Hindwi Pulishing Corportion Journl of Applied Mthemtics Volume 4, Article ID 38686, 6 pges http://dx.doi.org/.55/4/38686 Reserch Article Fejér nd Hermite-Hdmrd Type Inequlities for Hrmoniclly Convex Functions Feixing Chen nd Shnhe Wu School of Mthemtics nd Sttistics, Chongqing Three Gorges University, Wnzhou, Chongqing 44, Chin Deprtment of Mthemtics nd Computer Science, Longyn University, Longyn, Fujin 364, Chin Correspondence should e ddressed to Shnhe Wu; shnhewu@gmil.com Received June 4; Accepted 3 July 4; Pulished 6 August 4 Acdemic Editor: Yu-Ming Chu Copyright 4 F. Chen nd S. Wu. This is n open ccess rticle distriuted under the Cretive Commons Attriution License, which permits unrestricted use, distriution, nd reproduction in ny medium, provided the originl work is properly cited. We estlish Fejér type inequlity for hrmoniclly convex functions. Our results re the generliztions of some known results. Moreover, some properties of the mppings in connection with Hermite-Hdmrd nd Fejér type inequlities for hrmoniclly convex functions re lso considered.. Introduction Let f:i R R e convex function nd, I with <;then + ) f (t) dt f () +). () Inequlity () is known in the literture s the Hermite- Hdmrd inequlity. Fejér [] estlished the following weighted generliztion of inequlity (). Theorem. If f:[,] R is convex function, then the following inequlity holds: + ) dx dx f () +) dx, where p:[,] R is positive, integrle, nd symmetric with respect to x ( + )/. Some generliztions, refinements, vritions, nd improvementsof inequlities () nd() were investigted y Wu [], Chen nd Liu [3], Sriky nd Ogunmez [4], nd Xio et l. [5], respectively. () In [6], Drgomir proposed n interesting Hermite- Hdmrd type inequlity which refines the left hnd side of inequlityof () s follows. Theorem (see [6]). Let f e convex function defined on [, ]. ThenH is convex, incresing on [, ], ndforllt [, ],onehs + )H() H(t) H() where H (t) dx, (3) tx+( t) + )dx. (4) An nlogous result for convex functions which refines the right hnd side of inequlity ()wsotinedyyngnd Hong in [7] s follows. Theorem 3 (see [7]). Let f e convex function defined on [, ]. ThenF is convex, incresing on [, ], ndforllt [, ],onehs f () +) dxf() F(t) F(), (5)
Journl of Applied Mthemtics where F (t) ( ) [f (( +t )+( t )x) +f (( +t )+( t )x)]dx. (6) Yng nd Tseng in [8] estlished the following Fejér type inequlities, which is the generliztion of inequlities (3)nd (5)swellstherefinementoftheFejér inequlity (). Theorem 4 (see [8]). If f is convex on [, ], p : [, ] R is positive, integrle, nd symmetric out x (+)/. Then P nd Q re convex, incresing on [, ],ndforllt [,], one hs + ) dxp() P(t) P() where P (t) Q (t) dx Q() Q(t) Q() f () +) dx, (7) tx+( t) + x) dx, (8) [f (( +t )+( t )xx+ ) +f (( +t )+( t )xx+ )] dx. (9) In [9, ], İşcn nd Wu gve the definition of hrmonic convexity s follows. Definition 5. Let I R \{}e rel intervl. A function f:i R is sid to e hrmoniclly convex if xy f ( ) ty) + ( t), () tx + ( t) y for ll x, y I nd t [,]. If the inequlity in () is reversed, then f is sid to e hrmoniclly concve. The following Hermite-Hdmrd inequlity for hrmoniclly convex functions holds true. Theorem 6 (see [9]). Let f : I R \{} R e hrmoniclly convex function nd, I with <.If f L(, ), then one hs + ) x dx f () +). () In [], İşcn nd Wu estlished the following Hermite- Hdmrd inequlities for hrmoniclly convex functions vi the Riemnn-Liouville frctionl integrl. Theorem 7 (see []). Let f:i (, ) R e function such tht f L(, ), where, I with <.Iff is hrmoniclly convex function on[, ], then the following inequlities for frctionl integrls hold: (α+) )Γ ( α + ) [J α / (f g) ( ) f () +), where α>nd g(x) /x. +J α / + (f g)( )] () The Riemnn-Liouville frctionl integrls J α +f nd Jα f of order α>with re defined y J α x + Γ (α) J α Γ (α) x (x t) α f (t) dt, (t x) α f (t) dt, x>, x<, (3) where Γ(α) is the Gmm function defined y Γ(α) e t t α dt. In this pper, we estlish Fejér type inequlity for hrmoniclly convex functions; our min result includes, s specil cses, the inequlities given y Theorems 6 nd 7. Moreover, we investigte some properties of the mppings in connection to Hermite-Hdmrd nd Fejér type inequlities for hrmoniclly convex functions.. Fejér Type Inequlity for Hrmoniclly Convex Functions The following Fejér inequlityfor hrmonicllyconvex functions holds true. Theorem 8. Let f:i R \{} R e hrmoniclly convex function nd, I with <.Iff L (, ), then one hs + ) x dx x dx f () +) x dx, (4) where p : [,] R is nonnegtive nd integrle nd stisfies p( x ). (5) + x
Journl of Applied Mthemtics 3 Proof. Since f is hrmoniclly convex function on [, ],we hve, for ll x, y [, ], f ( xy x+y ) y)+x). (6) Choosing x /(t + ( t)) nd y /(t + ( t)), we hve + ) f (/ (t + ( t) )) +/ (t + ( t) )) f () +). (7) Since p is nonnegtive nd stisfies the condition of (5), we otin + t + ( t) ) ( t + ( t) t + ( t) ) +f ( t + ( t) )) t + ( t) f () +) p( t + ( t) ). (8) Integrting oth sides of the ove inequlities with respect to t over [, ],weotin + ) p( t + ( t) )dt ((f ( t + ( t) t + ( t) ) +f ( t + ( t) t + ( t) )) )dt f () +) The proof of Theorem 8 is completed. p( t + ( t) )dt. (9) Remrk 9. Putting p(x) in Theorem 8, weotin inequlity (). Remrk. Choosing α ( ) α {( x ) α +( x ) α }, (α >,<<), () in Theorem 8,itisesytooservethtp (/x) p (/( + x)). Since x dx x dx α ( ) α α ( ) α x {( x α ) +( α }dx x ) x {( x α ) +( α }dx x ) α ( α / ) {(u α / ) +( α }du u), x dx x dx α ( ) α x {( x α ) +( α }dx x ) α ( ) α / / u ){(u α ) +( α }du u) α ( ) α α ( ) α α ( ) α / { / u )(u α du ) / + α du} / u )( u) / { f g(u) (u α du / ) / + f g(u) ( α du} / u) {Γ(α) [J α / (f g)( )+Jα / + (f g)( )]} Γ (α+) ( α ) [J α / (f g)( )+Jα / + (f g) ( )], ()
4 Journl of Applied Mthemtics where g(x) /x, which implies tht inequlity (4) cne trnsformed to inequlity () under n pproprite selection of p(x). Remrk. In Theorem 8,tkingp (/x) ω (x),where< <, ω(x)is nonnegtive, integrle, nd symmetric with respect to x ( + )/.Theninequlity(4)ecomes + ) ω (x) dx )ω(x) dx x f () +) ω (x) dx. 3. Some Mppings in connection with Hermite-Hdmrd nd Fejér Inequlities for Hrmoniclly Convex Functions () Lemm. Let f:i R\{} R e hrmoniclly convex function nd, I with <,ndlet h (t) + t )+ ), (3) ++t t [, ].Thenh is convex, incresing on [, ],ndfor ll t [, ], f () +) )h(t). (4) + Proof. Firstly, for x, y [, ],wehve h(tx+( t) y) + [tx+( t) y] ) + ++[tx+( t) y] ) t (+ x) + ( t) (+ y) ) + t (++x) + ( t) (++y) ) t + x )+ t + y ) + t th(x) + ( t) h(y), ++x )+ t nd hence h is convex on [, ]. ++y ) (5) Next, if t [, ], it follows from the hrmonic convexity of f tht h (t) + t )+ ++t ) (/)(+ t) + (/)(++t) ) + ). It is esy to oserve tht h (t) + t )+ ++t ) ( +t + ( t ( ) +t ( ) f () + + ( ) +t ( ) f () + + t ) ) + +t ) ) ( ) t f () ( ) ( ) t f () ( ) (6) (7) f () +). Thus inequlity (4)holds. Finlly, for <t <t,sincehis convex, it follows from (4)tht h(t ) h(t ) t t h(t ) h() t h(t ) / (+)) t, (8) nd hence, h(t ) h(t ),whichmensthth is incresing on [, ].ThiscompletestheproofofLemm. Theorem 3. Let f:i R \{} R e hrmoniclly convex function nd, I with <.Iff L(, ) nd H is defined y H (t) ( ) + tx )dx + ( ) ++tx )dx ( t)((+)/) +tx )dx, then H is convex nd incresing on [, ],nd )H() H(t) H() + x dx. (9) (3)
Journl of Applied Mthemtics 5 Proof. It follows from Lemm tht h (t) + t )+ ++t ) (3) is convex nd incresing on [, ]. HenceH(t) is convex nd incresing on [, ]. Further,inequlity(3) cne deduced from (4). Theorem3 is proved. Theorem 4. Let f:i R \{} R e hrmoniclly convex function nd, I with <.Iff L(, ) nd G is defined y G (t) ( ) + ( t) x )dx + ( ) ( t) x )dx ( ) (+t) +( t) x )dx + ( ) (+t) +( t) x )dx, then G is convex nd incresing on [, ],nd x dxg() G(t) G() f () +). (3) (33) Proof. We note tht if f is convex nd g is liner, then the composition f gis convex. It follows from Lemm tht h (t) + t )+ ), (34) ++t nd k(t) ( t)xre incresing on [, ] nd [, ],respectively.hence, h (k (t)) + ( t) x )+ ( t) x ) (35) is convex nd incresing on [, ]. WeinferthtG is convex nd incresing on [, ]. Furthermore, inequlity (33) follows directly from (4). The proofof Theorem4 is completed. Theorem 5. Let f:i R \{} R e hrmoniclly convex function nd, I with <.Iff L (, ) nd P is defined y P (t) + tx + x )dx + ++tx ++x )dx ((+)/)( t) +tx x )dx, (36) where p : [,] R is nonnegtive nd integrle nd stisfies the condition of (5), thenp is convex nd incresing on [, ],nd + ) x dx P () P(t) P() Proof. From Lemm we otin tht x dx. (37) h (t) + t )+ ++t ) (38) is convex nd incresing on [, ].Since p (/(++x)) is nonnegtive nd stisfies p (/( + + x)) p (/( + x)),itfollowsthtp(t)is convex nd incresing on [, ], while inequlity (37) cnededucedfrom(4). Theorem 5 is proved. Theorem 6. Let f:i R \{} R e hrmoniclly convex function nd, I with <.Iff L (, ) nd Q is defined y Q (t) + ( t) x + x )dx + ( t) x x )dx (+t) +( t) x x+ )dx + (+t) +( t) x x+ )dx, (39) where p : [,] R is nonnegtive nd integrle nd stisfies the condition of (5),thenQ is convex nd incresing on [, ],nd x dxq() Q(t) Q() f () +) x dx. (4) Proof. By using the sme method s in the proof of Theorem 4,weotinfromLemm tht h (k (t)) + ( t) x )+ ( t) x ) (4) is convex nd incresing on [, ]. Sincep (/( + x)) is nonnegtive nd stisfies p (/(+x)) p (/( x)),
6 Journl of Applied Mthemtics we deduce tht Q(t) is convex nd incresing on [, ]. Inequlity (4) follows from(4) nd the identity p( + x )dx { p( x )dx + x dx. p( + x )dx p( x )dx} (4) [7] G. S. Yng nd M. C. Hong, A note on Hdmrd s inequlity, Tmkng Journl of Mthemtics,vol.8,no.,pp.33 37,997. [8] G. S. Yng nd K. L. Tseng, On certin integrl inequlities relted to Hermite-Hdmrd inequlities, Journl of Mthemticl Anlysis nd Applictions, vol.39,no.,pp.8 87, 999. [9] İ. İşcn, Hermite-Hdmrd nd Simpson-Like type inequlities for differentile hrmoniclly convex functions, Journl of Mthemtics,vol.4,ArticleID34635,pges,4. [] İ. İşcn nd S. Wu, Hermite-Hdmrd type inequlities for hrmoniclly convex functions vi frctionl integrls, Applied Mthemtics nd Computtion,vol.38,pp.37 44,4. This completes the proof of Theorem 6. Remrk 7. If we put α ( ) α {( x ) α +( x ) α }, (43) in inequlities (37) nd(4), respectively, we otin the refined versions of inequlity (). Conflict of Interests The uthors declre tht there is no conflict of interests regrding the puliction of this pper. Acknowledgments The present investigtion ws supported, in prt, y the Youth Project of Chongqing Three Gorges University of Chin (no. 3QN) nd, in prt, y the Foundtion of Scientific Reserch Project of Fujin Province Eduction Deprtment of Chin (no. JK49). References [] L. Fejér, Üer die Fourierreihen, II, Mth.Nturwiss. Anz Ungr. Akd. Wiss,vol.4,pp.369 39,96(Hungrin). [] S. Wu, On the weighted generliztion of the Hermite- Hdmrd inequlity nd its pplictions, Rocky Mountin Journl of Mthemtics,vol.39,no.5,pp.74 749,9. [3] F. X. Chen nd X. F. Liu, Refinements on the Hermite- Hdmrd inequlities for r-convex functions, Journl of Applied Mthemtics, vol. 3, Article ID 978493, 5 pges, 3. [4] M. Z. Sriky nd H. Ogunmez, On new inequlities vi Riemnn-Liouville frctionl integrtion, Astrct nd Applied Anlysis, vol., Article ID 48983, pges,. [5] Z. G. Xio, Z. H. Zhng, nd Y. D. Wu, On weighted Hermite- Hdmrd inequlities, Applied Mthemtics nd Computtion,vol.8,no.3,pp.47 5,. [6]S.S.Drgomir, TwomppingsinconnectiontoHdmrd s inequlities, Journl of Mthemticl Anlysis nd Applictions, vol.67,no.,pp.49 56,99.
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