu u u uu u 4 vv v uu u u F, ; 3; C4, u u uu u C 3C 4 C 5, 9 u u uu u u uu F,; ; u u uu F, ; ; C3 C 6, 3 u u uu u u u uu u u u uu u F,; 3; C 7, 3 F, ; 3; C4 u u uu u C 6C 7 C. 3 I we use73 in6, we hve5.this completes the proo. Corollry. In Theorem 7, one hs the ollowin. I one tkes, one hs the ollowin Hermite-Hdmrd-Fejér ineulity or hrmoniclly conve unctions which is relted the riht-hnd side o 6: C 3 d C 4 C 5 d C 6 C 7 C, c 6 BISKA Bilisim Technoloy

12 5 I. Iscn, M. Kunt nd N. Yzici: Hermite-Hdmrd-Fejer type ineulities... I one tkes, one hs the ollowin Hermite-Hdmrd type ineulity or hrmoniclly conve unction in rctionl interl orms which is relted the riht-hnd side o 5: C 3 Γ { } J/ h/ J / h/ C 4 C 5 C 6 C 7 C, 3 I one tkes nd, one hs the ollowin Hermite-Hdmrd type ineulity or hrmoniclly conve unction which is relted the riht-hnd side o 4: C 3 C 4 C 5 d C 6 C 7 C. We cn stte nother ineulity or > s ollows. Theorem. Let : I, R e dierentile unction on I such tht L,, where, I nd <. I,>, is hrmoniclly conve on,, :, R is continuous nd hrmoniclly symmetric with respect to /, then the ollowin ineulity or rctionl interls holds: where J / h/j / h/ J/ h/j / h/ Γ C p 9 3 C p 3 p C 9 p F p, p; p;, C p p F p,; p;, with <, h/,, nd /p/. 33 Proo. Usin, Hölder s ineulity nd the hrmoniclly conveity o, we hve J / h/j / h/ J/ h/j / h/ Γ u u uu u u uu uu uu c 6 BISKA Bilisim Technoloy

13 NTMSCI 4, No. 3, / 5 Γ u u p p uu u u p p p uu uu Γ u u p p p uu Γ u u p p uu p uu u u u u p p uu p u u p 3 u u p p 3 34 p uu Clcultin ollowin interls y Lemm, we hve nd similrly u u p p uu u u p p uu u p uu p v p p v u p u u p p dv p p F p, p; p; C9, 35 u p u p p uu uu v p v v p dv p v p p v p dv p p F p,; p; C. 36 I we use35 nd36 in34, we hve33.this completes the proo. Corollry 3. In Theorem, one hs the ollowin. I one tkes, one hs the ollowin Hermite-Hdmrd-Fejér ineulity or hrmoniclly conve unctions which is relted the riht-hnd side o 6: d d C p 9 3 C p 3, c 6 BISKA Bilisim Technoloy

14 5 I. Iscn, M. Kunt nd N. Yzici: Hermite-Hdmrd-Fejer type ineulities... I one tkes, one hs the ollowin Hermite-Hdmrd type ineulity or hrmoniclly conve unction in rctionl interl orms which is relted the riht-hnd side o 5: C p 9 3 Γ { } J/ h/ J / h/ C p 3, 3 I one tkes nd,one hs the ollowin Hermite-Hdmrd type ineulity or hrmoniclly conve unction which is relted the riht-hnd side o 4: d C p 9 3 C p 3. 3 Conclusions In this pper, Hermite-Hdmrd-Fejer type ineulities or hrmoniclly conve unctions in rctionl interl orms re iven. Also, n interl identity nd some trpezoidl Hermite-Hdmrd-Fejer type interl ineulities or hrmoniclly conve unctions in rctionl interl orms re otined. Competin Interests The uthors declre tht they hve no competin interests. Authors Contriutions All uthors hve contriuted to ll prts o the rticle. All uthors red nd pproved the inl mnuscript. Reerences M. Bomrdelli nd S. Vrošnec, Properties o h-conve unctions relted to the Hermite Hdmrd Fejér ineulities, Computers nd Mthemtics with Applictions 5 9, F. Chen nd S. Wu, Fejér nd Hermite-Hdmrd type ineulities or hrmoniclly conve unctions, Jurnl o pplied Mthemtics, volume 4, rticle id: Z. Dhmni, On Minkowski nd Hermite-Hdmrd interl ineulities vi rctionl intertion, Ann. Funct. Anl., L. Fejér, Uerdie Fourierreihen, II, Mth. Nturwise. Anz Unr. Akd., Wiss, 4 96, , in Hunrin. 5 J. Hdmrd, Étude sur les propriétés des onctions entières et en prticulier d une onction considérée pr Riemnn, J. Mth. Pures Appl., 5 93, İ. İşcn, New estimtes on enerliztion o some interl ineulities or s-conve unctions nd their pplictions, Int. J. Pure Appl. Mth., 64 3, İ. İşcn, Some new enerl interl ineulities or h-conve nd h-concve unctions, Adv. Pure Appl. Mth. 5 4, -9. doi:.55/pm-3-9. c 6 BISKA Bilisim Technoloy

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