OSTROWSKI TYPE FRACTIONAL INTEGRAL INEQUALITIES FOR MT-CONVEX FUNCTIONS

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1 Miskolc Mhemicl Noes HU e-issn Vol. 6 (5), No., pp OSTROWSKI TYPE FRACTIONAL INTEGRAL INEQUALITIES FOR MT-CONVEX FUNCTIONS WENJUN LIU Received Ferury, 4 Asrc. Some inequliies of Osrowski ype for MT-convex funcions vi frcionl inegrls re oined. These resuls no only generlize hose of [5], u lso provide new esimes on hese ypes of Osrowski inequliies for frcionl inegrls. Mhemics Sujec Clssificion: 6A33; 6A5; 6D7; 6D; 6D5 Keywords: Osrowski ype inequliy, MT-convex funcion, frcionl inegrl. INTRODUCTION The following resul is known in he lierure s he Osrowski inequliy (see [7, pge 468] or [8]), which gives n upper ound for he pproximion of he inegrl verge f./d y he vlue poin x Œ; : R Theorem. Le f W I! R; where I R is n inervl, e mpping differenile in he inerior I ı of I, nd le ; I ı wih <. If jf /j M for ll x Œ;, hen Z 6 f./d M. / 4 4 C x C. / 3 7 5; 8 x Œ; : (.) In recen yers, vrious generlizions, exensions nd vrins of such inequliies hve een oined (see [,4,5,8, 5,,4] nd he references cied herein). In [3] (see lso [5, 6]), Tunç nd Yidirim defined he following so-clled MTconvex funcion: Definiion. A funcion: I R! R is sid o elong o he clss of MT(I ), if i is nonnegive nd for ll x;y I nd.;/ sisfies he following inequliy: p p f C. /y/ p C p f.y/: (.) c 5 Miskolc Universiy Press

2 5 W. J. LIU In [5], Tunç derived some inequliies of Osrowski ype for MT-convex funcions. Theorem. Le f W Œ; Œ;/! R e differenile mpping on.;/ wih < such h f L Œ; : If jf j is MT-convex on Œ; nd jf /j M; x Œ; ; hen we hve Z M h / C. x/ i f./d (.3) 4. / for ech x Œ;. Theorem 3. Le f W Œ; Œ;/! R e differenile mpping on.;/ wih < such h f L Œ; : If jf j q is MT-convex on Œ;, q >, p Cq D nd jf /j M; x Œ; ; hen we hve Z f./d M q / C. x/ (.4). C p/ =p. / for ech x Œ;. Theorem 4. Le f W Œ; Œ;/! R e differenile mpping on.;/ wih < such h f L Œ; : If jf j q is MT-convex on Œ;, q nd jf /j M; x Œ; ; hen we hve Z C f./d M q / C. x/ q (.5). / for ech x Œ;. Frcionl clculus [7, 6, 9] ws inroduced he end of he nineeenh cenury y Liouville nd Riemnn, he sujec of which hs ecome rpidly growing re nd hs found pplicions in diverse fields rnging from physicl sciences nd engineering o iologicl sciences nd economics. We recll definiions nd preliminry fcs of frcionl clculus heory which will e used in his pper. Definiion. Le f L Œ; : The Riemnn-Liouville inegrls of order > wih re defined y nd JąC D. / Z x Z / f./d; x > JąCf nd J J D. x/ f./d; x < ;. / x respecively, where. / D R e u u du. Here, JC D J D : In he cse of D ; he frcionl inegrl reduces o he clssicl inegrl. f

3 OSTROWSKI TYPE INEQUALITIES FOR MT-CONVEX FUNCTIONS 5 Due o he wide pplicion of frcionl inegrls, some uhors exended o sudy frcionl inegrl inequliies, we refer he reder o he ppers [,3,6,9,] nd he reference cied herein. Moived y hese resuls, in he presen pper, we eslish some Osrowski ype inequliies for MT-convex funcions vi Riemnn-Liouville frcionl inegrls. So, new esimes on hese ypes of Osrowski inequliies vi frcionl inegrls re provided nd he resuls of [5] re generlized.. OSTROWSKI TYPE FRACTIONAL INTEGRAL INEQUALITIES FOR MT-CONVEX FUNCTIONS In his secion, we pply he following frcionl inegrl ideniy from Se [] o derive some new Osrowski ype frcionl inegrl inequliies for MT-convex funcions. Lemm. Le f W Œ;! R e differenile mpping on.;/ wih < : If f L Œ; ; hen for ll x Œ; nd >, one hs / C. x/ D / C Z f C.. C / J x //d f./ C J xc f./. x/ C Z f C. //d: (.) Using his lemm, we cn oin he following Osrowski ype frcionl inegrl inequliies for MT-convex funcions. Theorem 5. Le f W Œ; Œ;/! R e differenile mpping on.;/ wih < such h f L Œ; : If jf j is MT-convex on Œ; nd jf /j M; x Œ; ; hen he following inequliies for frcionl inegrls wih > nd x Œ; hold: / C. x/ M. C /. /. C /. C / J x f./ C J xc f./ / C C. x/ C : (.) Proof. From (.) nd since jf j is MT-convex, we hve / C. x/. C / J x f./ C J xc f./ C. / C Z x/ C Z f C. f C. //d //d

4 5 W. J. LIU C. M " / C Z x/ C Z " / C Z " x/ C Z C M. p D M Œ / C C. x/ C. / D M Œ / C C. x/ C. / D M Œ / C C. x/ C. / p f / C p p p p p f / C p p p # p C p d " p p # p C p d Z # f./ d # f./ d h C. / = C. / = i d C 3 ; C C ; 3. C /. / ;. C / where we hve used he Be funcion of Euler ype, which is defined s ;y/ D Z The proof is compleed. x. / y d D /.y/ ; 8 x;y > : C y/ Remrk. In Theorem 5, if we choose D, we ge he inequliy in Theorem. Theorem 6. Le f W Œ; Œ;/! R e differenile mpping on.;/ wih < such h f L Œ; : If jf j q is MT-convex on Œ;, q >, p Cq D nd jf /j M; x Œ; ; hen he following inequliies for frcionl inegrls wih > nd x Œ; hold: / C. x/ M. C p / =p. C / J x f./ C J xc f./ q / C C. x/ C : (.3) Proof. From Lemm nd using he well-known Hölder s inequliy, we hve / C. x/. C / J x f./ C J xc f./ / C Z f C. //d

5 OSTROWSKI TYPE INEQUALITIES FOR MT-CONVEX FUNCTIONS 53 C. C. x/ C Z / C x/ C f C. //d Z p d Z p f C. Z p d Z p f C. Since jf j q is MT-convex nd jf /j M, we ge Z Z " p f C. //q d p nd similrly Z M q Z f /q C " p p C f C. //q d M q : q //q d q //q d : p p p # p d D M q # f./q d By simple compuion, we hve Z p d D p C : Using hese resuls, we complee he proof of (.3). Remrk. In Theorem 6, if we choose D, we ge he inequliy in Theorem 3. Theorem 7. Le f W Œ; Œ;/! R e differenile mpping on.;/ wih < such h f L Œ; : If jf j q is MT-convex on Œ;, q nd jf /j M; x Œ; ; hen he following inequliies for frcionl inegrls wih > nd x Œ; hold: / C. x/. C / J x f./ C J xc f./ M. C / q. C /. /. C /! q / C C. x/ C : (.4) Proof. From Lemm nd using he well-known power men inequliy, we hve / C. x/. C / J x f./ C J xc f./ / C Z f C. //d

6 54 W. J. LIU C. C. x/ C Z / C x/ C Z Z f C. //d d Z q f C. d Z q f C. Since jf j q is MT-convex on Œ; nd jf /j M, we ge Z nd similrly f C. //q d Z " p p p f /q C p Z " p p # M q p C p Z # f./q d q //q d q //q d : d D. C /. / M q. C / f C. //q d. C /. / M q :. C / Using hese inequliies, we complee he proof of (.4). Remrk 3. In Theorem 7, if we choose D, we ge he inequliy in Theorem 4. ACKNOWLEDGEMENTS This work ws prly suppored y he Nionl Nurl Science Foundion of Chin (Grn No. 377), he Qing Ln Projec of Jingsu Province, nd he Trining Arod Projec of Ousnding Young nd Middle-Aged Universiy Techers nd Presidens. REFERENCES [] M. Alomri, M. Drus, S. S. Drgomir, nd P. Cerone, Osrowski ype inequliies for funcions whose derivives re s-convex in he second sense, Appl. Mh. Le., vol. 3, no. 9, pp. 7 76,. [] G. Ansssiou, M. R. Hooshmndsl, A. Ghsemi, nd F. Mofkhrzdeh, Mongomery ideniies for frcionl inegrls nd reled frcionl inequliies, JIPAM. J. Inequl. Pure Appl. Mh., vol., no. 4, pp. Aricle 97, 6, 9. [3] Z. Dhmni, New inequliies in frcionl inegrls, In. J. Nonliner Sci., vol. 9, no. 4, pp ,. [4] S. S. Drgomir, The Osrowski inegrl inequliy for mppings of ounded vriion, Bull. Ausrl. Mh. Soc., vol. 6, no. 3, pp , 999.

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8 56 W. J. LIU Auhor s ddress Wenjun Liu College of Mhemics nd Sisics, Nnjing Universiy of Informion Science nd Technology, Nnjing 44, Chin E-mil ddress: wjliu@nuis.edu.cn