Generalized Hermite-Hadamard Type Inequalities for p -Quasi- Convex Functions

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1 Ordu Üniv. Bil. Tek. Derg. Cilt:6 Syı: /Ordu Univ. J. Sci. Tech. Vol:6 No: QUASİ-KONVEKS FONKSİYONLAR İÇİN GENELLEŞTİRİLMİŞ HERMİTE-HADAMARD TİPLİ EŞİTSİZLİKLER Özet İm İŞCAN* Giresun Üniversitesi Fen Edeiyt Fkültesi Mtemtik Bölümü Giresun Türkiye Bu çlışmd yzr türevleneilir onksiyonlr için yeni genel ir özdeşlik verir ve u özdeşliği kullnrk -usi konveks onksiyonlr için zı yeni genelleştirilmiş Hermite-Hdmrd tili eşitsizlikler elde eder. Mthemtics Suject Clssiiction: 6D5 6A5 Anhtr Kelimeler: Hermite-Hdmrd eşitsizliği -usi konveks onksiyon Generlized Hermite-Hdmrd Tye neulities or -Qusi- Convex Functions Astrct İm İŞCAN* Giresun Üniversitesi Fen Edeiyt Fkültesi Mtemtik Bölümü Giresun Türkiye n this er the uthor gives new generl identity or dierentile unctions nd estlishes some new generlized Hermite-Hdmrd tye ineulities or -usi convex unctions y using this identity. Mthemtics Suject Clssiiction: 6D5 6A5 Keywords: Hermite-Hdmrd s ineulity -usi-convex unction *im.iscn@giresun.edu.tr 83

2 İ. İşcn ntroduction Let : R R e convex unction deined on the intervl with <. The ollowing ineulity o rel numers nd ( ) ( ) ( x) dx () holds. This doule ineulity is known in the literture s Hermite-Hdmrd integrl ineulity or convex unctions. Note tht some o the clssicl ineulities or mens cn e derived rom () or rorite rticulr selections o the ming. Both ineulities hold in the reversed direction i is concve. n Drgomir & Agrwl (998) gve the ollowing Lemm. By using this Lemm Drgomir otined the ollowing Hermite-Hdmrd tye ineulities or convex unctions: Lemm Let : R R e dierentile ming on <. L[ ] then the ollowing eulity holds: ( ) ( ) ( x) dx = nd ( t) ( t ( t) ). The notion o usi-convex unctions generlizes the notion o convex unctions. More recisely unction :[ ] R is sid usi-convex on [ ] i x () y su ( x) ( y) or ny x y ] nd. Clerly ny convex unction is usi-convex unction. Furthermore there exist usi-convex unctions which re not convex (see on 7). For some results which generlize imrove nd extend the ineulities() relted to usi-convex unctions we reer the reder to see (Alomri et l ; Alomri et l ; on 7; İşcn 3; 3; 3 İşcn et l 4Zehng 3) nd lenty o reerences therein. n (İşcn 4) the uthor gve deinition Hrmoniclly convex nd concve unctions s ollow. with () Deinition Let hrmoniclly convex i R\ e rel intervl. A unction : R is sid to e xy t ( y) ( t) ( x) tx ( t) y (3) 84

3 -Qusi-Konveks Fonksiyonlr İçin Genelleştirilmiş Hermite-Hdmrd Tili Eşitsizlikler or ll nd hrmoniclly concve. x y t ]. the ineulity in (3) is reversed then is sid to e Zhng et l (3) deined the hrmoniclly usi-convex unction nd sulied severl roerties o this kind o unctions. Deinition A unction : i or ll x y nd t ] is sid to e hrmoniclly usi-convex xy su tx ( t) y. ( x) ( y) We would like to oint out tht ny hrmoniclly convex unction on hrmoniclly usi-convex unction ut not conversely. For exmle the unction x(]; ( x) = ( x ) x 4]. is hrmoniclly usi-convex on (4] ut it is not hrmoniclly convex on (4]. n [6] Zhng nd Wn gve deinition o -convex unction s ollow: is Deinition 3 Let e -convex set. A unction : R is sid to e -convex unction or elongs to the clss PC() i or ll x y nd ]. / x () y ( x) () ( y) Remrk An intervl is sid to e -convex set i x () y or ll x y nd ] where = k or = n/ m n = r m = t nd k r t. Remrk e rel intervl nd R\ / x () y or ll x y nd ]. then / According to Remrk we cn give dierent version o the deinition o unction s ollow: -convex 85

4 İ. İşcn Deinition 4 [] Let : or ll concve. x y is sid to e -convex unction i nd ] e rel intervl nd / x () y ( x) () ( y) R\. A unction (4). the ineulity in (4) is reversed then is sid to e - According to Deinition 4 t cn e esily seen tht or -convexity reduces to ordinry convexity nd hrmoniclly convexity o unctions deined on resectively. = nd Exmle Let : R ( x) = x nd then nd g re oth -convex nd -concve unctions. = g : R g( x) = c c R n [4 Theorem 5] i we tke ollowing Theorem. R\ nd h ( t) = t then we hve the Theorem Let : R e -convex unction R\ with <. L[ ] then we hve nd / ( x) ( ) ( ) dx. x (5) n[] İşcn deined the -usi-convex unction nd sulied severl roerties o this kind o unctions s ollow: Deinition 5 Let e rel intervl nd R\ sid to e -usi-convex i. A unction : R is / tx ( t) y mx ( x) ( y) (6) or ll x y nd t ]. the ineulity in (6) is reversed then is sid to e - usi-concve. t cn e esily seen tht or r = nd r = -usi convexity reduces to ordinry usi convexity nd hrmoniclly usi convexity o unctions deined on resectively. Morever every -convex unction is -usi-convex unction. 86

5 -Qusi-Konveks Fonksiyonlr İçin Genelleştirilmiş Hermite-Hdmrd Tili Eşitsizlikler Exmle Let R x x R g : R g( x) = c c R then nd g re Proosition Let : ( ) = \ nd e rel intervl -usi-convex unctions. R\ nd : R is unction then ;. -usi-convex.. nd is -usi-convex nd nondecresing unction then is usi-convex. 3. -usi-concve nd nondecresing unction then is usiconcve. 4. -usiconcve. 5. -usi-convex usi-concve nd nonincresing unction then is usiconcve. 8. nd is usi-concve nd nonincresing unction then is -usiconcve. nd is usi-convex nd nondecresing unction then is nd is nd is usi-concve nd nondecresing unction then is nd is usi-convex nd nonincresing unction then is nd is -usi-convex nd nonincresing unction then is usi-convex. nd is Proosition : g : R R nd i we consider the unction / deined y g t) = t i nd only i g ( then is usi-convex on. is -usi-convex on For some results relted to -convex unctions nd its generliztions we reer the reder to see (Fng 4; İşcn 6; 6; 6Noor 5; Zhng et l 5). The min urose o this er is to estlish some new generl results connected with the right-hnd side o the ineulities (5) or -usi-convex unctions. Min Results n order to rove our min results we need the ollowing lemm: 87

6 İ. İşcn Lemm Let : R e dierentile ming on <. L[ ] R \ nd with > then the ollowing eulity holds: ( ) ( ) ( x) ( ) t dx = ( M / t ( )) x t ( t) / where M ( ) = t (t). t Proo: integrtion y rts we hve ( ) t = ( M / t ( )) t ( t) = ( ) t d ( M t ( )) ( ) t = ( M ( )) ( M ( )) t t ( ) ( ) = ( M t ( )) Setting x = t ( t) nd x dx = gives ( ) ( ) ( x) = dx. x which comletes the roo. Remrk 3 we tke = = = in Lemm then we otin the ineulity () in Lemm. Theorem Let : < R\ L R e dierentile unction on nd [ ]. ( ) ( ) ( x) dx x is -convex on [ ] then with 88

7 -Qusi-Konveks Fonksiyonlr İçin Genelleştirilmiş Hermite-Hdmrd Tili Eşitsizlikler mx ( ) ( ) C ( ; ) where C ( ) ( ; ) = ( )( ) A M A ( ) M A \ nd ( ) G C ( ; ) = A A M ln ( ) M / M t ( ) = t ( t) At = t ( t) G =. M = A A = / nd / t t Proo: From Lemm nd using the Hölder integrl ineulity nd o on [ ] we hve ( ) ( ) ( x) dx x ( ) t ( M / t ( )) t ( t) ( ) t mx ( ) ( ) / t ( t) t is esily check tht ( ) t = C / ( ; ). t ( t) -usi-convexity 89

8 İ. İşcn n Theorem i we ut unctions: then we otin the ollowing corollry or usi-convex Corollry Under the conditions o Theorem i we tke ( ) ( ) ( x) dx mx ( ) ( ) C ( ;). n Theorem i we ut hrmoniclly usi-convex unctions: then we hve then we otin the ollowing corollry or Corollry Under the conditions o Theorem i we tke ( ) ( ) ( x) dx x Theorem 3 Let : then we hve mx ( ) ( ) C ( ; ). R e dierentile unction on < R \ nd L[ ]. then where ( ) ( ) ( x) dx x is with -convex on [ ] or > = r K r D / / r / ( ) ( ; ; ) mx ( ) ( ) r r K ( r) = ( r )( ) L L R \ / ( ) D( ; ; ) = L ( ) = / ( ) = 9

9 -Qusi-Konveks Fonksiyonlr İçin Genelleştirilmiş Hermite-Hdmrd Tili Eşitsizlikler L = L := R \ ( )( ) L := is logrithmic men. ln ln is the - logrithmic men nd Proo: From Lemm nd using the Hölder integrl ineulity nd o on [ ] we hve ( ) ( ) ( x) dx x ( ) t t ( t) / ( M ( )) / / r r ( M t ( )) ( ) t / t ( t) / / r mx ( ) ( ) r ( ) t / t ( t) / / r / K. D mx ( ) ( ). t is esily check tht r t ( ) t = K ( r) -usi-convexity t ( t) / = D( ; ; ). n Theorem 3 i we ut then we otin the ollowing corollry or usi-convex unctions: Corollry 3 Under the conditions o Theorem i we tke then we hve 9

10 İ. İşcn ( ) ( ) ( x) dx x ( ) mx ( ) ( ). n Theorem 3 i we ut hrmoniclly usi-convex unctions: / r K r / then we otin the ollowing corollry or Corollry 4 Under the conditions o Theorem i we tke ( ) ( ) ( x) dx x then we hve / / r / K ( r ) D ( ; ; ) mx ( ) ( ). Reerences Alomri M. W. Drus M. nd Kirmci U. S. (). Reinements o Hdmrd-tye ineulities or usi-convex unctions with lictions to trezoidl ormul nd to secil mens. Comuters nd Mthemtics with Alictions 59: 5-3. Alomri M. nd Hussin S. (). Two ineulities o Simson tye or usi-convex unctions nd lictions. Alied Mthemtics E-Notes : -7. Drgomir S.S. nd Agrwl R.P. (998). Two neulities or Dierentile Mings nd Alictions to Secil Mens o Rel Numers nd to Trezoidl Formul. Al. Mth. Lett. (5): Fng Z. B. nd Shi R. (4). On the ( h) -convex unction nd some integrl ineulities. J. neul. Al. 4 (45): 6 ges. on D.A. (7). Some estimtes on the Hermite-Hdmrd ineulity through usi-convex unctions. Annls o University o Criov Mth. Com. Sci. Ser. 34: İşcn İ. (3). Generliztion o dierent tye integrl ineulities vi rctionl integrls or unctions whose second derivtives solute vlues re usi-convex Konurl journl o Mthemtics ():

11 -Qusi-Konveks Fonksiyonlr İçin Genelleştirilmiş Hermite-Hdmrd Tili Eşitsizlikler İşcn İ. (3). New generl integrl ineulities or usi-geometriclly convex unctions vi rctionl integrls Journl o neulities nd Alictions 3(49): 5 ges. İşcn İ. (3). On generliztion o some integrl ineulities or usi-convex unctions nd their lictions nterntionl Journl o Engineering nd Alied sciences (EAAS) 3(): İşcn İ. (4). Hermite-Hdmrd tye ineulities or hrmoniclly convex unctions. Hcet. J. Mth. Stt. 43(6): İşcn İ. (6). Ostrowski tye ineulities or -convex unctions. Reserchgte doi:.34/rg Aville online t htts:// İşcn İ. (6). Hermite-Hdmrd nd Simson-like tye ineulities or dierentile -usiconvex unctions. Reserchgte doi:.34/rg Aville online t htts:// İşcn İ. (6). Hermite-Hdmrd tye ineulities or -convex unctions. Reserchgte doi:.34/rg Aville online t htts:// İşcn İ. nd Numn S. (4). Ostrowski tye ineulities or hrmoniclly usi-convex unctions. Electronic Journl o Mthemticl Anlysis nd Alictions () July 4: Noor M.A. Noor K.. nd tikhr S. (5). Nonconvex Functions nd ntegrl neulities. Punj University Journl o Mthemtic 47(): 9-7. Zhng T.-Y. Ji A.-P. And Qi F. (3). ntegrl ineulities o Hermite-Hdmrd tye or hrmoniclly usi-convex unctions. Proc. Jngjeon Mth. Soc. 6 (3): Zhng K. S. nd Wn J. P. (7). -convex unctions nd their roerties. Pure Al. Mth. 3():

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Hermite-Hadamard and Simpson-like Type Inequalities for Differentiable p-quasi-convex Functions Filomt 3:9 7 5945 5953 htts://doi.org/.98/fil79945i Pulished y Fculty of Sciences nd Mthemtics University of Niš Seri Aville t: htt://www.mf.ni.c.rs/filomt Hermite-Hdmrd nd Simson-like Tye Ineulities for