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
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
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
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
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
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
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
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
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), 869 877. F. Chen nd S. Wu, Fejer nd Hermite-Hdmrd type ineulities for hrmoniclly convex functions, Jurnl of pplied Mthemtics, volume 4, rticle id:38686. 3 Z. Dhmni, On Minkowski nd Hermite-Hdmrd integrl ineulities vi frctionl integrtion, Ann. Funct. Anl. ) ), 5-58. 4 L. Fejér, Uerdie Fourierreihen, II, Mth. Nturwise. Anz Ungr. Akd., Wiss, 4 96), 369-39, 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., 58 893), 7-5. 6 İ. İşcn, New estimtes on generliztion of some integrl ineulities for s-convex functions nd their pplictions, Int. J. Pure Appl. Mth., 864) 3), 77-746. 7 İ. İşcn, Some new generl integrl ineulities for h-convex nd h-concve functions, Adv. Pure Appl. Mth. 5) 4), -9. doi:.55/pm-3-9. 8 İ. İşcn, Generliztion of different type integrl ineulitiesfor s-convex functions vi frctionl integrls, Applicle Anlysis, 3. doi:.8/368.3.85785. 9 İ. İşcn, On generliztion of different type integrl ineulities for s-convex functions vi frctionl integrls, Mthemticl Sciences nd Applictions E-Notes, ) 4), 55-67. İ. İşcn, S. Wu, Hermite-Hdmrd type ineulities for hrmoniclly convex functions vi frctionl integrls, Appl. Mth. Comput., 38 4) 37-44. İ. İşcn, Hermite-Hdmrd type ineulities for hrmoniclly convex functions, Hcet. J. Mth. Stt., 43 6) 4), 935-94 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
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