ON AN INTEGRATION-BY-PARTS FORMULA FOR MEASURES

Volume 8 (2007), Issue 4, Article 93, 13 pp. ON AN INTEGRATION-BY-PARTS FORMULA FOR MEASURES A. ČIVLJAK, LJ. DEDIĆ, AND M. MATIĆ AMERICAN COLLEGE OF MANAGEMENT AND TECHNOLOGY ROCHESTER INSTITUTE OF TECHNOLOGY DON FRANA BULICA 6, 20000 DUBROVNIK CROATIA civljk@cmt.hr DEPARTMENT OF MATHEMATICS FACULTY OF NATURAL SCIENCES, MATHEMATICS AND EDUCATION UNIVERSITY OF SPLIT TESLINA 12, 21000 SPLIT CROATIA ljubn@pmfst.hr DEPARTMENT OF MATHEMATICS FACULTY OF NATURAL SCIENCES, MATHEMATICS AND EDUCATION UNIVERSITY OF SPLIT TESLINA 12, 21000 SPLIT CROATIA mmtic@pmfst.hr Received 21 June, 2007; ccepted 15 November, 2007 Communicted by W.S. Cheung ABSTRACT. An integrtion-by-prts formul, involving finite Borel mesures supported by intervls on rel line, is proved. Some pplictions to Ostrowski-type nd Grüss-type inequlities re presented. Key words nd phrses: Integrtion-by-prts formul, Hrmonic sequences, Inequlities. 2000 Mthemtics Subject Clssifiction. 26D15, 26D20, 26D99. 1. INTRODUCTION In the pper [4], S.S. Drgomir introduced the notion of w 0 -Appell type sequence of functions s sequence w 0, w 1,..., w n, for n 1, of rel bsolutely continuous functions defined on [, b], such tht w k = w k 1,.e. on [, b], k = 1,..., n. 210-07

2 A. ČIVLJAK, LJ. DEDIĆ, AND M. MATIĆ For such sequence the uthor proved generlistion of Mitrinović-Pečrić integrtion-byprts formul (1.1) where nd A n = b w 0 (t)g(t)dt = A n + B n, n ( 1) [ k 1 w k (b)g (k 1) (b) w k ()g (k 1) () ] b B n = ( 1) n w n (t)g (n) (t)dt, for every g : [, b] R such tht g (n 1) is bsolutely continuous on [, b] nd w n g (n) L 1 [, b]. Using identity (1.1) the uthor proved the following inequlity b (1.2) w 0 (t)g(t)dt A n w n p g (n) q, for w n L p [, b], g (n) L p [, b], where p, q [1, ] nd 1/p + 1/q = 1, giving explicitly some interesting specil cses. For some similr inequlities, see lso [5], [6] nd [7]. The im of this pper is to give generliztion of the integrtion-by-prts formul (1.1), by replcing the w 0 -Appell type sequence of functions by more generl sequence of functions, nd to generlize inequlity (1.2), s well s to prove some relted inequlities. 2. INTEGRATION-BY-PARTS FORMULA FOR MEASURES For, b R, < b, let C[, b] be the Bnch spce of ll continuous functions f : [, b] R with the mx norm, nd M[, b] the Bnch spce of ll rel Borel mesures on [, b] with the totl vrition norm. For µ M[, b] define the function ˇµ n : [, b] R, n 1, by Note tht nd ˇµ n (t) = ˇµ n (t) = ˇµ n (t) 1 (n 2)! 1 (n 1)! t (t )n 1 (n 1)! [,t] (t s) n 1 dµ(s). (t s) n 2ˇµ 1 (s)ds, n 2 µ, t [, b], n 1. The function ˇµ n is differentible, ˇµ n(t) = ˇµ n 1 (t) nd ˇµ n () = 0, for every n 2, while for n = 1 ˇµ 1 (t) = dµ(s) = µ([, t]), [,t] which mens tht ˇµ 1 (t) is equl to the distribution function of µ. A sequence of functions P n : [, b] R, n 1, is clled µ-hrmonic sequence of functions on [, b] if P n(t) = P n 1 (t), n 2; P 1 (t) = c + ˇµ 1 (t), t [, b], for some c R. The sequence (ˇµ n, n 1) is n exmple of µ-hrmonic sequence of functions on [, b]. The notion of µ-hrmonic sequence of functions hs been introduced in [2]. See lso [1]. J. Inequl. Pure nd Appl. Mth., 8(4) (2007), Art. 93, 13 pp. http://jipm.vu.edu.u/

INTEGRATION-BY-PARTS FORMULA 3 Remrk 2.1. Let w 0 : [, b] R be n bsolutely integrble function nd let µ M[, b] be defined by dµ(t) = w 0 (t)dt. If (P n, n 1) is µ-hrmonic sequence of functions on [, b], then w 0, P 1,..., P n is w 0 - Appell type sequence of functions on [, b]. For µ M[, b] let µ = µ + µ be the Jordn-Hhn decomposition of µ, where µ + nd µ re orthogonl nd positive mesures. Then we hve µ = µ + + µ nd µ = µ ([, b]) = µ + + µ = µ + ([, b]) + µ ([, b]). The mesure µ M[, b] is sid to be blnced if µ([, b]) = 0. This is equivlent to µ + = µ = 1 2 µ. Mesure µ M[, b] is clled n-blnced if ˇµ n (b) = 0. We see tht 1-blnced mesure is the sme s blnced mesure. We lso write m k (µ) = t k dµ(t), k 0 for the k-th moment of µ. Lemm 2.2. For every f C[, b] nd µ M[, b] we hve f(t)dˇµ 1 (t) = f(t)dµ(t) µ({})f(). Proof. Define I, J : C[, b] M[, b] R by I(f, µ) = nd J(f, µ) = Then I nd J re continuous biliner functionls, since f(t)dˇµ 1 (t) f(t)dµ(t) µ({})f(). I(f, µ) f µ, J(f, µ) 2 f µ. Let us prove tht I(f, µ) = J(f, µ) for every f C[, b] nd every discrete mesure µ M[, b]. For x [, b] let µ = δ x be the Dirc mesure t x, i.e. the mesure defined by f(t)dδ x (t) = f(x). If < x b, then ˇµ 1 (t) = δ x ([, t]) = { 0, t < x 1, x t b nd by simple clcultion we hve I(f, δ x ) = f(t)dˇµ 1 (t) = f(x) = f(t)dδ x (t) 0 = f(t)dδ x (t) δ x ({})f() = J(f, δ x ). J. Inequl. Pure nd Appl. Mth., 8(4) (2007), Art. 93, 13 pp. http://jipm.vu.edu.u/

4 A. ČIVLJAK, LJ. DEDIĆ, AND M. MATIĆ Similrly, if x =, then ˇµ 1 (t) = δ ([, t]) = 1, t b nd by similr clcultion we hve I(f, δ ) = f(t)dˇµ 1 (t) = 0 = f() f() = f(t)dδ (t) δ ({})f() = J(f, δ x ). Therefore, for every f C[, b] nd every x [, b] we hve I(f, δ x ) = J(f, δ x ). Every discrete mesure µ M[, b] hs the form µ = k 1 c k δ xk, where (c k, k 1) is sequence in R such tht c k <, k 1 nd {x k ; k 1} is subset of [, b]. By using the continuity of I nd J, for every f C[, b] nd every discrete mesure µ M[, b] we hve ( I(f, µ) = I f, ) c k δ xk = c k I(f, δ xk ) k 1 k 1 = ( c k J(f, δ xk ) = J f, ) c k δ xk k 1 k 1 = J(f, µ). Since the Bnch subspce M[, b] d of ll discrete mesures is wekly dense in M[, b] nd the functionls I(f, ) nd J(f, ) re lso wekly continuous we conclude tht I(f, µ) = J(f, µ) for every f C[, b] nd µ M[, b]. Theorem 2.3. Let f : [, b] R be such tht f (n 1) hs bounded vrition for some n 1. Then for every µ-hrmonic sequence (P n, n 1) we hve (2.1) f(t)dµ(t) = µ({})f() + S n + R n, where (2.2) S n = nd n ( 1) [ k 1 P k (b)f (k 1) (b) P k ()f (k 1) () ] (2.3) R n = ( 1) n P n (t)df (n 1) (t). J. Inequl. Pure nd Appl. Mth., 8(4) (2007), Art. 93, 13 pp. http://jipm.vu.edu.u/

INTEGRATION-BY-PARTS FORMULA 5 Proof. By prtil integrtion, for n 2, we hve R n = ( 1) n P n (t)df (n 1) (t) By Lemm 2.2 we hve R 1 = = ( 1) [ n P n (b)f (n 1) (b) P n ()f (n 1) () ] ( 1) n P n 1 (t)f (n 1) (t)dt = ( 1) n [ P n (b)f (n 1) (b) P n ()f (n 1) () ] + R n 1. P 1 (t)df(t) = [P 1 (b)f(b) P n ()f()] + = [P 1 (b)f(b) P n ()f()] + = [P 1 (b)f(b) P n ()f()] + f(t)dp 1 (t) f(t)dˇµ 1 (t) Therefore, by itertion, we hve n R n = ( 1) [ k P k (b)f (k 1) (b) P k ()f (k 1) () ] + which proves our ssertion. f(t)dµ(t) µ({})f(). f(t)dµ(t) µ({})f(), Remrk 2.4. By Remrk 2.1 we see tht identity (2.1) is generliztion of the integrtion-byprts formul (1.1). Corollry 2.5. Let f : [, b] R be such tht f (n 1) hs bounded vrition for some n 1. Then for every µ M[, b] we hve f(t)dµ(t) = Šn + Řn, where nd Š n = n ( 1) k 1ˇµ k (b)f (k 1) (b) Ř n = ( 1) n ˇµ n (t)df (n 1) (t). Proof. Apply the theorem bove for the µ-hrmonic sequence (ˇµ n, n 1) nd note tht ˇµ n () = 0, for n 2. Corollry 2.6. Let f : [, b] R be such tht f (n 1) hs bounded vrition for some n 1. Then for every x [, b] we hve n (x b) k 1 f(x) = f (k 1) (b) + R n (x), (k 1)! J. Inequl. Pure nd Appl. Mth., 8(4) (2007), Art. 93, 13 pp. http://jipm.vu.edu.u/

6 A. ČIVLJAK, LJ. DEDIĆ, AND M. MATIĆ where R n (x) = ( 1)n (t x) n 1 df (n 1) (t). (n 1)! [x,b] Proof. Apply Corollry 2.5 for µ = δ x nd note tht in this cse for k 1. ˇµ k (t) = (t x)k 1, x t b, nd ˇµ k (t) = 0, t < x, (k 1)! Corollry 2.7. Let f : [, b] R be such tht f (n 1) hs bounded vrition for some n 1. Further, let (c m, m 1) be sequence in R such tht c m < nd let {x m ; m 1} [, b]. Then c m f(x m ) = n (x m b) k 1 c m f (k 1) (b) + c m R n (x m ), (k 1)! where R n (x m ) is from Corollry 2.6. Proof. Apply Corollry 2.5 for the discrete mesure µ = c mδ xm. 3. SOME OSTROWSKI-TYPE INEQUALITIES In this section we shll use the sme nottions s bove. Theorem 3.1. Let f : [, b] R be such tht f (n 1) is L-Lipschitzin for some n 1. Then for every µ-hrmonic sequence (P n, n 1) we hve b (3.1) f(t)dµ(t) µ({})f() S n L P n (t) dt, where S n is defined by (2.2). Proof. By Theorem 2.3 we hve R n = which proves our ssertion. b P n (t)df (n 1) (t) L P n (t) dt, Corollry 3.2. If f is L-Lipschitzin, then for every c R nd µ M[, b] we hve b f(t)dµ(t) µ([, b])f(b) c [f(b) f()] L c + ˇµ 1 (t) dt. Proof. Put n = 1 in the theorem bove nd note tht P 1 (t) = c + ˇµ 1 (t), for some c R. Corollry 3.3. If f is L-Lipschitzin, then for every c 0 nd µ 0 we hve f(t)dµ(t) µ([, b])f(b) c [f(b) f()] L [c(b ) + ˇµ 2 (b)] L(b )(c + µ ). J. Inequl. Pure nd Appl. Mth., 8(4) (2007), Art. 93, 13 pp. http://jipm.vu.edu.u/

INTEGRATION-BY-PARTS FORMULA 7 Proof. Apply Corollry 3.2 nd note tht in this cse b c + ˇµ 1 (t) dt = b [c + ˇµ 1 (t)] dt = c(b ) + ˇµ 2 (b) c(b ) + (b ) µ = (b )(c + µ ). Corollry 3.4. Let f be L-Lipschitzin, (c m, m 1) sequence in [0, ) such tht c m <, nd let {x m ; m 1} [, b]. Then for every c 0 we hve [ c m [f(b) f(x m )] + c [f(b) f()] L c(b ) + ] c m (b x m ) [ L(b ) c + ] c m. Proof. Apply Corollry 3.3 for the discrete mesure µ = c mδ xm. Corollry 3.5. If f is L-Lipschitzin nd µ 0, then f(t)dµ(t) µ([, x])f() µ((x, b])f(b) for every x [, b]. Proof. Apply Corollry 3.2 for c = ˇµ 1 (x). Then nd b ˇµ 1 (x) + ˇµ 1 (t) dt = c + ˇµ 1 (b) = µ((x, b]), x L [(2x b)ˇµ 1 (x) 2ˇµ 2 (x) + ˇµ 2 (b)], ˇµ 1 (x) = µ([, x]) (ˇµ 1 (x) ˇµ 1 (t)) dt + b = (2x b)ˇµ 1 (x) 2ˇµ 2 (x) + ˇµ 2 (b). x (ˇµ 1 (t) ˇµ 1 (x)) dt Corollry 3.6. Let f : [, b] R be such tht f (n 1) is L-Lipschitzin for some n 1. Then for every µ M[, b] we hve b f(t)dµ(t) Šn L (b )n ˇµ n (t) dt L µ, n! where Šn is from Corollry 2.5. Proof. Apply the theorem bove for the µ-hrmonic sequence (ˇµ n, n 1). J. Inequl. Pure nd Appl. Mth., 8(4) (2007), Art. 93, 13 pp. http://jipm.vu.edu.u/

8 A. ČIVLJAK, LJ. DEDIĆ, AND M. MATIĆ Corollry 3.7. Let f : [, b] R be such tht f (n 1) is L-Lipschitzin for some n 1. Then for every x [, b] we hve n f(x) (x b) k 1 f (k 1) (b x)n (b) L. (k 1)! n! Proof. Apply Corollry 3.6 for µ = δ x nd note tht in this cse for k 1. ˇµ k (t) = (t x)k 1, x t b, nd ˇµ k (t) = 0, t < x, (k 1)! Corollry 3.8. Let f : [, b] R be such tht f (n 1) is L-Lipschitzin, for some n 1. Further, let (c m, m 1) be sequence in R such tht c m < nd let {x m ; m 1} [, b]. Then c m f(x m ) n (x m b) k 1 c m f (k 1) (b) (k 1)! L c m (b x m ) n n! L n! (b )n c m. Proof. Apply Corollry 3.6 for the discrete mesure µ = c mδ xm. Theorem 3.9. Let f : [, b] R be such tht f (n 1) hs bounded vrition for some n 1. Then for every µ-hrmonic sequence (P n, n 1) we hve f(t)dµ(t) µ({})f() S n mx P n(t) (f (n 1) ), t where b (f (n 1) ) is the totl vrition of f (n 1) on [, b]. Proof. By Theorem 2.3 we hve R n = P n (t)df (n 1) (t) mx P n(t) t which proves our ssertion. (f (n 1) ), Corollry 3.10. If f is function of bounded vrition, then for every c R nd µ M[, b] we hve f(t)dµ(t) µ([, b])f(b) c [f(b) f()] mx c + ˇµ 1(t) (f). t Proof. Put n = 1 in the theorem bove. J. Inequl. Pure nd Appl. Mth., 8(4) (2007), Art. 93, 13 pp. http://jipm.vu.edu.u/

INTEGRATION-BY-PARTS FORMULA 9 Corollry 3.11. If f is function of bounded vrition, then for every c 0 nd µ 0 we hve f(t)dµ(t) µ([, b])f(b) c [f(b) f()] [c + µ ] (f). Proof. In this cse we hve mx c + ˇµ 1 (t) = c + ˇµ 1 (b) = c + µ. t Corollry 3.12. Let f be function of bounded vrition, (c m, m 1) sequence in [0, ) such tht c m < nd let {x m ; m 1} [, b]. Then for every c 0 we hve [ c m [f(b) f(x m )] + c [f(b) f()] c + ] c m (f). Proof. Apply Corollry 3.11 for the discrete mesure µ = c mδ xm. Corollry 3.13. If f is function of bounded vrition nd µ 0, then we hve f(t)dµ(t) µ([, x])f() µ((x, b])f(b) Proof. Apply Corollry 3.11 for c = ˇµ 1 (x). Then mx c + ˇµ 1 (t) = mx ˇµ 1 (t) ˇµ 1 (x) t t 1 2 [ˇµ 1(b) ˇµ 1 () + ˇµ 1 () + ˇµ 1 (b) 2ˇµ 1 (x) ] = mx{ˇµ 1 (x) ˇµ 1 (), ˇµ 1 (b) ˇµ 1 (x)} (f). = 1 2 [ˇµ 1(b) ˇµ 1 () + ˇµ 1 () + ˇµ 1 (b) 2ˇµ 1 (x) ]. Corollry 3.14. Let f : [, b] R be such tht f (n 1) hs bounded vrition for some n 1. Then for every µ M[, b] we hve f(t)dµ(t) Šn mx ˇµ n(t) (f (n 1) ) t where Šn is from Corollry 2.5. (b )n 1 (n 1)! µ (f (n 1) ), Proof. Apply the theorem bove for the µ-hrmonic sequence (ˇµ n, n 1). J. Inequl. Pure nd Appl. Mth., 8(4) (2007), Art. 93, 13 pp. http://jipm.vu.edu.u/

10 A. ČIVLJAK, LJ. DEDIĆ, AND M. MATIĆ Corollry 3.15. Let f : [, b] R be such tht f (n 1) hs bounded vrition for some n 1. Then for every x [, b] we hve n f(x) (x b) k 1 f (k 1) (b x)n 1 (b) (f (n 1) ). (k 1)! (n 1)! Proof. Apply Corollry 3.14 for µ = δ x nd note tht in this cse mx ˇµ n (t) = t (b x)n 1. (n 1)! Corollry 3.16. Let f : [, b] R be such tht f (n 1) hs bounded vrition for some n 1. Further, let (c m, m 1) be sequence in R such tht c m < nd let {x m ; m 1} [, b]. Then c m f(x m ) n (x m b) k 1 c m f (k 1) (b) (k 1)! 1 (f (n 1) ) c m (b x m ) n 1 (n 1)! (b )n 1 (f (n 1) ) c m (n 1)! Proof. Apply Corollry 3.14 for the discrete mesure µ = c mδ xm. Theorem 3.17. Let f : [, b] R be such tht f (n) L p [, b] for some n 1. Then for every µ-hrmonic sequence (P n, n 1) we hve f(t)dµ(t) µ({})f() S n P n q f (n) p, where p, q [1, ] nd 1/p + 1/q = 1. Proof. By Theorem 2.3 nd the Hölder inequlity we hve R n = P n (t)df (n 1) (t) = P n (t)f (n) (t)dt ( b ) 1 ( P n (t) q q b dt f (n) (t) p dt = P n q f (n) p. Remrk 3.18. We see tht the inequlity of the theorem bove is generliztion of inequlity (1.2). ) 1 p J. Inequl. Pure nd Appl. Mth., 8(4) (2007), Art. 93, 13 pp. http://jipm.vu.edu.u/

INTEGRATION-BY-PARTS FORMULA 11 Corollry 3.19. Let f : [, b] R be such tht f (n) L p [, b] for some n 1, nd µ M[, b]. Then f(t)dµ(t) Šn ˇµ n q f (n) p where p, q [1, ] nd 1/p + 1/q = 1. (b ) n 1+1/q (n 1)! [(n 1)q + 1] 1/q µ f (n) p, Proof. Apply the theorem bove for the µ-hrmonic sequence (ˇµ n, n 1). Corollry 3.20. Let f : [, b] R be such tht f (n) L p [, b], for some n 1. Further, let (c m, m 1) be sequence in R such tht c m < nd let {x m ; m 1} [, b]. Then c m f(x m ) n c m (x m b) k 1 (k 1)! f (n) p (n 1)! [(n 1)q + 1] 1/q (b )n 1+1/q f (n) p (n 1)! [(n 1)q + 1] 1/q where p, q [1, ] nd 1/p + 1/q = 1. f (k 1) (b) c m (b x m ) n 1+1/q c m, Proof. Apply the theorem bove for the discrete mesure µ = c mδ xm. 4. SOME GRÜSS-TYPE INEQUALITIES Let f : [, b] R be such tht f (n) L [, b], for some n 1. Then for some rel constnts m n nd M n. m n f (n) (t) M n, t [, b],.e. Theorem 4.1. Let f : [, b] R be such tht f (n) L [, b], for some n 1. Further, let (P k, k 1) be µ-hrmonic sequence such tht P n+1 () = P n+1 (b), for tht prticulr n. Then f(t)dµ(t) µ({})f() S n M n m n 2 b P n (t) dt. Proof. Apply Theorem 2.3 for the specil cse when f (n 1) is bsolutely continuous nd its derivtive f (n), existing.e., is bounded.e. Define the mesure ν n by Then ν n ([, b]) = b dν n (t) = P n (t) dt. P n (t) dt = P n+1 () P n+1 (b) = 0, J. Inequl. Pure nd Appl. Mth., 8(4) (2007), Art. 93, 13 pp. http://jipm.vu.edu.u/

12 A. ČIVLJAK, LJ. DEDIĆ, AND M. MATIĆ which mens tht ν n is blnced. Further, nd by [1, Theorem 2] which proves our ssertion. R n = ν n = b b M n m n 2 = M n m n 2 P n (t) dt P n (t) f (n) (t)dt ν n b P n (t) dt, Corollry 4.2. Let f : [, b] R be such tht f (n) L [, b], for some n 1. Then for every (n + 1)-blnced mesure µ M[, b] we hve f(t)dµ(t) Šn M n m b n ˇµ n (t) dt 2 where Šn is from Corollry 2.5. M n m n 2 (b ) n µ, n! Proof. Apply Theorem 4.1 for the µ-hrmonic sequence (ˇµ k, k 1) nd note tht the condition P n+1 () = P n+1 (b) reduces to ˇµ n+1 (b) = 0, which mens tht µ is (n + 1)-blnced. Corollry 4.3. Let f : [, b] R be such tht f (n) L [, b] for some n 1. Further, let (c m, m 1) be sequence in R such tht c m < nd let {x m ; m 1} [, b] stisfy the condition c m (b x m ) n = 0. Then c m f(x m ) n (x m b) k 1 c m f (k 1) (b) (k 1)! M n m n c m (b x m ) n 2n! M n m n (b ) n c m. 2n! Proof. Apply Corollry 4.2 for the discrete mesure µ = c mδ xm. J. Inequl. Pure nd Appl. Mth., 8(4) (2007), Art. 93, 13 pp. http://jipm.vu.edu.u/

INTEGRATION-BY-PARTS FORMULA 13 Corollry 4.4. Let f : [, b] R be such tht f (n) L [, b] for some n 1. Then for every µ M[, b], such tht ll k-moments of µ re zero for k = 0,..., n, we hve f(t)dµ(t) M n m b n ˇµ n (t) dt 2 M n m n (b ) n µ. 2 n! Proof. By [1, Theorem 5], the condition m k (µ) = 0, k = 0,..., n is equivlent to ˇµ k (b) = 0, k = 1,..., n + 1. Apply Corollry 4.2 nd note tht in this cse Šn = 0. Corollry 4.5. Let f : [, b] R be such tht f (n) L [, b] for some n 1. Further, let (c m, m 1) be sequence in R such tht c m < nd let {x m ; m 1} [, b]. If then c m = c m x m = = c m x n m = 0, c m f(x m ) M n m n c m (b x m ) n 2n! M n m n (b ) n c m. 2n! Proof. Apply Corollry 4.4 for the discrete mesure µ = c mδ xm. REFERENCES [1] A. ČIVLJAK, LJ. DEDIĆ AND M. MATIĆ, Euler-Grüss type inequlities involving mesures, submitted. [2] A.ČIVLJAK, LJ. DEDIĆ AND M. MATIĆ, Euler hrmonic identities for mesures, Nonliner Functionl Anl. & Applics., 12(1) (2007). [3] Lj. DEDIĆ, M. MATIĆ, J. PEČARIĆ AND A. AGLIĆ ALJINOVIĆ, On weighted Euler hrmonic identities with pplictions, Mth. Inequl. & Appl., 8(2), (2005), 237 257. [4] S.S. DRAGOMIR, The generlised integrtion by prts formul for Appell sequences nd relted results, RGMIA Res. Rep. Coll., 5(E) (2002), Art. 18. [ONLINE: http://rgmi.vu.edu.u/ v5(e).html]. [5] P. CERONE, Generlised Tylor s formul with estimtes of the reminder, RGMIA Res. Rep. Coll., 5(2) (2002), Art. 8. [ONLINE: http://rgmi.vu.edu.u/v5n2.html]. [6] P. CERONE, Perturbted generlised Tylor s formul with shrp bounds, RGMIA Res. Rep. Coll., 5(2) (2002), Art. 6. [ONLINE: http://rgmi.vu.edu.u/v5n2.html]. [7] S.S. DRAGOMIR AND A. SOFO, A perturbed version of the generlised Tylor s formul nd pplictions, RGMIA Res. Rep. Coll., 5(2) (2002), Art. 16. [ONLINE: http://rgmi.vu.edu. u/v5n2.html]. J. Inequl. Pure nd Appl. Mth., 8(4) (2007), Art. 93, 13 pp. http://jipm.vu.edu.u/