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

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1 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 Asrc. In his noe, we oin new some ineuliies of Simpson s ype sed on convexiy. Some pplicions for specil mens of rel numers re lso given.. Inroducion The following ineuliy is one of he es-known resuls in he lierure s Simpson s ineuliy. Theorem. Le f [; ]! R e four imes coninuously di erenile mpping on ; nd f 4 f 4 x < Then, he following ineuliy holds sup x; f f f Z f For recen re nemens, 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 [; ]! R is di erenile mpping whose derivive is coninuous on ; nd f L [; ]. Then he following ineuliy Z f f. f kf k holds, where kf k R jf xj dx The ound of. for L-Lipschizin mpping ws given in [] y 5 L Also, he following ineuliy ws oined in []. Mhemics Sujec Clssi cion. D5, D. Key words nd phrses. Simpson s ineuliy, convex funcion.? corresponding uhor.

2 MEHMET ZEKI SARIKAYA?, ERHAN. SET, AND M. EMIN OZDEMIR Theorem. Suppose f [; ]! R is n soluely coninuous mpping on [; ] whose derivive elongs o L p [; ] Then he following ineuliy holds, Z f f. f kf k p where p In [] Alomri e. l. oined some ineuliies for funcions whose second derivives solue vlues re usi-convex connecing wih he Hermi-Hdmrd ineuliy on he sis of he following Lemm. Lemm. Le f I R! R e wice di erenile mpping on I wih f L [; ] ; hen. f f Z Z f d In [4], Hussin e. l. prove some ineuliies reled o Hermie-Hdmrd s ineuliy for s-convex funcions y used he ove lemm. Theorem 4. Le f I [;! R e wice di erenile mpping on I such h f L [; ] where ; I wih < If jf j is s convex on [; ] for some xed s [; ] nd ; hen he following ineuliy holds Z f f jf j jf j.4 p s s where p Remrk. If we ke s in.4, hen Z f f jf j jf j.5 The min im of his pper is o eslish new Simpson s ype ineuliies for he clss of funcions whose derivives in solue vlue cerin powers re convex funcions.. Min Resuls In order o prove our min heorems, we need he following Lemm. Lemm. Le f I R! R e wice di erenile mpping on I such h f L [; ] ; where ; I wih < ; hen he following euliy holds Z. f 4f f Z k f d

3 INEQUALITIES OF SIMPSON S TYPE where 8 >< k > Proof. By de niion of k, we hve. I Z Z Z k f I I ; ; ; ; d f Inegring y prs wice, we cn se. I 4 f nd similrly,.4 I 4 f Z 4 f Z d f f d " f f 4 f Adding. nd.4, Z 5 f d " f f I I I f f f Z d f f Z f d d # # d Using he chnge of he vrile x for [; ] nd muliplying he oh sides y ; we oin. which complees he proof. The nex heorems give new re nemen of he Simpson ineuliy for wice di erenile funcions

4 4 MEHMET ZEKI SARIKAYA?, ERHAN. SET, AND M. EMIN OZDEMIR Theorem 5. Le f I R! R e wice di erenile mpping on I such h f L [; ] ; where ; I wih < If jf j is convex on [; ] ; hen he following ineuliy holds.5 f 4f f Z [jf j jf j] Proof. From Lemm nd y used convexiy of jf j ; we ge Z f 4f f where nd Z jk j jf Z Z J J J J Z Z By simple compuion, nd J J Z Z Z j d [ jf j jf j] d [ jf j jf j] d [ jf j jf j] d [ jf j jf j] d [ jf j jf j] d [ jf j jf j] d 5 7 jf j 5 7 jf j Z 5 7 jf j 5 7 jf j which complees he proof. [ jf j jf j] d [ jf j jf j] d

5 INEQUALITIES OF SIMPSON S TYPE 5 Corollry. In Theorem 5, if f f f, hen we hve Z f [jf j jf j] Remrk. We noe h he oined midpoin ineuliy.5 is eer hn he ineuliy.. A similr resuls is emodied in he following heorem. Theorem. Le f I R! R e wice di erenile mpping on I such h f L [; ] ; where ; I wih < If jf j is convex on [; ] nd ; hen he following ineuliy holds where p f 4f Z f 5 7 jf j 5 7 jf j 5 7 jf j 5 7 jf j Proof. Suppose h From Lemm nd using he well known power men ineuliy, we hve Z f 4f f Z jk j jf Z Z 8 < Z Z Z Z j d jf j d jf j d d! jf j d d!! 9 jf j d! ;

6 MEHMET ZEKI SARIKAYA?, ERHAN. SET, AND M. EMIN OZDEMIR Since jf j is convex, herefore we hve. Z Z Z Z jf j d jf j jf j d jf j jf j d jf j 5 7 jf j 5 7 jf j jf j d nd.7 Z Z Z Z 5 7 jf j 5 7 jf j jf j d jf j jf j d jf j jf j d jf j jf j d From. nd.7, we hve 8 < f 4f Z f d! 5 7 jf j 5 7 jf j Z d! 5 7 jf j 5 7 jf j Z

7 INEQUALITIES OF SIMPSON S TYPE jf j 5 7 jf j 5 7 jf j 5 7 jf j where we use he fc h Z The proof is complee. d Z d Remrk. In Theorem, if, hen we hve he ineuliy of.5. Corollry. In Theorem, if f f f, hen we hve Z f 5 7 jf j 5 7 jf j 5 7 jf j 5 7 jf j Corollry. In Theorem, if f f f nd p, hen we hve Z f 5 7 jf j 5 7 jf j 5 7 jf j 5 7 jf j. Applicions o Specil Mens We shll consider he following specil mens The rihmeic men A A; ; ; ; The hrmonic men H H ; ; ; > ; c The logrihmic men 8 < if L L ; ln ln if d The p logrihmic men, ; > ;

8 8 MEHMET ZEKI SARIKAYA?, ERHAN. SET, AND M. EMIN OZDEMIR 8 h >< L p L p ; > p p p i p if if, p R f ; g ; ; >. 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 ove mens. Proposiion. Le ; R, < < nd n N, n > Then, we hve A n ; n An ; L n n ; nn n n Proof. The sserion follows from Theorem 5 pplied o convex mpping f x x n ; x [; ] nd n N Proposiion. Le ; R, < < Then, for ll p >, we hve H ; A ; L ; Proof. The sserion follows from Theorem pplied o he convex mpping f x x; x [; ] References [] M. Alomri, M. Drus nd S.S. Drgomir, New ineuliies of Hermie-Hdmrd ype for funcions whose second derivives solue vlues re usi-convex, RGMIA Res. Rep. Coll., 9, Supplemen, Aricle 7. [Onlinehp// [] 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. [Onlinehp// [] S.S. Drgomir, R.P. Agrwl nd P. Cerone, On Simpson s ineuliy nd pplicions, J. of Ineul. Appl., 5, [4] S. Hussin, M.I. Bhi nd M. Il, Hdmrd-ype ineuliies for s-convex funcions I, Punj Univ. Jour. of Mh., Vol.4, pp5-, 9. [5] B.Z. Liu, An ineuliy of Simpson ype, Proc. R. Soc. A, 4 5, [] J. Peµcrić, F. Proschn nd Y.L. Tong, Convex funcions, pril ordering nd sisicl pplicions, Acdemic Press, New York, 99.

9 INEQUALITIES OF SIMPSON S TYPE 9 Deprmen of Mhemics,Fculy of Science nd Ars, Afyon Kocepe Universiy, Afyon, Turkey E-mil ddress sriky@ku.edu.r Aürk Universiy, K.K. Educion Fculy, Deprmen of Mhemics, 54, Cmpus, Erzurum, Turkey E-mil ddress erhnse@yhoo.com Grdue School of Nurl nd Applied Sciences, A¼Gr Irhim Çeçen Universiy, A¼Gr, Turkey E-mil ddress emos@uni.edu.r