Hermite-Hadamard-Fejer Type Inequalities for s Convex Function in the Second Sense via Fractional Integrals

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1 Filom 30: 06), DOI 0.98/FIL63S Published by Fculy o Sciences nd Mhemics, Universiy o Niš, Serbi Avilble : hp:// Hermie-Hdmrd-Fejer Type Ineuliies or s Convex Funcion in he Second Sense vi Frcionl Inegrls Erhn Se, İmd İşcn b, Hsn Hüseyin Kr Deprmen o Mhemics, Fculy o Ars nd Sciences, Ordu Universiy, 500, Ordu, Turkey b Deprmen o Mhemics, Fculy o Ars nd Sciences, Giresun Universiy, Giresun, Turkey Absrc. In his pper, we esblished Hermie-Hdmrd-Fejer ype ineuliies or s convex uncions in he second sense vi rcionl inegrls. The some resuls presened here would provide exnsions o hose given in erlier works.. Inroducion The ollowing ineuliy is well known in he lierure s he Hermie-Hdmrd inegrl ineuliy[0]: ) b x) dx b ) b) where : I R R is convex uncion on he inervl I o rel numbers nd.b I wih < b. A uncion : [, b] R R is sid o be convex i whenever x, y [, b] nd [0, ] he ollowing ineuliy holds x ) y ) x) ) y ). In [9], Fejér gve generlizion o he ineuliies ) s he ollowing: I : [, b] R is convex uncion, nd : [, b] R is nonnegive, inegrble nd symmeric bou b, hen ) ) b b x) dx x) x) dx ) b) x) dx. ) I, hen we re lking bou he Hermie Hdmrd ineuliies.more bou hose ineuliies cn be ound in number o ppers nd monogrphies or exmple, see [7]-[9]). In [], Hudzik nd Mligrd considered mong ohers he clss o uncions which re s-convex in he second sense. 00 Mhemics Subjec Clssiicion. 6D07, 6D5. Keywords. s convex uncion, Hermie-Hdmrd ineuliy, Hermie-Hdmrd-Fejer ineuliy, Riemnn-Liouville rcionl inegrl. Received: Sepember 04; Acceped: 0 April 05 Communiced by Drgn S. Djordjević Emil ddresses: erhnse@yhoo.com Erhn Se), imdi@yhoo.com İmd İşcn), h.huseyin.kr6@gmil.com Hsn Hüseyin Kr)

2 E. Se, İ. İşcn, H. H. Kr / Filom 30: 06), Deiniion.. A uncion : [0, ) R is sid o be s-convex in he second sense i λx λ) y ) λ s x) λ) s y ). or ll x, y [0, ), λ [0, ] nd or some ixed s 0, ]. I cn be esily seen h s, s-convexiy reduces o ordinry convexiy o uncions deined on [0, ). In [7], Drgomir nd Fizprick proved Hdmrd s ineuliy which holds or s-convex uncions in he second sense. Theorem.. Suppose h : [0, ) [0, ) is n s-convex uncions in he second sense, where s 0, ), nd le, b [0, ), < b. I L [, b], hen he ollowing ineuliies hold: s ) b x) dx b ) b) s We give some necessry deiniions nd mhemicl preliminries o rcionl clculus heory which re used hrough his pper. Deiniion.3. Le L [, b]. The Riemnn Liouville inegrls J nd J o order > 0 wih 0 re b deined by inegrls hold: nd J x) Γ ) x x ) ) d, x > J b x) x) ) d, x < b Γ ) x respecively where Γ ) e u du. Here is J 0 0 x) J0 x) x) b In he cse o, he rcionl inegrl reduces o he clssicl inegrl. The recen resuls nd he properies concerning his operor cn be ound []-[6]) In [5], Srıky e.l. represened Hermie-Hdmrd s ineuliies in rcionl inegrl orms s ollows. Theorem.4. Le : [, b] R be posiive uncion wih 0 0. In [], İşcn gve he ollowing Hermie-Hdmrd-Fejer inegrl ineuliies vi rcionl inegrls: Theorem.5. Le : [, b] R be convex uncion wih 0.

3 E. Se, İ. İşcn, H. H. Kr / Filom 30: 06), Se e l. esblished some ineuliies conneced wih he le-hnd side o he ineuliy ) used he ollowing lemm. Lemm.6. [9] Le : [, b] R be dierenible mpping on, b) wih < b nd le : [, b] R. I, L [, b], hen he ollowing ideniy or rcionl inegrls holds: b ) [ ] [ ] J ) b ) J b) J ) ) J ) b ) b ) b b) k ) ) d 4) ) Γ ) where k ) s ), [ ], b b s), [ b, b]. Se e l. proved he ollowing hree heorems. Theorem.7. [9] Le : I R be dierenible mpping on I nd L [, b] wih 0. b ) [,b], Γ ) ) [ ) b) ] Theorem.8. [9] Le : I R be dierenible mpping on I nd L [, b] wih , hen he ollowing ineuliy or rcionl inegrls holds: ] b J ) b ) J b) J ) ) J ) b ) b ) b b) ) wih > 0. b ) [,b], / ) ) / Γ ) { [ 3) ) ) b) ] / [ ) ) 3) b) ] } / Theorem.9. [9] Le : I R be dierenible mpping on I nd L [, b] wih , hen he ollowing ineuliy or rcionl inegrls holds: ] b J ) b ) J b) J ) ) J ) b ) b ) b b) 7) ) b ) [3 / p ) ) b) / ) 3 b) ) ] / / Γ ) where /p /. 6)

4 We recll he ollowing uncion: The incomplee Be uncion deined by ) x B x, β ) β d. 0 E. Se, İ. İşcn, H. H. Kr / Filom 30: 06), In his pper, moived by he recen resuls given in [],[9], we esblish Hermie-Hdmrd-Fejer ype ineuliies or s-convex uncions in he second sense vi rcionl inegrl. An ineresing eure o our resuls is h hey provide new esimes on hese ypes o ineuliies or rcionl inegrls.. Min Resuls Now, by using he Lemm.6 we prove our min heorems. Theorem.. Le : I [0, ) R be dierenible mpping on I o nd L [, b] wih < b nd : [, b] R is coninious. I is s convex on [, b] or some ixed s 0, ], hen he ollowing ineuliy or rcionl inegrls holds: ] b J ) b ) J b) J ) ) J ) b ) b ) b b) 8) ) b ) [,b], { Γ ) B /, s ) } [ ) b) ]. s s ) Proo. Since is s convex on [, b] or some ixed s 0, ], we know h or [, b] ) b b ) b b b b ) s ) ) s b) b From Lemm.6 we hve ) [ ] b J ) b ) J b) b ) b Γ ) s ) ) d b [, b b ], b ) s Γ ) [ b,b], b ) s Γ ) [, b ], b ) s Γ ) b b [ ] J ) ) J ) b ) b b) ) b s) ) d s ) ds) b ) s ) ) s b) ) d b b s) ds) ) s ) ) s b) ) d ) [ b ) s ) ) s b) ] d [ b,b], b ) s b ) [ b ) s ) ) s b) ] d Γ ) b [, b ], b ) s Γ ) ) b ) b ) s d b) b ) s d [ b,b], b ) s Γ ) ) b ) s d b) b ) ) s d b b

5 E. Se, İ. İşcn, H. H. Kr / Filom 30: 06), [, [ b ], b ) s ) b ) s B /, s ) b) b ) s ] Γ ) s s ) [ [ b,b], b ) s ) b ) s Γ ) s s ) b) ] b ) s B /, s ) b ) s { b ) s ) B/, s ) b) ) Γ ) [, b ], s s ) [ b ),b], s s ) b) )} B/, s ) b ) [,b], Γ ) B /, s ) [ ) b) [ ] ) b) ] s s ) b ) [,b], { Γ ) B /, s ) } [ ) b) ] s s ) where nd b b ) s d b ) b ) s d b ) s d b b ) s s s ) b ) ) s d b ) s B /, s ). Remrk.. In Theorem 7, i we choose s, hen 8) reduces ineuliy 5) o Theorem 4. Theorem.3. Le : I [0, ) R be dierenible mpping on I o nd L [, b] wih , hen he ollowing ineuliy or rcionl inegrls holds: ] b J ) b ) J b) J ) ) J ) b ) b ) b b) ) b ) [,b], ) ) / ) / s Γ ) { s ) 3) ) s ) ) b) ) / s ) ) ) s ) 3) b) ) / }. 9) Proo. Since is s convex on [, b] or some ixed s 0, ], we know h or [, b] ) b b ) b b ) s b ) ) s b) b b

6 E. Se, İ. İşcn, H. H. Kr / Filom 30: 06), Using Lemm.6, power men ineuliy nd convexiy o, i ollows h b Γ ) ) [ J ) b ) J b b Γ ) [, b ], Γ ) b [ b,b], Γ ) b s ) b ] [ ] b) J ) ) J ) ) b ) b b) ) / b d s ) / b s) b d b / b s ) ds d / b s) b ds d b [, b b ], b ) Γ ) )) / b ) s/ [ b b,b], b ) s/ b ) d / b s) ) / d s ) ds ) / d b s) ds ) / d ) b ) s ) ) s b) ) d b ) s ) b ) ) s b) ) / d b ) ) / Γ ) ) [, b ], [b ) s B b ) s/ /, s ) ) b ) s s s ) [ [ b,b], b ) s/ b ) s s s ) b ) [,b], b) ] / ) b ) s B /, s ) b) ] / s ) / ) / s Γ ) ) s s ) B /, s ) b) ) } / { s s ) B /, s ) ) b) ) / / b ) [,b], ) ) / s ) / Γ ) { s ) 3) ) s ) ) b) ) / s ) ) ) s ) 3) b) ) / }. Remrk.4. In Theorem 8, i we choose s, hen 0) reduces ineuliy 6) o Theorem 5. Theorem.5. Le : I [0, ) R be dierenible mpping on I o nd L [, b] wih , hen he ollowing ineuliy

7 E. Se, İ. İşcn, H. H. Kr / Filom 30: 06), or rcionl inegrls holds: ] b J ) b ) J b) J ) ) J ) b ) b ) b b) ) b ) [,b], s p ) ) /p s ) / Γ ) [ ) s ) b) ) / ) b) s )) ] / 0) where p. Proo. Using Lemm.6, Hölder s ineuliy nd he s-convex o i ollows h ] b J ) b ) J b) J ) ) J ) b ) b ) b b) ) b Γ) s ) b d b s ) b ) d b p /p b Γ ) s ) d ) / d p /p Γ ) s ) b b d ) / d b b p /p b Γ ) s ) d ) / d Here we use b b b b ) [,b], s p ) ) /p s ) / Γ ) [ ) s ) b) ) / ) b) s )) / ]. p s ) d ) d ) d b b [ ) s ) b) ] s s ) b [ ) b) s )]. s s ) b ) d / p s ) b ) p d p p ) p Remrk.6. In Theorem 9, i we choose s, hen 0) reduces ineuliy 7) o Theorem 6. Reerences [] G. Ansssiou, M.R. Hooshmndsl, A. Ghsemi nd F. Mokhrzdeh, Monogomery ideniies or rcionl inegrls nd reled rcionl ineuliies, J. Ine. Pure nd Appl. Mh., 04) 009), Ar. 97. [] S. Belrbi nd Z. Dhmni, On some new rcionl inegrl ineuliies, J. Ine. Pure nd Appl. Mh., 03) 009), Ar. 86.

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