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1 Bnch J. Mth. Anl. 4 (2010), no. 1, Bnch Journl of Mthemticl Anlysis ISSN: (electronic) GENERALIZATIONS OF OSTROWSKI INEQUALITY VIA BIPARAMETRIC EULER HARMONIC IDENTITIES FOR MEASURES AMBROZ ČIVLJAK1 AND LJUBAN DEDIĆ2 Dedicted to Professor Lrs-Erik Persson on the occsion of his 65th birthdy Communicted by Z. Páles Abstrct. Some generliztions of Ostrowski inequlity re given by using biprmetric Euler identities involving rel Borel mesures nd hrmonic sequences of functions. 1. Introduction The following Ostrowski inequlity, see [5], is well known: f(x) 1 b [ ( ) f(t)dt b 1 x +b 2 ] (b )M, x b, (b ) 2 where f : [, b] R is differentible function such tht f (x) M, for every x [, b]. The constnt 1 is the best possible. In other words, Ostrowski s 4 inequlity gives us n estimte for the devition of the vlues of smooth function from its men vlue. It hs been generlized in recent yers in number of wys. In this pper we shll present some new generliztions of Ostrowskitype inequlities by using biprmetric Euler identities which involve rel Borel mesures nd hrmonic sequences of functions. Dte: Received: 25 September 2009; Revised: 1 Mrch 2010; Accepted: 11 April Corresponding uthor Mthemtics Subject Clssifiction. Primry 26D15. Secondry 28A25, 26D20, 26D99. Key words nd phrses. Ostrowski inequlity, hrmonic sequences, biprmetric Euler identities. 170

2 BIPARAMETRIC EULER HARMONIC IDENTITIES 171 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. In the rest of the pper we use the nottion F (s)dµ(s) to denote the Lebesgue integrl of F over [, b] with respect to the mesure µ, while for given function ϕ : [, b] R of bounded vrition F (s)dϕ(s) denotes Lebesgue Stieltjes integrl of F over [, b] with respect to ϕ. Also, by b F (s)ds we denote the usul Lebesgue integrl of F over [, b]. For µ M[, b] define the function ˇµ n : [, b] R, n 1, by 1 ˇµ n (t) (t s) n 1 dµ(s). (n 1)! For n 1, ˇµ 1 (t) [,t] [,t] dµ(s) µ([, t]), t b, which mens tht ˇµ 1 is equl to the distribution function of µ. Substituting ˇµ n (s) 1 (n 1)! [,s] (s u)n 1 dµ(u) in t ˇµ n(s)ds nd using the Fubini theorem we esily get the formul ˇµ n+1 (t) t ˇµ n (s)ds, t b, n 1. It mens tht for n 1, ˇµ n+1 is differentible t lmost ll points of [, b] nd ˇµ n+1 ˇµ n lmost everywhere on [, b] with respect to Lebesgue mesure. Substituting ˇµ 1 (s) dµ(u) in t (t s)n 2ˇµ [,s] 1 (s)ds nd using the Fubini theorem once gin we esily get the following formul ˇµ n (t) 1 (n 2)! t (t s) n 2ˇµ 1 (s)ds, t b, n 2. From this formul we get immeditely tht ˇµ n () 0, n 2. Also, note tht function g(s) (t s) n 1 is nonincresing on [, t] so tht from the first expression for ˇµ n (t) we get the estimte ˇµ n (t) (t )n 1 (n 1)! µ, t b, n 1, where µ denotes the totl vrition of µ. A sequence of functions P n : [, b] R, n 1, is clled µ-hrmonic sequence of functions on [, b] if for some c R, nd P n+1 (t) P n+1 () + P 1 (t) c + ˇµ 1 (t), t b, t P n (s)ds, t b, n 1. Since P n+1, n 1 is defined s n indefinite Lebesgue integrl of P n, it is well known tht P n+1, n 1 is bsolutely continuous function, P n+1 P n,.e. on [, b] with respect to Lebesgue mesure,

3 172 A. ČIVLJAK, LJ. DEDIĆ nd for every f C[, b] we hve f(t)dp n+1 (t) b f(t)p n (t)dt, n 1. The sequence (ˇµ n, n 1) is n exmple of µ-hrmonic sequence of functions on [, b]. Assume tht (P n, n 1) is µ-hrmonic sequence of functions on [, b]. Define P n, for n 1, to be periodic function of period 1, relted to P n s nd P n(t) P n( + (b )t) (b ) n, 0 t < 1, P n(t + 1) P n(t), t R. Thus, for n 2, P n is continuous on R\Z nd hs jump of α n P n() P n (b) (b ) n t every k Z, whenever α n 0. Note tht for n 1, ( Pn+1) P n.e. on R. Let f : [, b] R be such tht f (n 1) is continuous function of bounded vrition on [, b] for some n 1. In recent pper [1] the following identity hs been proved: µ([, b])f(x) f x (t)dµ(t) + S n (x) + R n (x), (1.1) where S m (x) m P k (x) [ f (k 1) (b) f (k 1) () ] + m [P k () P k (b)] f (k 1) (x), for 1 m n, with convention S 1 (x) P 1 (x) [f(b) f()], nd f( + x t), t x f x (t) f(b + x t), x < t b, while R n (x) (b ) n Pn k2 ( ) x t df (n 1) (t) b for every x [, b]. Identity (1.1) is clled the generlized Euler hrmonic identity. It hs been used in [1] to prove some generliztions of Ostrowski s inequlity. The reder cn find further references to some recent results on generliztions nd pplictions of Euler identities in [2], [4] nd [3]. The ( im of this pper is to generlize formul (1.1) by replcing the sequence (Pn x t b ), n 1) with more generl sequence of functions, nd using them to prove some further generliztions of Ostrowski s inequlity.

4 BIPARAMETRIC EULER HARMONIC IDENTITIES Biprmetric Euler hrmonic identities For µ M[, b] let (P n, n 1) be µ-hrmonic sequence of functions on [, b]. For x, y [, b], x y, define function K n : [, b] 3 R, for n 1, by Pn (b + x y + t), t + y x K n (x, y, t) P n (x y + t), + y x < t b, (2.1) for y x < b, nd Pn (t), t < b K n (, b, t) P n (), t b. (2.2) Thus, for n 2, K n (x, y, ) is continuous on [, b] \ + y x} nd hs jump of P n () P n (b) t +y x. Note tht K n (x, y, ), n 1 is function of bounded vrition nd for n 1 K n+1(x, y, ) K n (x, y, ).e. on [, b] with respect to Lebesgue mesure. Also note tht K n (x, y, ) K n (x, y, b) P n (b + x y), n 1. Lemm 2.1. For every f C[, b] nd n 2 we hve b f(t)dk n (x, y, t) f(t)k n 1 (x, y, t)dt + f( + y x) [P n () P n (b)]. Proof. Follows directly from properties of Lebesgue-Stieltjes integrl of continuous function f over [, b] with respect to K n, nd given properties of the function K n. Nmely, the function K n (x, y, ), n 2 is lmost everywhere differentible on [, b] nd its derivtive is equl to K n 1 (x, y, ).e. on [, b] with respect to Lebesgue mesure. Further, it hs jump t +y x of mgnitude P n () P n (b), which proves our ssertion. Lemm 2.2. For every µ M[, b] nd f C[, b] we hve f(t)dk 1 (x, y, t) f x,y (t)dµ(t) f( + y x)µ([, b]), (2.3) where f x,y (t) f(y x + t), t b + x y f( b + y x + t), b + x y < t b. (2.4) Proof. Define I, J : C[, b] M[, b] R by I(f, µ) f(t)dk 1 (x, y, t) nd J(f, µ) f x,y (t)dµ(t) f( + y x)µ([, b]). Then I nd J re continuous biliner functionls with 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 µ M[, b]. Since P 1 (t) c + µ([, t]), t b, for some constnt c, nd obviously the integrl on the left hnd side of (2.3) is independent of the choice of the constnt

5 174 A. ČIVLJAK, LJ. DEDIĆ c, we my ssume tht c 0. Therefore, from (2.1) nd (2.2) we esily see tht for n 1 µ([, b + x y + t]), t + y x K 1 (x, y, t) µ([, x y + t]), + y x < t b, (2.5) for y x < b, nd µ([, t]), t < b K 1 (, b, t) µ(}), t b. (2.6) (1) For z [, b] let µ δ z be the Dirc mesure t z, i.e., the mesure defined by f(t)dδ z (t) f(z). If z [, b] nd z b + x y, from (2.5) nd (2.6) we get 0, + y x < t < z + y x K 1 (x, y, t) 1, ( t + y x) or (z + y x t b), for y x < b, nd Now, by simple clcultion we hve K 1 (, b, t) 1, t b. I(f, δ z ) f(y x + z) f( + y x) f(y x + t)dδ z (t) f( + y x)δ z ([, b]) for y x < b, nd f x,y (t)dδ z (t) f( + y x)δ z ([, b]) J(f, δ z ), I(f, δ z ) I(f, δ ) 0 f(b) f( + b ) f(b)dδ (t) f( + b )δ z ([, b]) f,b (t)dδ (t) f( + y x)δ ([, b]) J(f, δ ) J(f, δ z ), for y x b. Similrly, if z [, b] nd b + x y < z b, from (2.5) nd (2.6) we find K 1 (x, y, t) 0, ( t < + y x b + z) or ( + y x < t b) 1, + y x b + z t + y x, for y x < b, nd K 1 (, b, t) 0, ( t < z) or (t b) 1, z t < b.

6 BIPARAMETRIC EULER HARMONIC IDENTITIES 175 Now, by nlogous clcultion we hve I(f, δ z ) f( b + y x + z) f( + y x) f( b + y x + t)dδ z (t) f( + y x)δ z ([, b]) for y x < b, nd f x,y (t)dδ z (t) f( + y x)δ z ([, b]) J(f, δ z ), I(f, δ z ) f(z) f(b) f(t)dδ z (t) f( + b )δ z ([, b]) f,b (t)dδ z (t) f( + y x)δ z ([, b]) J(f, δ z ), for y x b. Therefore, for every f C [, b] nd every z [, b] we hve I(f, δ z ) J(f, δ z ). (2) Every discrete mesure µ M[, b], with finite support, is liner combintion of Dirc mesures, i.e., it hs the form µ n c kδ xk, for some rel numbers c k, nd x k [, b]. By linerity of I nd J, we get I(f, µ) I(f, c k δ xk ) c k I(f, δ xk ) c k J(f, δ xk ) J(f, c k δ xk ) J(f, µ). for every f C[, b] nd every discrete mesure µ M[, b] with finite support. (3) Let T be the miniml topology on M[, b] such tht liner functionls µ F dµ re continuous, for every bounded Borel function F : [, b] R. By the definition we see tht T contins the wek topology on M[, b] nd is contined in the wek topology on M[, b]. Further, the curve x δ x is bounded nd T - mesurble since x F dδ x F (x) is mesurble by ssumption. Therefore, the integrl δ x dµ(x) exists in the T topology, for every µ M[, b]. It is esy to see tht this integrl is equl to µ, i.e. δx dµ(x) µ, for every mesure µ M[, b], which mens tht µ is T -limit of sequence of discrete mesures with finite support. Thus, we conclude tht the subspce of ll discrete mesures with finite support is T -dense in M[, b], nd therefore the functionls I(f, ) nd J(f, ) re equl, for every f C[, b], since they re equl on dense subspce nd they re T -continuous. This completes the proof. Theorem 2.3. For µ M[, b] let (P n, n 1) be µ-hrmonic sequence of functions on [, b] nd f : [, b] R such tht f (n 1) is continuous function of bounded vrition for some n 1. Then we hve f x,y (t)dµ(t) µ(})f( + y x) + S n (x, y) R n (x, y),

7 176 A. ČIVLJAK, LJ. DEDIĆ for every x, y [, b], x y, where f x,y (t) is defined by (2.4), S n (x, y) ( 1) k P k (b + x y) [ f (k 1) (b) f (k 1) () ] + ( 1) k f (k 1) ( + y x) [P k (b) P k ()] nd R n (x, y) ( 1) n K n (x, y, t) df (n 1) (t). Proof. For 1 k n consider the integrl R k (x, y) ( 1) k K k (x, y, t) df (k 1) (t). Integrting by prts we get R k (x, y) ( 1) k K k (x, y, t) f (k 1) (t) b (2.7) ( 1) k f (k 1) (t)dk k (x, y, t). For every k 2, by Lemm 2.1, we get R k (x, y) ( 1) k P k (b + x y) [ f (k 1) (b) f (k 1) () ] ( 1) k f (k 1) ( + y x) [P k () P k (b)] b ( 1) k f (k 1) (t)k k 1 (x, y, t) dt ( 1) k P k (b + x y) [ f (k 1) (b) f (k 1) () ] (2.8) +( 1) k f (k 1) ( + y x) [P k (b) P k ()] +R k 1 (x, y), since K k (x, y, ) K k (x, y, b) P k (b + x y). By Lemm 2.2, for k 1, (2.7) becomes R 1 (x, y) P 1 (b + x y) [f(b) f()] + f(t)dk 1 (x, y, t) P 1 (b + x y) [f(b) f()] f( + y x)µ([, b]) (2.9) + f x,y (t)dµ(t)

8 BIPARAMETRIC EULER HARMONIC IDENTITIES 177 where f x,y (t) is defined by (2.4). From (2.8) nd (2.9) it follows, by itertion R n (x, y) since ( 1) k P k (b + x y) [ f (k 1) (b) f (k 1) () ] + ( 1) k f (k 1) ( + y x) [P k (b) P k ()] f( + y x)µ(}) + f x,y (t)dµ(t) f( + y x)µ([, b]) f( + y x) [P 1 (b) P 1 () + µ(})], which proves our ssertion. Corollry 2.4. Let f : [, b] R be such tht f (n 1) is continuous function of bounded vrition for some n 1. Then we hve f x,y (t)dµ(t) + Šn(x, y) Řn(x, y). for every x, y [, b], x y, where nd Š n (x, y) Ǩ n (x, y, t) for y x < b, while ( 1) k ˇµ k (b + x y) [ f (k 1) (b) f (k 1) () ] + ( 1) k f (k 1) ( + y x)ˇµ k (b), Ř n (x, y) ( 1) n Ǩ n (x, y, t) df (n 1) (t) ˇµn (b + x y + t), t + y x ˇµ n (x y + t), + y x < t b Ǩ n (, b, t) ˇµn (t), t < b ˇµ n (), t b. Proof. Apply the theorem bove to the specil cse P n ˇµ n, n 1, nd note tht ˇµ k () 0 for k 2. Corollry 2.5. Let f : [, b] R be such tht f (n 1) is continuous function of bounded vrition for some n 1. Then we hve b f(t)dt + S n (x, y) R n (x, y).

9 178 A. ČIVLJAK, LJ. DEDIĆ for every x, y [, b], x y, where ( 1) S k n (x, y) (b + x y) [ k f (k 1) (b) f (k 1) () ] k! ( 1) k + (b ) k f (k 1) ( + y x), k! R n (x, y) ( 1) n K n (x, y, t) df (n 1) (t) nd 1 K n (x, y, t) (b 2 + x y + n! t)n, t + y x 1 (x y + t n! )n, + y x < t b for y x < b, while K n (, b, t) 1 n! (t )n, < t < b 0, (t ) or (t b). Proof. Apply Corollry 2.4 in the specil cse when µ is the Lebesgue mesure on [, b]. In this cse (t )k ˇµ k (t), k 1 k! nd b b f x,y (t)dµ(t) f x,y (t)dt f(t)dt. 3. Generliztions of Ostrowski s inequlity In this section we use the identity obtined in Theorem 2.3 to prove number of Ostrowski-type inequlities which hold for clss of functions f whose derivtives f (n 1) re either L-Lipschitzin on [, b] or continuous nd of bounded vrition on [, b]. Anlogous results re obtined for clss of functions f possessing derivtives f (n) in L p [, b], 1 p. Throughout this section we use the sme nottions s bove. Lemm 3.1. For every µ-hrmonic sequence (P n, n 1) nd f C[, b] we hve b f(k n (x, y, t))dt b f(p n (t))dt. Proof. Follows from (2.1) nd (2.2) using simple clcultions, b f(k n (x, y, t))dt +y x b b+x y f(p n (b + x y + t))dt + f(p n (t))dt + b+x y b f(p n (t))dt +y x b f(p n (x y + t))dt f(p n (t))dt.

10 BIPARAMETRIC EULER HARMONIC IDENTITIES 179 Theorem 3.2. Let f : [, b] R be such tht f (n 1) is L-Lipschitzin on [, b] for some n 1. Then b f x,y (t)dµ(t) µ(})f( + y x) + S n (x, y) L P n (t) dt, for every x, y [, b], x y. Proof. By Lemm 3.1 we hve R n (x, y) L L b b K n (x, y, t) df (n 1) (t) K n (x, y, t) dt P n (t) dt. Therefore, our ssertion follows from Theorem 2.3. Corollry 3.3. If f is L-Lipschitzin on [, b], then for every x, y [, b], x y, nd c R we hve f x,y (t)dµ(t) µ([, b])f( + y x) [c + ˇµ 1 (b + x y)] [f(b) f()] L b c + ˇµ 1 (t) dt. Proof. Put n 1 in the theorem bove. Corollry 3.4. If f is L-Lipschitzin on [, b] nd µ 0, then for every x, y, z [, b], x y, we hve f x,y (t)dµ(t) µ([, b])f( + y x) [ˇµ 1 (b + x y) ˇµ 1 (z)] [f(b) f()] L [(2z b)ˇµ 1 (z) 2ˇµ 2 (z) + ˇµ 2 (b)]. Proof. Put c ˇµ 1 (z) in Corollry 3.3 nd note tht in this cse b ˇµ 1 (t) ˇµ 1 (z) dt (2z b)ˇµ 1 (z) 2ˇµ 2 (z) + ˇµ 2 (b). Corollry 3.5. Let f : [, b] R be such tht f (n 1) is L-Lipschitzin on [, b] for some n 1. Then for µ 0 we hve f x,y (t)dµ(t) + Šn(x, y) Lˇµ (b )n n+1(b) L µ, n! for every x, y [, b], x y.

11 180 A. ČIVLJAK, LJ. DEDIĆ Proof. Apply the theorem bove to the µ-hrmonic sequence (ˇµ n, n 1). Then b ˇµ n (t) dt b ˇµ n (t)dt ˇµ n+1 (b) (b )n n! µ. Corollry 3.6. Let f : [, b] R be such tht f (n 1) is L-Lipschitzin on [, b] for some n 1. Then we hve b f(t)dt + S n (x, y) )n+1 L(b, (n + 1)! for every x, y [, b], x y. Proof. Apply the corollry bove to the Lebesgue mesure on [, b]. Corollry 3.7. Let f : [, b] R be such tht f (n 1) is L-Lipschitzin on [, b] for some n 1. Then f(y x + z) + T n (x, y, z) L (b z)n, n! for every x, y, z [, b], x y nd z b + x y, where T n (x, y, z) k (b + x y z)k 1 [ ( 1) f (k 1) (b) f (k 1) () ] (k 1)! k (b z)k 1 + ( 1) f (k 1) ( + y x). (k 1)! Proof. Apply Corollry 3.5 to µ δ z, z b + x y. Corollry 3.8. Let f : [, b] R be such tht f (n 1) is L-Lipschitzin on [, b] for some n 1. Then f( b + y x + z) + T 2 n (x, y, z) (b z) n L, n! for every x, y, z [, b], x y nd b + x y < z b, where Tn(x, 2 k (b z)k 1 y, z) ( 1) f (k 1) ( + y x). (k 1)! Proof. Apply Corollry 3.5 to µ δ z, b + x y < z b. Theorem 3.9. Let f : [, b] R be such tht f (n 1) is continuous function of bounded vrition on [, b] for some n 1. Then f x,y (t)dµ(t) µ(})f( + y x) + S n (x, y) sup P n (t) V b (f (n 1) ), for every x, y [, b], x y. t

12 Proof. We hve BIPARAMETRIC EULER HARMONIC IDENTITIES 181 R n (x, y) K n (x, y, t) df (n 1) (t) sup K n (x, y, t) V b (f (n 1) ) t sup P n (t) V b (f (n 1) ). t Therefore, our ssertion follows from Theorem 2.3. Corollry If f is continuous function of bounded vrition on [, b], then for every x, y [, b], x y, nd c R we hve f x,y (t)dµ(t) µ([, b])f( + y x) for every x, y [, b], x y. [c + ˇµ 1 (b + x y)] [f(b) f()] sup c + ˇµ 1 (t) V b (f), t Proof. Put n 1 in the theorem bove. Corollry If f is continuous function of bounded vrition on [, b] nd µ 0, then for every x, y, z [, b], x y, we hve f x,y (t)dµ(t) µ([, b])f( + y x) [ˇµ 1 (b + x y) ˇµ 1 (z)] [f(b) f()] 1 2 [ˇµ 1(b) ˇµ 1 () + ˇµ 1 () + ˇµ 1 (b) 2ˇµ 1 (z) ] V b (f). Proof. Put c ˇµ 1 (z) in Corollry Then sup c + ˇµ 1 (t) sup ˇµ 1 (t) ˇµ 1 (z) t t 1 2 [ˇµ 1(b) ˇµ 1 () + ˇµ 1 () + ˇµ 1 (b) 2ˇµ 1 (z) ]. Corollry Let f : [, b] R be such tht f (n 1) is continuous function of bounded vrition on [, b] for some n 1. Then for µ 0 we hve f x,y (t)dµ(t) + Šn(x, y) ˇµ n (b)v b (f (n 1) ) for every x, y [, b], x y. (b )n 1 V b (f (n 1) ) µ, (n 1)!

13 182 A. ČIVLJAK, LJ. DEDIĆ Proof. Apply the theorem bove to the µ-hrmonic sequence (ˇµ n, n 1). Then sup ˇµ n (t) ˇµ n (b) t (b )n 1 (n 1)! µ. Corollry Let f : [, b] R be such tht f (n 1) is continuous function of bounded vrition on [, b] for some n 1. Then b f(t)dt + S n (x, y) (b )n V b (f (n 1) ), n! for every x, y [, b], x y. Proof. Apply the corollry bove to the Lebesgue mesure on [, b]. Corollry Let f : [, b] R be such tht f (n 1) is continuous function of bounded vrition on [, b] for some n 1. Then f(y x + z) + T n (x, y, z) (b z)n 1 V b (f (n 1) ), (n 1)! for every x, y, z [, b], x y nd z b + x y, where T n (x, y, z) is from Corollry 3.7. Proof. Apply Corollry 3.12 to µ δ z, z b + x y. Corollry Let f : [, b] R be such tht f (n 1) is continuous function of bounded vrition on [, b] for some n 1. Then f( b + y x + z) + T 2 n (x, y, z) (b z) n 1 V b (f (n 1) ), (n 1)! for every x, y, z [, b], x y nd b + x y < z b, where T 2 n(x, y, z) is from Corollry 3.8. Proof. Apply Corollry 3.12 to µ δ z, b + x y < z b. Theorem Let f : [, b] R be such tht f (n) is integrble for some n 1. Then f x,y (t)dµ(t) µ(})f( + y x) + S n (x, y) sup P n (t) f (n) 1, for every x, y [, b], x y. t Proof. Note tht in this cse nd pply Theorem 3.9. V b (f (n 1) ) b f (n) (t) dt f (n) 1,

14 BIPARAMETRIC EULER HARMONIC IDENTITIES 183 Theorem Let f : [, b] R be such tht f (n) L [, b] for some n 1. Then b f x,y (t)dµ(t) µ(})f( + y x) + S n (x, y) P n (t) dt f (n), for every x, y [, b], x y. Proof. In this cse f (n 1) is L-Lipschitzin with L f (n). Theorem Let f : [, b] R be such tht f (n) L p [, b] for some n 1 nd 1 < p <. Then f x,y (t)dµ(t) µ(})f( + y x) + S n (x, y) P n q f (n) p, for every x, y [, b], x y, where 1/p + 1/q 1. Proof. By pplying the Hölder inequlity we hve R n (x, y) which proves our ssertion. b K n (x, y, t) f (n) (t) dt ( b 1/q K n (x, y, t) dt) q f (n) p ( b 1/q P n (t) dt) q f (n) p, Corollry Let f : [, b] R be such tht f (n) L p [, b] for some n 1 nd 1 < p <. Then f x,y (t)dµ(t) + Šn(x, y) µ f (n) p (b ) n 1+1/q (n 1)! [(n 1)q + 1], 1/q for every x, y [, b], x y, where 1/p + 1/q 1. Proof. Apply the theorem bove to the µ-hrmonic sequence (ˇµ n, n 1) nd note tht b [ ] q µ b ˇµ n (t) q dt (t ) (n 1)q dt (n 1)! [ ] q µ (b ) (n 1)q+1 (n 1)! (n 1)q + 1. Corollry Let f : [, b] R be such tht f (n) L p [, b] for some n 1 nd 1 < p <. Then b f(t)dt + S n (x, y) f (n) p (b ) n+1/q n! [nq + 1], 1/q for every x, y [, b], x y, where 1/p + 1/q 1.

15 184 A. ČIVLJAK, LJ. DEDIĆ Proof. Apply the theorem bove to the Lebesgue mesure on [, b]. Corollry Let f : [, b] R be such tht f (n) L p [, b] for some n 1 nd 1 < p <. Then f(y x + z) + T n (x, y, z) f (n) p (b z) n 1+1/q (n 1)! [(n 1)q + 1], 1/q for every x, y, z [, b], x y nd z b + x y, where T n (x, y, z) is from Corollry 3.7. Proof. Apply Corollry 3.19 to µ δ z, z b + x y. Corollry Let f : [, b] R be such tht f (n) L p [, b] for some n 1 nd 1 < p <. Then f( b + y x + z) + T 2 n (x, y, z) f (n) p (b z) n 1+1/q (n 1)! [(n 1)q + 1], 1/q for every x, y, z [, b], x y nd b + x y < z b, where T 2 n(x, y, z) is from Corollry 3.8. Proof. Apply Corollry 3.19 to µ δ z, b + x y < z b. References [1] A. Čivljk, Lj. Dedić nd M. Mtić, Euler hrmonic identities for mesures, Nonliner Funct. Anl. Appl. 12 (2007), no. 3, [2] Lj. Dedić, M. Mtić nd J. Pečrić, On generliztions of Ostrowski inequlity vi some Euler-type identities, Mth. Inequl. Appl. 3 (2000), no. 3, [3] Lj. Dedić, M. Mtić, J. Pečrić nd A. Aglić Aljinović, On weighted Euler hrmonic identities with pplictions, Mth. Inequl. Appl. 8 (2005), no. 2, [4] Lj. Dedić, M. Mtić, J. Pečrić nd A. Vukelić, On generliztions of Ostrowski inequlity vi Euler hrmonic identities, J. Inequl. Appl., 7 (2002), no. 6, [5] A. Ostrowski, Über die Absolutbweichung einer differentiebren Funktion von ihrem Integrlmittelwert, Comment. Mth. Helv. 10 (1938), Americn College of Mngement nd Technology, Rochester Institute of Technology, Don Frn Bulić 6, Dubrovnik, Croti. E-mil ddress: civljk@cmt.hr 2 Deprtment of mthemtics, University of Split, Teslin 12, Split, Croti. E-mil ddress: ljubn@pmfst.hr