Filomt 3:5 (7), 4989 4997 htts://doi.org/.98/fil75989c Published by Fculty o Sciences nd Mthemtics, University o Niš, Serbi Avilble t: htt://www.m.ni.c.rs/ilomt Some New Ineulities o Simson s Tye or s-convex Functions vi Frctionl Integrls Jinhu Chen,b, Xinjiu Hung, Dertment o Mthemtics, Nnchng University, Nnchng, 333, Jingxi,P. R. Chin b School o Mthemtics nd Sttistics, Centrl South University, Chngsh, 483, Hunn, P. R. Chin Abstrct. In this er, we estblish some new ineulities o Simson s tye bsed on s-convexity vi rctionl integrls. Our results generlize the results obtined by Sriky et l. [.. Introduction nd Preliminries It is well known tht the ollowing ineulity, nmed Simson s ineulity, is one o the best known results in the literture. [ ( ) () (b) b (x)dx 3 b 88 (4) (b ) 4, where : [, b R be our times continuously dierentible ming on (, b) nd (4) = su x (,b) (4) (x) <. In [, the clss o unctions which re s-convex in the second sense hs been introduced by Breckner s the ollowing. Deinition.. Let s be rel number s (,. A unction : [, ) R, is sid to be s-convex (in the second sense), or belongs to the clss K s, i holds or ll x, y [, ) nd λ [,. (λx ( λ)y) λ s (x) ( λ) s (y) It cn be esily seen tht or s =, s-convexity reduces to ordinry convexity o unctions deined on [, ). In [3, Drgomir nd Fitztrick roved vrint o Hermite-Hdmrd s ineulity which holds or s-convex unctions in the second sense. In [3, the uthors roved some new integrl ineulities o these clsses o unctions vi (h (α, m))-logrithmiclly convexity. Mthemtics Subject Clssiiction. D5, D Keywords. Simson s tye; s-convex unctions; rctionl integrls. Received: 3 August ; Acceted: 9 December Communicted by Drgn S. Djordjević This work suorted by the Ntionl Nturl Science Foundtion o Chin (443 nd 53) nd suorted rtly by the Provincil Nturl Science Foundtion o Jingxi, Chin (BAB9) nd the Science nd Technology Project o Eductionl Commission o Jingxi Province, Chin (53) nd Hunn Provincil Innovtion Foundtion or Postgrdutes (No. CXB37). Corresonding uthor: Xinjiu Hung Emil ddresses: cjh9889@3.com (Jinhu Chen), xjhungxwen@3.com (Xinjiu Hung)
J. Chen, X. Hung / Filomt 3:5 (7), 4989 4997 499 Theorem.. Suose tht : [, ) [, ) is n s-convex unction in the second sense, where s (, ), nd let, b [, ), < b. I L ([, b), then the ollowing ineulities hold: s ( b b (x)dx The ollowing Lemm is roved by Sriky et l.(see [). () (b). () s Lemm.3. Let : I R be n bsolutely continuous ming on I o such tht L ([, b), where, b I o with < b. Then the ollowing eulity holds: [ () 4 ( = b [ ( t 3 (b) b Using Lemm.3, Sriky et l. unctions in the second sense. b (x)dx ) b t ) ( 3 t ) t b ). in [ estblished the ollowing results which hold or s-convex Theorem.4. Let : I [, ) R be dierentible ming on I o such tht L ([, b), where, b I o with < b. I is s-convex on [, b, or some ixed s (,, then the ollowing ineulity holds: [ (b) b (x)dx b (b ) (s 4)s 5 s 3 s [ () (b) (3). s (s )(s ) Theorem.5. Let : I [, ) R be dierentible ming on I o such tht L ([, b), where, b I o with < b. I is s-convex on [, b, or some ixed s (, nd >, then the ollowing ineulity holds: [ (b) b (x)dx b ( (b ) ) {( (b) ( b ) ) ( () ( b ) } (4), 3( ) s s where =. Theorem.. Let : I [, ) R be dierentible ming on I o such tht L ([, b), where, b I o with < b. I is s-convex on [, b, or some ixed s (, nd >, then the ollowing ineulity holds: [ (b) b (x)dx b ( (b ) ) {( ( s ) (b) () ) ( ( s ) () (b) ) } (5) 3( ) s (s ) s, (s ) where =. Theorem.7. Let : I [, ) R be dierentible ming on I o such tht L ([, b), where, b I o with < b. I is s-convex on [, b, or some ixed s (, nd >, then the ollowing ineulity holds: [ (b) b (x)dx b (b ) ( 5 {( 3 ) 5 s (s 4) s (s 7)3 s (b) (s )3 s ) 3 s (s )(s ) 3 s (s )(s ) () () ( (s )3 s 3 s (s )(s ) (b) 5s (s 4) s (s 7)3 s ) } (), 3 s (s )(s ) ()
J. Chen, X. Hung / Filomt 3:5 (7), 4989 4997 499 where =. In the ollowing, we will give some necessry deinitions nd mthemticl reliminries o rctionl clculus theory which re used urther in this er. For more detils, one cn consult [4-5. Deinition.8. Let L ([, b). The Riemnn-Liouville integrls J α nd Jα o order α > with re b deined by J α (x) = x (x t) α (t), x > Γ(α) nd J α b (x) = b (t x) α (t), x < b Γ(α) x resectively, where Γ(α) = e t u α du is Euler gmm unction. Here J (x) = J (x) = (x). b In the cse o α =, the rctionl integrl reduces to the clssicl integrl. For some recent results connected with rctionl integrl ineulities, see [-. For more rctionl integrl lictions, lese see [4- The im o this er is to estblish some new ineulities or s-convex unctions in the second sense vi Riemnn-Liouville rctionl integrl. Our results generlize the results obtined by Sriky [ nd rovide new estimtes on these tyes o ineulities or rctionl integrls.. Min Results In this section, we introduce some ineulities vi rctionl integrls. First, new identity is resented s ollows: Lemm.. Let : I R be n bsolutely continuous ming on I o such tht L ([, b), where, b I o with < b. Then the ollowing eulity holds: () 4 ( (b) α Γ(α ) (b ) α J α b ( Jα ( = b [ (t α ) 3 b t ) ( 3 tα ) t ) b. (7) Proo. It suices to note tht [ (t α I = ) 3 b t ) ( 3 tα ) t ) b ( t α = ) 3 b t ) ( 3 tα ) t ) b = I I. (8)
J. Chen, X. Hung / Filomt 3:5 (7), 4989 4997 499 Integrting by rts ( t α I = ) 3 b t ) = ( t α b ) ( d ( t 3 b t ) ) = { [( t α b 3 ) ( t b t ) ( t b t )d(tα } 3 ) = [ b (b) 3 ( ( t b t )d(tα 3 ) = [ b (b) 3 ( α t α ( t b t ) = [ b (b) 3 ( α b ( x b ) α (x) dx b b b = [ b (b) 3 ( α ( b b )α (x b α (x)dx = [ b (b) 3 ( Γ(α ) ) ( b b )α (x Γ(α) b α (x)dx = [ b (b) 3 ( α Γ(α ) (b ) α J α b (, (9) nd similrly we hve, ( I = 3 tα ) t ) b = [ b () 3 ( α Γ(α ) (b ) α J α (. () From (8), (9) nd (), we hve conclusion (7). This comletes the roo. Remrk.. In Lemm., i α =, then we obtin Lemm.3. Using this lemm, we cn obtin the ollowing rctionl integrl ineulities which new result o Simson s ineulity or s-convex unctions. Theorem.3. Let : I [, ) R be dierentible ming on I o such tht L ([, b), where, b I o with < b. I is s-convex on [, b, or some ixed s (,, then the ollowing ineulity holds: where [ [ () 4 ( (b) α Γ(α ) (b ) α J α b ( Jα ( b s [ () (b) {I 3 (α, s) I 4 (α, s)}, () I 3 (α, s) = I 4 (α, s) = ( 3 ) α ( 3 tα [( t)s ( t) s, ( tα ( 3 ) α 3 )[( t)s ( t) s. ()
J. Chen, X. Hung / Filomt 3:5 (7), 4989 4997 4993 Proo. From Lemm. nd since is s-convex on [, b, we get (b) α Γ(α ) (b ) α J α b ( Jα ( b [ t α b t ) 3 tα t b) b [ t α b ( t ) 3 tα ( t b)) = b [ t α b ( t ) [ 3 tα ( t b)) b [ t α ( t s (b) ( t s () [ 3 tα ( t s () ( t s (b) b [ () (b) { tα s } [( t) s ( t) s 3 b [ () (b) { ( 3 ) α s b s [ () (b) {I 3 (α, s) I 4 (α, s)}, ( 3 tα [( t)s ( t) s ( tα ( 3 ) α } 3 )[( t)s ( t) s where I 3 (α, s) nd I 4 (α, s) re deined in (). This comletes the roo. Corollry.4. Let : I [, ) R be dierentible ming on I o such tht L ([, b), where, b I o with < b. I is convex on [, b, then the ollowing ineulity holds: (b) α Γ(α ) (b ) α J α b ( Jα ( b [ () (b) {I 3 (α, ) I 4 (α, )}. 4 (3) Proo. Setting s = in (), we get the reuired result. Remrk.5. In Theorem.3, i α =, then we obtin Theorem.4. In the ollowing theorem, we shll roose new uer bound or the right-hnd side o Simsons ineulity or s-convex ming with rctionl integrl tye. Theorem.. Let : I [, ) R be dierentible ming on I o such tht L ([, b), where, b I o with < b. I is s-convex on [, b, or some ixed s (, nd >, then the ollowing ineulity holds: (b) α Γ(α ) (b ) α J α b ( Jα ( (b ) ( 3 ) {( (b) ( b ) ( () ( b ) }, s s (4) where =.
J. Chen, X. Hung / Filomt 3:5 (7), 4989 4997 4994 Proo. From Lemm. nd Hölder s ineulity, we get (b) α Γ(α ) (b ) α J α b ( Jα ( = b [ t α b t ) 3 tα t b) b { 3 ) ( b t ) ) 3 ) ( t ) b) }. Since is s-convex on [, b, by using in (), we get nd b t ) (b) ( b s t b) () ( b. s Hence (b) α Γ(α ) (b ) α J α b ( Jα ( (b ) ( 3 ) {( (b) ( b ) ( () ( b ) }. s s This comletes the roo. Corollry.7. Let : I [, ) R be dierentible ming on I o such tht L ([, b), where, b I o with < b. I is convex on [, b nd >, then the ollowing ineulity holds: (b) α Γ(α ) (b ) α J α b ( Jα ( (b ) ( 3 ) {( (b) ( b ) ( () ( b ) } (5), where =. Proo. Setting s = in (4), we get the reuired result. Remrk.8. In Theorem., i α =, then we obtin Theorem.5. Next, we shll give nother versions o Simson s tye ineulity or s-convex unctions with rctionl integrl tye s ollows: Theorem.9. Let : I [, ) R be dierentible ming on I o such tht L ([, b), where, b I o with < b. I is s-convex on [, b, or some ixed s (, nd >, then the ollowing ineulity holds: (b) α Γ(α ) (b ) α J α b ( Jα ( (b ) ( 3 ) {( ( s ) (b) () ) ( ( s ) () (b) ) } () s (s ) s, (s ) where =.
J. Chen, X. Hung / Filomt 3:5 (7), 4989 4997 4995 Proo. From Lemm. nd Hölder s ineulity, we get (b) α Γ(α ) (b ) α J α b ( Jα ( = b [ t α b t ) 3 tα t b) b { 3 ) ( b t ) ) 3 ) ( t ) b) }. (7) Since is s-convex on [, b, we know tht or t [, nd s (, b t ) ( t s (b) ( t s (). (8) From (7) nd (8), we hve (b) α Γ(α ) (b ) α J α b ( Jα ( b { 3 ) ( b t ) ) 3 ) ( t ) b) }. b { 3 ) ( ( t s (b) ( t ) s () 3 ) ( ( t s () ( t ) } s (b) = b 3 ) {( ( t s (b) ( t ) s () ( ( t s () ( t ) } s (b) (b ) ( 3 ) {( ( s ) (b) () ) ( ( s ) () (b) ) } s (s ) s. (s ) This comletes the roo. Corollry.. Let : I [, ) R be dierentible ming on I o such tht L ([, b), where, b I o with < b. I is convex on [, b nd >, then the ollowing ineulity holds: (b) α Γ(α ) (b ) α J α b ( Jα ( (b ) ( 3 ) {( 3 (b) () ) ( 3 () (b) ) }, 4 4 where =. Proo. Setting s = in (), we get the reuired result. Remrk.. In Theorem.9, i α =, then we obtin Theorem.. (9)
J. Chen, X. Hung / Filomt 3:5 (7), 4989 4997 499 Theorem.. Let : I [, ) R be dierentible ming on I o such tht L ([, b), where, b I o with < b. I is s-convex on [, b, or some ixed s (, nd >, then the ollowing ineulity holds: (b) α Γ(α ) (b ) α J α b ( Jα ( b () I 5(α, s) {I (α, s) I7 (α, s) }, where = nd I 5 (α, s) = 3 ) = ( 3 tα ), I (α, s) = I 7 (α, s) = tα 3 ( ( t s (b) ( t s () ), 3 tα ( ( t s () ( t s (b) ). Proo. Using Lemm., the ower men ineulity nd Hölder s ineulity, we hve (b) α Γ(α ) (b ) α J α b ( Jα ( = b [ t α b t ) 3 tα t b) b { ( 3 ) tα 3 ( t b t ) ) ( 3 ) tα 3 ( t b t ) } ) Since is s-convex on [, b, we know tht or t [, nd s (, b t ) t ( s (b) ( t s (). (3) From () nd (3), we hve (b) α Γ(α ) (b ) α J α b ( Jα ( b { ( 3 ) tα 3 [ ( t s (b) ( t s () ) ( 3 ) tα 3 [ ( t s () ( t s (b) ) } = b I 5(α, s) {I (α, s) I7 (α, s) } where I 5 (α, s), I (α, s) nd I 7 (α, s) re deined in (). This comletes the roo. Corollry.3. Let : I [, ) R be dierentible ming on I o such tht L ([, b), where, b I o with < b. I is convex on [, b nd >, then the ollowing ineulity holds: (b) α Γ(α ) (b ) α J α b ( Jα ( b (4) I 5(α, ) {I (α, ) I7 (α, ) }, where =. Proo. Setting s = in (), we get the reuired result. Remrk.4. In Theorem., i α =, then we obtin Theorem.7. () ()
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