ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX

Transcription

1 Journl of Applied Mhemics, Sisics nd Informics JAMSI), 9 ), No. ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX MEHMET ZEKI SARIKAYA, ERHAN. SET AND M. EMIN OZDEMIR Absrc In his noe, we obin new some ineuliies of Simpson s ype bsed on convexiy. Some pplicions for specil mens of rel numbers re lso given. Mhemics Subjec Clssificion : 6D5, 6D Addiionl Key Words nd Phrses: Simpson s ineuliy, convex funcion. INTRODUCTION The following Theorem describes he known ] in he lierure s Simpson s ineuliy. Theorem.. Le f :, b] R be four imes coninuously differenible mpping on, b) nd f 4) =sup x,b) f 4) x) <. Then, he following ineuliy holds: )] f)+fb) + b +f b fx)dx b f 4) b ) For recen refinemens, counerprs, generlizions nd new Simpson s ype ineuliies, see ],],5]). In ], Drgomir e. l. proved he following some recen developmens on Simpson s ineuliy for which he reminder is expressed in erms of lower derivives hn he fourh. Theorem.. Suppose f :, b] R is differenible mpping whose derivive is coninuous on, b) nd f L, b]. Then he following ineuliy )] f)+fb) + b +f b fx)dx b b f ) holds, where f = b f x) dx. The bound of ) for L-Lipschizin mpping ws given in ] by 5 6L b ). Also, he following ineuliy ws obined in ]. DOI.48/jmsi--4 Universiy of SS. Cyril nd Mehodius in Trnv Downlod De /5/8 :6 AM

2 M. Z. SARIKAYA, ERHAN SET, M. E. OZDEMIR Theorem.. Suppose f :, b] R is n bsoluely coninuous mpping on, b] whose derivive belongs o L p, b]. Then he following ineuliy holds, )] f)+fb) + b +f b fx)dx b where p + =. 6 + ] + b ) f +) p ) In ] Alomri e. l. obined some ineuliies for funcions whose second derivives bsolue vlues re usi-convex connecing wih he Hermi-Hdmrd ineuliy on he bsis of he following Lemm. Lemm.4. Le f : I R R be wice differenible mpping on I wih f L, b], hen f)+fb) b b fx)dx = b ) ) f + ) b) d. ) In 4], Hussin e. l. prove some ineuliies reled o Hermie-Hdmrd s ineuliy for s-convex funcions by used he bove lemm. Theorem.5. Le f : I, ) R be wice differenible mpping on I such h f L, b] where, b I wih <b.if f is s convex on, b] for some fixed s, ] nd, hen he following ineuliy holds: f)+fb) b b fx)dx b ) 6 p where p + = Remrk.6. If we ke s =in 4), hen f)+fb) b b fx)dx b ) f ) + f b) s+)s+) f ) + f b) ] 4) ]. 5) The min im of his pper is o esblish new Simpson s ype ineuliies for he clss of funcions whose derivives in bsolue vlue cerin powers re convex funcions.. MAIN RESULTS In order o prove our min heorems, we need he following Lemm. Lemm.. Le f : I R R be wice differenible mpping on I such h f L, b], where, b I wih <b,hen he following euliy holds: 6 f)+4f +b ) + fb) ] b b fx)dx =b ) k ) f b + ) ) d 6) 8 Downlod De /5/8 :6 AM

3 INEQUALITIES OF SIMPSON S TYPE where k) = Proof. By definiion of k), we hve I = k ) f b + ) ) d ),, ) ) ),, ]. = ) f b + ) ) d + ) ) f b + ) ) d ) = I + I. Inegring by prs wice, we cn se: I = 4b ) f +b )+ b = 4b ) f +b )+ nd similrly, b ) 6 )f b + ) ) d +b f )+ 6 f) ] 8) f b + ) ) d I = 4b ) f +b )+ b = 4b ) f +b )+ 5 6 )f b + ) ) d +b b ) f )+ 6 fb) ] f b + ) ) d. 9) Adding 8) nd 9), I = I + I = b ) 6 f)+ f +b )+ 6 fb) f b + ) ) d ]. Using he chnge of he vrible x = b + ) for, ] nd muliplying he boh sides by b ), we obin 6) which complees he proof. The nex heorems give new refinemen of Simpson s ineuliy for wice differenible funcions: Theorem.. Le f : I R R be wice differenible mpping on I such h f L, b], where, b I wih <b.if f is convex on, b], hen he following ineuliy holds: 6 f)+4f +b ) ] + fb) b b fx)dx b ) 6 f ) + f b) ]. ) 9 Downlod De /5/8 :6 AM

4 M. Z. SARIKAYA, ERHAN SET, M. E. OZDEMIR Proof. From Lemm. nd by using he convexiy of f, we ge 6 f)+4f +b ) ] + fb) b b fx)dx where nd b ) k ) f b + ) ) d { b ) ) f b) + ) f ) ] d + ) } ) f b) + ) f ) ] d =b ) J + J ) J = J = By simple compuion, ) f b) + ) f ) ] d ) f ) b) + ) f ) ] d. J = + ) f b) + ) f ) ] d ) f b) + ) f ) ] d nd = 59 5 f b) + 5 f ) J = ) ) f b) + ) f ) ] d + ) ) f b) + ) f ) ] d = 5 f b) f ) which complees he proof. An immedie conseuence of Theorem. is he following Corollry: Corollry.. Le f : I R R be wice differenible mpping on I such h f L, b], where, b I wih <b.if f) =f +b )=fb) nd f is convex on,b], hen he following ineuliy holds: b ) + b b ) fx)dx f f ) + f b) ]. b 6 Remrk.4. We noe h he obined midpoin ineuliy ) is beer hn he ineuliy ). 4 Downlod De /5/8 :6 AM

5 INEQUALITIES OF SIMPSON S TYPE A similr resul is embodied in he following heorem. Theorem.5. Le f : I R R be wice differenible mpping on I such h f L, b], where, b I wih <b.if f is convex on, b] nd, hen he following ineuliy holds: 6 f)+4f +b ) ] + fb) b b fx)dx b ) ) 6 { 59 5 f b) + 5 f ) ) + 5 f b) f ) ) } where p + =. Proof. Suppose h. From Lemm., we hve 6 f)+4f +b ) ] + fb) b b fx)dx b ) k ) f b + ) ) d { =b ) ) f b + ) ) d + ) } ) f b + ) ) d. Using he Hölder s ineuliy for funcions ) nd ) f b + ) ) for he firs inegrl nd he funcions ) ) nd ) ) f b + ) ) 4 Downlod De /5/8 :6 AM

6 M. Z. SARIKAYA, ERHAN SET, M. E. OZDEMIR for he second inegrl, from he bove relion we ge he ineuliies: 6 f)+4f +b ) ] + fb) b b fx)dx { b ) ) d) + ) f b + ) ) d) ) ) ) d ) ) f b + ) ) d ) }. Since f is convex, herefore we hve ) f b + ) ) d ) f b) + ) f ) ] d = )] f b) + ) f ) ] d ) + )] f b) + ) f ) ] d = 59 5 f b) + 5 f ) nd ) ) f b + ) ) d ) ) f b) + ) f ) ) d = ) ) f b) + ) f ) ) d ) + ) ) f b) + ) f ) ) d = 5 f b) f ) 4 Downlod De /5/8 :6 AM

7 INEQUALITIES OF SIMPSON S TYPE From ) nd ), we hve 6 f)+4f +b ) ] + fb) b b fx)dx { b ) ) d) 59 5 f b) + 5 f ) ) + ) ) ) d 5 f b) f ) ) } =b ) ) 6 { 59 5 f b) + 5 f ) ) + 5 f b) f ) ) } where we use he fc h ) d = The proof is complee. ) ) d = 6. Corollry.6. Le f : I R R be wice differenible mpping on I such h f L, b], where, b I wih <b.if f) =f +b )=fb) nd f is convex on, b] nd, hen he following ineuliy holds: b b fx)dx f ) +b b ) ) 6 { 59 5 f b) + 5 f ) ) + 5 f b) f ) ) } where p + =. Remrk.. By seing = in Theorem.5 nd Corollry.6, we obin Theorem. nd Corollry. respecively. Corollry.8. Le f : I R R be wice differenible mpping on I such h f L, b], where, b I wih <b.if f) =f +b )=fb) nd f is convex on,b], hen he following ineuliy holds: b b fx)dx f ) +b b ) ) 6 { 59 5 f b) + 5 f ) ) + 5 f b) f ) ) }. 4 Downlod De /5/8 :6 AM

8 M. Z. SARIKAYA, ERHAN SET, M. E. OZDEMIR. APPLICATIONS TO SPECIAL MEANS We shll consider he following specil mens: ) The rihmeic men: A = A, b) := +b,,b, b) The hrmonic men: H = H, b) := b,,b >, + b c) The logrihmic men: if = b L = L, b) :=,, b >, b ln b ln if b d) The p logrihmic men L p = L p, b) := b p+ p+ p+)b ) ] p if b if = b, p R {, } ;, b >. I is well known h L p is monoonic nondecresing over p R wih L := L nd L := I. In priculr, we hve he following ineuliies H L A. Now, using he resuls of Secion, some new ineuliies is derived for he bove mens. Proposiion.. Le, b R, <<bnd n N, n>. Then, we hve A n,b n )+ An, b) L n n, b) ) nn )b n + b n ]. 6 Proof. The sserion follows from Theorem. pplied o convex mpping f x) =x n,x, b] nd n N. Proposiion.. Le, b R, <<b.then, for ll >, we hve H, b)+ A, b) L, b) ) b ) 6 { 59 5 b + ) b + 59 } ) 5. Proof. The sserion follows from Theorem.5 pplied o he convex mpping f x) =/x, x, b]. 44 Downlod De /5/8 :6 AM

9 INEQUALITIES OF SIMPSON S TYPE REFERENCES M. Alomri, M. Drus nd S.S. Drgomir, New ineuliies of Hermie-Hdmrd ype for funcions whose second derivives bsolue vlues re usi-convex, RGMIA Res. Rep. Coll., 9), Supplemen, Aricle. Online:hp:// M. Alomri, M. Drus nd S.S. Drgomir, New ineuliies of Simpson s ype for s- convex funcions wih pplicions, RGMIA Res. Rep. Coll., 4) 9), Aricle 9. Online:hp:// S.S. Drgomir, R.P. Agrwl nd P. Cerone, On Simpson s ineuliy nd pplicions, J. of Ineul. Appl., 5), S. Hussin, M.I. Bhi nd M. Ibl, Hdmrd-ype ineuliies for s-convex funcions I, Punjb Univ. Jour. of Mh., Vol.4, pp:5-6, 9). B.Z. Liu, An ineuliy of Simpson ype, Proc. R. Soc. A, 46 5), J. Pečrić, F. Proschn nd Y.L. Tong, Convex funcions, pril ordering nd sisicl pplicions, Acdemic Press, New York, 99. Mehme Zeki Sriky, Deprmen of Mhemics, Fculy of Science nd Ars, Düzce Universiy, Düzce, Turkey e-mil: srikymz@gmil.com Erhn SET, Deprmen of Mhemics, Fculy of Science nd Ars, Düzce Universiy, Düzce, Turkey e-mil: erhnse@yhoo.com M. Emin Ozdemir, Aürk Universiy, K.K. Educion Fculy, Deprmen of Mhemics, 54, Cmpus, Erzurum, Turkey e-mil: emos@uni.edu.r Received My 45 Downlod De /5/8 :6 AM