On New Inequalities of Hermite-Hadamard-Fejer Type for Harmonically Quasi-Convex Functions Via Fractional Integrals

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1 X th Interntionl Sttistics Dys Conference ISDC 6), Giresun, Turkey On New Ineulities of Hermite-Hdmrd-Fejer Type for Hrmoniclly Qusi-Convex Functions Vi Frctionl Integrls Mehmet Kunt * nd İmdt İşcn Deprtment of Mthemtics, Fculty of Sciences, Krdeniz Technicl University, 68, Trzon, Turkey. Deprtment of Mthemtics, Fculty of Sciences nd Arts, Giresun University, 8, Giresun, Turkey. mkunt@ktu.edu.tr, imdt.iscn@giresun.edu.tr Astrct In this pper, some Hermite-Hdmrd-Fejer type integrl ineulities for hrmoniclly usi-convex functions in frctionl integrl forms re otined. Keywords: Hermite-Hdmrd ineulity, Hermite-Hdmrd-Fejer ineulity, Riemnn-Liouville frctionl integrl, Hrmoniclly usi-convex function.. Introduction Let f: I R R e convex function defined on the intervl I of rel numers nd, I with <. The ineulity f + ) f)+f) fx)dx is well known in the literture s Hermite-Hdmrd s ineulity 5. The most well-known ineulities relted to the integrl men of convex function f re the Hermite Hdmrd ineulities or their weighted versions, the so-clled Hermite- Hdmrd-Fejér ineulities. In 4, Fejér estlished the following Fejér ineulity which is the weighted generliztion of Hermite-Hdmrd ineulit ): Theorem. Let f:, R e convex function. Then the ineulity f + ) gx)dx f)+f) fx)gx)dx gx)dx ) holds, where g:, R is nonnegtive,integrle nd symmetric to + )/. For some results which generlize, improve, nd extend the ineulities ) nd ) see, 6, 7, 6, 8. Following definitions nd mthemticl preliminries of frctionl clculus theory re used further in this pper. Definition.. Let f L,. The Riemnn-Liouville integrls J + f nd J f of oder > with re defined y J + fx) = x Γ) x t) ft)dt, nd J fx) = Γ) x t x) ft)dt, x > x < ) 35

2 X th Interntionl Sttistics Dys Conference ISDC 6), Giresun, Turkey respectively, where Γ) is the Gmm function defined y Γ) = J + fx) = J fx) = fx). e t t dt nd Becuse of the wide ppliction of Hermite-Hdmrd type ineulities nd frctionl integrls, mny reserchers extend their studies to Hermite-Hdmrd type ineulities involving frctionl integrls not limited to integer integrls. Recently, more nd more Hermite-Hdmrd ineulities involving frctionl integrls hve een otined for different clsses of functions; see 3, 8, 9, 7, 9,. Definition.. A function f: I, ), ) is sid to e hrmoniclly usiconvex, if xy f ) sup{fx), fy)} tx + t)y for ll x, y I nd t,. In, İşcn gve definition of hrmoniclly convex functions nd estlished following Hermite-Hdmrd type ineulity for hrmoniclly convex functions s follows: Definition 3. Let I R\{} e rel intervl. A function f: I R is sid to e hrmoniclly convex, if f xy tx+ t)y ) tfy) + t)fx) 3) for ll x, y I nd t,. If the ineulity in 3) is reversed, then f is sid to e hrmoniclly concve. Theorem.. Let f: I R\{} R e hrmoniclly convex function nd, I with <. If f L, then the following ineulities holds: f + ) fx) x f)+f) dx. 4) In 4 Ltif et. l. gve the following definition: Definition 4. A function g:, R\{} R is sid to e hrmoniclly symmetric with respect to + if gx) = g + ) x holds for ll x,. In Chn nd Wu represented Hermite-Hdmrd-Fejer ineulity for hrmoniclly convex functions s follows: Theorem 3. Let f: I R\{} R e hrmoniclly convex function nd, I with <. If f L, nd g:, R\{} R is nonnegtive, integrle nd hrmoniclly symmetric with respect to +, then f + ) gx) x dx fx)gx) x dx f)+f) gx) x dx. 5) In 3, Kunt nd İşcn presented, respectively, Hermite-Hdmrd ineulity in frctionl integrl forms for hrmoniclly convex functions, Hermite-Hdmrd-Fejer ineulity in frctionl integrl forms for hrmoniclly convex functions s follows: Theorem 4. Let f: I, ) R e function such tht f L,, where, I with <. If f is hrmoniclly convex function on,, then the following ineulities for 36

3 X th Interntionl Sttistics Dys Conference ISDC 6), Giresun, Turkey frctionl integrls hold: f ) Γ+) + J + ) { + +J + f h)/) f h)/) } f)+f) 6) with > nd hx) = /x, x,. Theorem 5. Let f:, R e hrmoniclly convex function with < nd f L,. If g:, R is nonnegtive, integrle nd hrmoniclly symmetric with respect to, then the following ineulities for frctionl integrls holds: + f + ) J + J + + f)+f) + g h)/) + J + fg h)/) + J + J + + g h)/) + J + g h)/) fg h)/) with > nd hx) = /x, x,. g h)/) 7) Lemm. 3. Let f: I, ) R e differentile function on I such tht f L,, where, I nd <. If g:, R is integrle nd hrmoniclly symmetric with respect to, then the following eulity for frctionl integrls holds: + f + ) J + J + + = Γ) + g h)/) + J + fg h)/) + J t g h)/) fg h)/) s ) g h)s)ds) f h) t)dt 8) t s) g h)s)ds) f h) t)dt with > nd hx) = /x, x,. In this pper, we hve some Hermite-Hdmrd-Fejer type integrl ineulities for hrmoniclly usi-convex functions in frctionl integrl forms.. Min results Throughout this section, we tke g = sup t, gt), for the continuous function g:, R. Theorem 6. Let f: I, ) R e differentile function on I such tht f L,, where, I nd <. If f is hrmoniclly usi-convex on,, g:, R is continuous nd hrmoniclly symmetric with respect to, then the following ineulity + for frctionl integrls holds: 37

4 X th Interntionl Sttistics Dys Conference ISDC 6), Giresun, Turkey f + ) J + g h)/) + J + g h)/) + J + fg h)/) + J + fg h)/) + g ) Γ+) ) C )sup{f ), f )} 9) where C ) = u u + u)) du with < nd hx) = /x, x,. Proof. From Lemm we hve u) u + u)) du f + ) J + g h)/) + J + g h)/) + J + fg h)/) + J + fg h)/) + Γ) g Γ) = g Γ) t t s ) g h)s)ds) f h) t)dt t t s) g h)s)ds) f h) t)dt s ) ds) f h) t)dt t ) Setting t = u+ u) s) ds) f h) t)dt t f ) dt t t) t f t ) dt, nd dt = ) du gives f + ) J + g h)/) + J + g h)/) + J + fg h)/) + J + fg h)/) + g ) Γ+) ) u u+ u)) f ) du u+ u) ) u) u+ u)) f ) du u+ u) 38

5 X th Interntionl Sttistics Dys Conference ISDC 6), Giresun, Turkey Since f is hrmoniclly usi-convex on,, we hve f ) u+ u)t sup{f ), f )} ) If we use.3) in.), we hve f + ) J + g h)/) + J + g h)/) + J + fg h)/) + J + fg h)/) + g ) Γ + ) ) u u + u)) sup{f ), f )}du u) u + u)) sup{f ), f )}du g ) Γ + ) ) sup{f ), f )} Since u u+ u)) C ) = du u u+ u)) u) u+ u)) du du ) u) u+ u)) If we use 3) in ) we hve 9). This completes the proof. Corollry. In Theorem 6; du 3) ) If we tke = we hve the following Hermite-Hdmrd-Fejer ineulity for hrmoniclly usi-convex functions which is relted to the left-hnd side of 5): f + ) gx) x dx fx)gx) dx g ) C )sup{f ), f )}, x ) If we tke gx) = we hve following Hermite-Hdmrd ineulity for hrmoniclly usi-convex functions in frctionl integrl forms which is relted to the left-hnd side of 6): Γ + ) f ) + J + f h) ) + ) +J + f h) { ) } ) C )sup{f ), f )}, 3) If we tke = nd gx) = we hve the following Hermite-Hdmrd type ineulity for hrmoniclly usi-convex functions which is relted to the left-hnd side of 4): f + ) fx) x dx )C )sup{f ), f )}. 39

6 X th Interntionl Sttistics Dys Conference ISDC 6), Giresun, Turkey Theorem 7. Let f: I, ) R e differentile function on I such tht f L,, where, I nd <. If f,, is hrmoniclly usi-convex on,, g:, R is continuous nd hrmoniclly symmetric with respect to, then the + following ineulity for frctionl integrls holds: f) + f) J /+ J /+ g h)/) + J / fg h)/) + J / fg h)/) g h)/) g ) Γ+) ) C )sup{f ), f ) } 4) where C ) is the sme in Theorem 6, > nd hx) = /x, x,. Proof. Using ), power men ineulity nd the hrmoniclly usi-convexity of f, it follows tht f + ) J + g h) + ) + J + g h) ) J + fg h) ) + J + fg h) ) + g ) Γ + ) ) g ) Γ + ) ) g ) Γ + ) ) + u u + u)) du) u + u)) f ) du u + u) u) u + u)) f u + u) ) du u u u + u)) du) + u) u + u)) du) u u + u)) f u + u) ) du) u) u + u)) du) u) u + u)) f u + u) ) du) u u + u)) sup{f ), f ) }du) u) u + u)) sup{f ), f ) }du) 3

7 X th Interntionl Sttistics Dys Conference ISDC 6), Giresun, Turkey = g ) Γ + ) ) sup{f ), f ) } u u + u)) du = g ) Γ+) ) C )sup{f ), f ) }. This completes the proof. Corollry. In Theorem 7; u) u + u)) du ) If we tke = we hve the following Hermite-Hdmrd-Fejer ineulity for hrmoniclly usi-convex functions which is relted to the left-hnd side of 5): f + ) gx) x dx fx)gx) dx x g ) C )sup{f ), f ) }, ) If we tke gx) = we hve following Hermite-Hdmrd ineulity for hrmoniclly usi-convex functions in frctionl integrl forms which is relted to the left-hnd side of 6): Γ + ) f ) + J + f h) ) + ) +J + f h) { ) } ) C )sup{f ), f ) }, 3) If we tke = nd gx) = we hve the following Hermite-Hdmrd ineulity for hrmoniclly usi-convex functions which is relted to the left-hnd side of 4): f + ) fx) x dx )C )sup{f ), f ) } We cn stte nother ineulity for > s follows: Theorem 8. Let f: I, ) R e differentile function on I such tht f L,, where, I nd <. If f, >, is hrmoniclly usi-convex on,, g:, R is continuous nd hrmoniclly symmetric with respect to, then the + following ineulity for frctionl integrls holds: f) + f) J g h) + ) + J g h) ) J fg h) + ) + J fg h) ) g ) Γ+) ) C p ) + C 3 p ) sup{f ),f ) } u where C ) = p du, C u+ u)) 3) = p /x, x, nd /p + / =. u) p u+ u)). 5) p du, with >, hx) = 3

8 X th Interntionl Sttistics Dys Conference ISDC 6), Giresun, Turkey Proof. Using ), Hölder s ineulity nd the hrmoniclly usi-convexity of f, it follows tht f + ) J + g h)/) + J + g h)/) + J + fg h)/) + J + fg h)/) + g ) Γ + ) ) g ) Γ + ) ) u p u + u)) + u + u)) f ) du u + u) u) u + u)) f u + u) ) du p du) u) p du) u + u)) p g ) Γ + ) ) u p u + u)) + p p du) u) p du) u + u)) p = g ) Γ + ) ) Since u p u+ u)) p du) p p p p + u f u + u) ) du) f u + u) ) du) sup{f ), f ) }du) sup{f ), f ) }du) sup{f ), f ) } u) p du) p u+ u)) 6) p u C ) = p u+ u)) p du 7) C 3 ) = u) p u+ u)) p du 8) If we use 7) nd 8) in 6), we hve 5). This completes the proof. Corollry 3. In Theorem 8; ) If we tke = we hve the following Hermite-Hdmrd-Fejer ineulity for hrmoniclly usi-convex functions which is relted to the left-hnd side of 5): 3

9 X th Interntionl Sttistics Dys Conference ISDC 6), Giresun, Turkey f + ) gx) x dx fx)gx) dx x g ) C p ) + C p 3 ) sup{f ), f ) } ) If we tke gx) = we hve following Hermite-Hdmrd ineulity for hrmoniclly usi-convex functions in frctionl integrl forms which is relted to the left-hnd side of 6): f Γ + ) ) + ) { J + + +J + f h)/) f h)/) } ) C p ) + C p 3 ) sup{f ), f ) } 3) If we tke = nd gx) = we hve the following Hermite-Hdmrd ineulity for hrmoniclly usi-convex functions which is relted to the left-hnd side of 4): f + ) fx) x dx References ) C p ) + C p 3 ) sup{f ), f ) } M. Bomrdelli nd S. Vrošnec, Properties of h-convex functions relted to the Hermite Hdmrd Fejér ineulities, Computers nd Mthemtics with Applictions 58 9), F. Chen nd S. Wu, Fejer nd Hermite-Hdmrd type ineulities for hrmoniclly convex functions, Jurnl of pplied Mthemtics, volume 4, rticle id: Z. Dhmni, On Minkowski nd Hermite-Hdmrd integrl ineulities vi frctionl integrtion, Ann. Funct. Anl. ) ), L. Fejér, Uerdie Fourierreihen, II, Mth. Nturwise. Anz Ungr. Akd., Wiss, 4 96), , in Hungrin). 5 J. Hdmrd, Étude sur les propriétés des fonctions entières et en prticulier d une fonction considérée pr Riemnn, J. Mth. Pures Appl., ), İ. İşcn, New estimtes on generliztion of some integrl ineulities for s-convex functions nd their pplictions, Int. J. Pure Appl. Mth., 864) 3), İ. İşcn, Some new generl integrl ineulities for h-convex nd h-concve functions, Adv. Pure Appl. Mth. 5) 4), -9. doi:.55/pm İ. İşcn, Generliztion of different type integrl ineulitiesfor s-convex functions vi frctionl integrls, Applicle Anlysis, 3. doi:.8/ İ. İşcn, On generliztion of different type integrl ineulities for s-convex functions vi frctionl integrls, Mthemticl Sciences nd Applictions E-Notes, ) 4), İ. İşcn, S. Wu, Hermite-Hdmrd type ineulities for hrmoniclly convex functions vi frctionl integrls, Appl. Mth. Comput., 38 4) İ. İşcn, Hermite-Hdmrd type ineulities for hrmoniclly convex functions, Hcet. J. Mth. Stt., 43 6) 4), A. A. Kils, H. M. Srivstv, J. J. Trujillo, Theory nd pplictions of frctionl differentil eutions. Elsevier, Amsterdm 6). 3 M. Kunt, İ. İşcn, On new ineulities of Hermite-Hdmrd-Fejer type for hrmoniclly convex functions,., 33

10 X th Interntionl Sttistics Dys Conference ISDC 6), Giresun, Turkey vi frctionl integrls, RGMIA Reserch Report Collection, 85), Article 9, 7 pp. 4 M. A. Ltif, S. S. Drgomir nd E. Momonit, Some Fejer type ineulities for hrmoniclly-convex functions with pplictions to specil mens, RGMIA Reserch Report Collection, 85), Article 4, 7 pp. 5 A. P. Prudnikov, Y. A. Brychkov, O. J. Mrichev, Integrl nd series, Elementry Functions, vol., Nuk, Moscow, M.Z. Sr ky, On new Hermite Hdmrd Fejér type integrl ineulities, Stud. Univ. Bes-Bolyi Mth. 573) ), M.Z. Sr ky, E. Set, H. Yld z nd N. Bsk, Hermite-Hdmrd s ineulities for frctionl integrls nd relted frctionl ineulities, Mthemticl nd Computer Modelling, 579) 3), K.-L. Tseng, G.-S. Yng nd K.-C. Hsu, Some ineulities for differentile mppings nd pplictions to Fejér ineulity nd weighted trpezoidl formul, Tiwnese journl of Mthemtics, 54) ), J. Wng, X. Li, M. Fec kn nd Y. Zhou, Hermite-Hdmrd-type ineulities for Riemnn-Liouville frctionl integrls vi two kinds of convexity, Appl. Anl., 9) ), J. Wng, C. Zhu nd Y. Zhou, New generlized Hermite-Hdmrd type ineulities nd pplictions to specil mens, J. Ineul. Appl., 335) 3), 5 pges. T. Y. Zhng, A. P. Ji, F. Qi, Integrl ineulities of Hermite-Hdmrd type for hrmoniclly usi-convex functions. Proc. Jngjeon Mth. Soc.,63) 3),