Hermite-Hadamard-Fejér type inequalities for convex functions via fractional integrals

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1 Sud. Univ. Beş-Bolyi Mh. 6(5, No. 3, Hermie-Hdmrd-Fejér ype inequliies for convex funcions vi frcionl inegrls İmd İşcn Asrc. In his pper, firsly we hve eslished Hermie Hdmrd-Fejér inequliy for frcionl inegrls. Secondly, n inegrl ideniy nd some Hermie- Hdmrd-Fejér ype inegrl inequliies for he frcionl inegrls hve een oined. The some resuls presened here would provide exensions of hose given in erlier works. Mhemics Sujec Clssificion (: 6A5, 6A33, 6D. Keywords: Hermie-Hdmrd inequliy, Hermie-Hdmrd-Fejér inequliy, Riemnn-Liouville frcionl inegrl, convex funcion.. Inroducion Le f : I R R e convex funcion defined on he inervl I of rel numers nd, I wih <. The inequliy ( + f f( + f( f(xdx (. is well known in he lierure s Hermie-Hdmrd s inequliy [4]. The mos well-known inequliies reled o he inegrl men of convex funcion f re he Hermie Hdmrd inequliies or is weighed versions, he so-clled Hermie-Hdmrd-Fejér inequliies. In [3], Fejér eslished he following Fejér inequliy which is he weighed generlizion of Hermie-Hdmrd inequliy (.: Theorem.. Le f : [, ] R e convex funcion. Then he inequliy ( + f( + f( f g(xdx f(xg(xdx g(xdx (. holds, where g : [, ] R is nonnegive,inegrle nd symmeric o ( + /.

2 356 İmd İşcn For some resuls which generlize, improve, nd exend he inequliies (. nd (. see [, 5, 6, 7,, 6]. We give some necessry definiions nd mhemicl preliminries of frcionl clculus heory which re used hroughou his pper. Definiion.. Le f L [, ]. The Riemnn-Liouville inegrls J+f nd J f of order > wih re defined y nd J +f(x J f(x x x (x f(d, x > ( x f(d, x < respecively, where is he Gmm funcion defined y e d nd J +f(x J f(x f(x. Becuse of he wide pplicion of Hermie-Hdmrd ype inequliies nd frcionl inegrls, mny reserchers exend heir sudies o Hermie-Hdmrd ype inequliies involving frcionl inegrls no limied o ineger inegrls. Recenly, more nd more Hermie-Hdmrd inequliies involving frcionl inegrls hve een oined for differen clsses of funcions; see [, 8, 9,, 4, 5, 7, 8]. In [4], Srıky e. l. represened Hermie Hdmrd s inequliies in frcionl inegrl forms s follows. Theorem.3. Le f : [, ] R e posiive funcion wih < nd f L [, ]. If f is convex funcion on [, ], hen he following inequliies for frcionl inegrls hold ( + Γ( + [ f ( J + f( + J f( ] f( + f( (.3 wih >. In [4] some Hermie-Hdmrd ype inegrl inequliies for frcionl inegrl were proved using he following lemm. Lemm.4. Le f : [, ] R e differenile mpping on (, wih <. If f L [, ] hen he following equliy for frcionl inegrls holds: f( + f( Γ( + [ ( J + f( + J f( ] (.4 [( ] f ( + ( d.

3 Hermie-Hdmrd-Fejér ype inequliies 357 Theorem.5. Le f : [, ] R e differenile mpping on (, wih <. If f is convex on [, ] hen he following inequliy for frcionl inegrls holds: f( + f( Γ( + [ ( J + f( + J f( ] (.5 ( ( + [ f ( + f ( ]. Lemm.6 ([, 8]. For < nd <, we hve (. In his pper, we firsly represened Hermie-Hdmrd-Fejér inequliy in frcionl inegrl forms which is he weighed generlizion of Hermie-Hdmrd inequliy (.3. Secondly, we oined some new inequliies conneced wih he righhnd side of Hermie-Hdmrd-Fejér ype inegrl inequliy for he frcionl inegrls.. Min resuls Throughou his secion, le g sup [,] g(x, for he coninuous funcion g : [, ] R. Lemm.. If g : [, ] R is inegrle nd symmeric o ( + / wih <, hen wih >. J +g( J g( [ J + g( + J g( ] Proof. Since g is symmeric o (+/, we hve g ( + x g(x, for ll x [, ]. Hence, in he following inegrl seing x + nd dx d gives J+g( This complees he proof. ( x g(xdx ( g( + d ( g(d J g(. Theorem.. Le f : [, ] R e convex funcion wih < nd f L [, ]. If g : [, ] R is nonnegive,inegrle nd symmeric o ( + /, hen he following

4 358 İmd İşcn inequliies for frcionl inegrls hold ( + [J f +g( + J g( ] [ J+ (fg ( + J (fg ( ] (. f( + f( [ J + g( + J g( ] wih >. Proof. Since f is convex funcion on [, ], we hve for ll [, ] ( ( + + ( + + ( f f f ( + ( + f ( + (. (. Muliplying oh sides of (. y g ( + ( hen inegring he resuling inequliy wih respec o over [, ], we oin ( + f g ( + ( d + [f ( + ( + f ( + ( ] g ( + ( d f ( + ( g ( + ( d f ( + ( g ( + ( d. Seing x + (, nd dx ( d gives ( + ( f (x g (x dx { } ( (x f ( + x g (x dx + (x f (x g (x dx { } ( ( x f (x g ( + x dx + (x f (x g (x dx { } ( ( x f (x g (x dx + (x f (x g (x dx. Therefore, y Lemm. we hve ( + ( f [J +g( + J g( ] [ ( J + (fg ( + J (fg ( ] nd he firs inequliy is proved. For he proof of he second inequliy in (. we firs noe h if f is convex funcion, hen, for ll [, ], i yields f ( + ( + f ( + ( f( + f(. (.3

5 Hermie-Hdmrd-Fejér ype inequliies 359 Then muliplying oh sides of (.3 y g ( + ( nd inegring he resuling inequliy wih respec o over [, ], we oin + f ( + ( g ( + ( d f ( + ( g ( + ( d [f( + f(] g ( + ( d i.e. ( [ J + (fg ( + J (fg ( ] ( ( f( + f( [J +g( + J g( ] The proof is compleed. Remrk.3. In Theorem., (i if we ke, hen inequliy (. ecomes inequliy (. of Theorem.. (ii if we ke g(x, hen inequliy (. ecomes inequliy (.3 of Theorem.3. Lemm.4. Le f : [, ] R e differenile mpping on (, wih < nd f L [, ]. If g : [, ] R is inegrle nd symmeric o (+/ hen he following equliy for frcionl inegrls holds ( f( + f( [J +g( + J g( ] [ J+ (fg ( + J (fg ( ] [ ] ( s g(sds (s g(sds f (d (.4 wih >. Proof. I suffices o noe h I [ ( s g(sds ( I + I. ( s g(sds f (d + (s g(sds ( ] f (d (s g(sds f (d

6 36 İmd İşcn By inegrion y prs nd Lemm. we ge ( I ( s g(sds f( ( ( s g(sds f( nd similrly I Thus, we cn wrie ( g(f(d ( (fg(d [ f(j+g( J+(fg( ] [ f( [ J + g( + J g( ] ] J+(fg( ( (s g(sds f( ( g(f(d ( (s g(sds f( ( (fg(d [ f( [ J + g( + J g( ] ] J (fg (. I I + I {( f( + f( [J +g( + J g( ] [ J+ (fg ( + J (fg ( ]}. Muliplying he oh sides y ( we oin (.4 which complees he proof. Remrk.5. In Lemm.4, if we ke g(x, hen equliy (.4 ecomes equliy (.4 of Lemm.4. Theorem.6. Le f : I R R e differenile mpping on I nd f L [, ] wih <. If f is convex on [, ] nd g : [, ] R is coninuous nd symmeric o ( + /, hen he following inequliy for frcionl inegrls holds ( f( + f( [J +g( + J g( ] [ J+ (fg ( + J (fg ( ] wih >. ( + g ( + Γ( + ( [ f ( + f ( ] (.5 Proof. From Lemm.4 we hve ( f( + f( [J +g( + J g( ] [ J+ (fg ( + J (fg ( ] ( s g(sds (s g(sds f ( d. (.6

7 Hermie-Hdmrd-Fejér ype inequliies 36 Since f is convex on [, ], we know h for [, ] ( f ( f + f ( + f (, (.7 nd since g : [, ] R is symmeric o ( + / we wrie (s g(sds hen we hve + ( s g( + sds + ( s g(sds, ( s g(sds (s g(sds + ( s g(sds + ( s g(s ds, [ ], + ( s g(s ds, [. (.8 +, ] + A cominion of (.6, (.7 nd (.8, we ge ( f( + f( [J +g( + J g( ] [ J+ (fg ( + J (fg ( ] ( + + ( ( s g(s ds f ( + f ( Since + + ( g ( Γ( { + ( s g(s ds ( f ( + f ( d d [( ( ] (( f ( + ( f ( d [( ( ] (( f ( + ( f ( d } ( [( ( ] ( d [( ( ] ( d ( + ( + ( (.

8 36 İmd İşcn nd + + [( ( ] ( d [( ( ] ( d ( + ( + ( + + (. Hence, if we use (. nd (. in (.9, we oin he desired resul. This complees he proof. Remrk.7. In Theorem.6, if we ke g(x, hen equliy (.5 ecomes equliy (.5 of Theorem.5. Theorem.8. Le f : I R R e differenile mpping on I nd f L [, ] wih <. If f q, q >, is convex on [, ] nd g : [, ] R is coninuous nd symmeric o ( + /, hen he following inequliy for frcionl inegrls holds ( f( + f( [J +g( + J g( ] [ J+ (fg ( + J (fg ( ] (. ( + ( g ( f ( q + f ( q ( /q ( + Γ( + where > nd /p + /q. /q Proof. Using Lemm.4, Hölder s inequliy, (.8 nd he convexiy of f q, i follows h ( f( + f( [J +g( + J g( ] [ J+ (fg ( + J (fg ( ] ( + /q ( s g(sds d ( + + [ + + [ + ( + /q ( s g(sds f ( q d ( + ( s g(s ds d ( s g(s ds d ] /q ( + ( s g(s ds f ( q d

9 + { + Hermie-Hdmrd-Fejér ype inequliies ( + /q g ( /q Γ( + ( s g(s ds f ( q d ( ( + + ] /q [ ] /q [( ( ] ( ( f ( q + ( f ( q d + [( ( ] ( /q ( f ( q + ( f ( d} q (.3 + where i is esily seen h ( + + ( s ds ( + ( + [ ]. ( d + ( s ds d + + Hence, if we use (. nd (. in (.3, we oin he desired resul. This complees he proof. We cn se noher inequliy for q > s follows: Theorem.9. Le f : I R R e differenile mpping on I nd f L [, ] wih <. If f q, q >, is convex on [, ] nd g : [, ] R is coninuous nd symmeric o ( + /, hen he following inequliies for frcionl inegrls hold: ( (i f( + f( [J +g( + J g(] [ J+ (fg ( + J (fg (] /p g ( + ( /p ( f ( q + f ( q /q (p + /p Γ( + p (.4 wih >. ( (ii f( + f( [J +g( + J g(] [ J+ (fg ( + J (fg (] g ( + (p + /p Γ( + for <, where /p + /q. ( f ( q + f ( q /q (.5

10 364 İmd İşcn Proof. (i Using Lemm.4, Hölder s inequliy, (.8 nd he convexiy of f q, i follows h ( f( + f( [J +g( + J g( ] [ J+ (fg ( + J (fg ( ] ( p + /p ( /q ( s g(sds d f ( d q. ( + /p g Γ( + ( [( ( ] p d + [( ( ] p d + ( f ( q + f ( q d /q ( g ( + Γ( + ( f ( q + f ( q Here we use ( g ( + Γ( + ( f ( q + f ( q [( ] p d + [ ( ] p d /p /q (.6 [( p p ] d + /q [ p ( p ] d g ( [ ( + ] /p ( f ( q + f ( q /q Γ( + p + p. for [, /] nd for [/, ], which follows from [( ] p ( p p [ ( ] p p ( p (A B q A q B q, /p for ny A B nd q. Hence he inequliy (.4 is proved. (ii The inequliy (.5 is esily proved using (.6 nd Lemm.6. Remrk.. In Theorem.9, if we ke, hen equliy (.5 ecomes equliy in [8, Corollry 3].

11 Hermie-Hdmrd-Fejér ype inequliies 365 References [] Bomrdelli, M., Vrošnec, S., Properies of h-convex funcions reled o he Hermie Hdmrd Fejér inequliies, Compuers nd Mhemics wih Applicions, 58(9, [] Dhmni, Z., On Minkowski nd Hermie-Hdmrd inegrl inequliies vi frcionl vi frcionl inegrion, Ann. Func. Anl. ((, [3] Fejér, L., Uerdie Fourierreihen, II, Mh., Nurwise. Anz Ungr. Akd. Wiss, 4(96, , (in Hungrin. [4] Hdmrd, J., Éude sur les propriéés des foncions enières e en priculier d une foncion considérée pr Riemnn, J. Mh. Pures Appl., 58(893, 7-5. [5] [6] [7] [8] [9] [] İşcn, İ., New esimes on generlizion of some inegrl inequliies for (, m-convex funcions, Conemp. Anl. Appl. Mh., ((3, İşcn, İ., New esimes on generlizion of some inegrl inequliies for s-convex funcions nd heir pplicions, In. J. Pure Appl. Mh., 86(4(3, İşcn, İ., Some new generl inegrl inequliies for h-convex nd h-concve funcions, Adv. Pure Appl. Mh., 5((4, -9, doi:.55/pm-3-9. İşcn, İ., Generlizion of differen ype inegrl inequliiesfor s-convex funcions vi frcionl inegrls, Applicle Anlysis, 93(9(4, İşcn, İ., New generl inegrl inequliies for qusi-geomericlly convex funcions vi frcionl inegrls, J. Inequl. Appl., 3(49(3, 5 pges. İşcn, İ., On generlizion of differen ype inegrl inequliies for s-convex funcions vi frcionl inegrls, Mhemicl Sciences nd Applicions E-Noes, ((4, [] Prudnikov, A.P., Brychkov, Y.A., Mrichev, O.I., Inegrl nd series. In: Elemenry Funcions, vol., Nuk, Moscow, 98. [] Srıky, M.Z., On new Hermie Hdmrd Fejér ype inegrl inequliies, Sud. Univ. Beş-Bolyi Mh. 57(, no. 3, [3] Srıky, M.Z., Ogunmez, H., On new inequliies vi Riemnn-Liouville frcionl inegrion, Asrc n Applied Anlysis, (, pges, Aricle ID doi:.55//48983 [4] Srıky, M.Z., Se, E., Yldız, H., Bşk, N., Hermie-Hdmrd s inequliies for frcionl inegrls nd reled frcionl inequliies, Mhemicl nd Compuer Modelling, 57(9(3, [5] Se, E., New inequliies of Osrowski ype for mpping whose derivives re s-convex in he second sense vi frcionl inegrls, Compuers nd Mh. wih Appl., 63(, [6] Tseng, K.-L., Yng, G.-S., Hsu, K.-C., Some inequliies for differenile mppings nd pplicions o Fejér inequliy nd weighed rpezoidl formul, Tiwnese journl of Mhemics, 5(4(, [7] Wng, J., Li, X., Fečkn, M., Zhou, Y., Hermie-Hdmrd-ype inequliies for Riemnn-Liouville frcionl inegrls vi wo kinds of convexiy, Appl. Anl., 9((, doi:.8/ [8] Wng, J., Zhu, C., Zhou, Y., New generlized Hermie-Hdmrd ype inequliies nd pplicions o specil mens, J. Inequl. Appl., 3(35(3, 5 pges.

12 366 İmd İşcn İmd İşcn Giresun Universiy, Fculy of Sciences nd Ars, Deprmen of Mhemics, Giresun, Turkey e-mil:

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