Received21June,2007;accepted15November,2007CommunicatedbyW.S.Cheung w = w , a.e.on [ a,b ] ,k =1 ,...,n. [ a,b ] , suchthat w ,w , ... ,w , for n ≥ 1 , ofrealabsolutelycontinuousfunctionsdefinedon Inthepaper[4],S.S.Dragomirintroducedthenotionofa w -Appellt

ON AN INTEGRATION-BY-PARTS FORMULA FOR MEASURES

A. ˇCIVLJAK, LJ. DEDI ´C, AND M. MATI ´C

AMERICANCOLLEGE OFMANAGEMENT ANDTECHNOLOGY

ROCHESTERINSTITUTE OFTECHNOLOGY

DONFRANABULICA6, 20000 DUBROVNIK

CROATIA

acivljak@acmt.hr DEPARTMENT OFMATHEMATICS

FACULTY OFNATURALSCIENCES, MATHEMATICS ANDEDUCATION

UNIVERSITY OFSPLIT

TESLINA12, 21000 SPLIT

CROATIA

ljuban@pmfst.hr DEPARTMENT OFMATHEMATICS

FACULTY OFNATURALSCIENCES, MATHEMATICS ANDEDUCATION

UNIVERSITY OFSPLIT

TESLINA12, 21000 SPLIT

CROATIA

mmatic@pmfst.hr

Received 21 June, 2007; accepted 15 November, 2007 Communicated by W.S. Cheung

ABSTRACT. An integration-by-parts formula, involving finite Borel measures supported by in- tervals on real line, is proved. Some applications to Ostrowski-type and Grüss-type inequalities are presented.

Key words and phrases: Integration-by-parts formula, Harmonic sequences, Inequalities.

2000 Mathematics Subject Classification. 26D15, 26D20, 26D99.

1. INTRODUCTION

In the paper [4], S.S. Dragomir introduced the notion of aw0-Appell type sequence of func- tions as a sequencew₀, w₁,. . ., w_n,forn ≥ 1,of real absolutely continuous functions defined on[a, b],such that

w_k⁰ =wk−1, a.e. on[a, b], k = 1, . . . , n.

210-07

For such a sequence the author proved a generalisation of Mitrinovi´c-Peˇcari´c integration-by- parts formula

(1.1)

Z b

a

w₀(t)g(t)dt =A_n+B_n, where

A_n=

n

X

k=1

(−1)^k−1

w_k(b)g^(k−1)(b)−w_k(a)g^(k−1)(a) and

B_n= (−1)ⁿ Z b

a

w_n(t)g⁽ⁿ⁾(t)dt,

for everyg : [a, b]→Rsuch thatg⁽ⁿ⁻¹⁾ is absolutely continuous on[a, b]andw_ng⁽ⁿ⁾ ∈L₁[a, b].

Using identity (1.1) the author proved the following inequality (1.2)

Z b

a

w₀(t)g(t)dt−A_n

≤ kw_nk_pkg⁽ⁿ⁾k_q,

forw_n ∈ L_p[a, b], g⁽ⁿ⁾ ∈ L_p[a, b], wherep, q ∈ [1,∞] and1/p+ 1/q = 1, giving explicitly some interesting special cases. For some similar inequalities, see also [5], [6] and [7]. The aim of this paper is to give a generalization of the integration-by-parts formula (1.1), by replacing the w0-Appell type sequence of functions by a more general sequence of functions, and to generalize inequality (1.2), as well as to prove some related inequalities.

2. INTEGRATION-BY-PARTSFORMULA FOR MEASURES

Fora, b∈R, a < b,letC[a, b]be the Banach space of all continuous functionsf : [a, b]→R with the max norm, andM[a, b]the Banach space of all real Borel measures on[a, b]with the total variation norm. Forµ∈M[a, b]define the functionµˇ_n: [a, b]→R, n≥1,by

ˇ

µ_n(t) = 1 (n−1)!

Z

[a,t]

(t−s)ⁿ⁻¹dµ(s).

Note that

ˇ

µ_n(t) = 1 (n−2)!

Z t

a

(t−s)ⁿ⁻²µˇ₁(s)ds, n≥2 and

|µˇ_n(t)| ≤ (t−a)ⁿ⁻¹

(n−1)! kµk, t ∈[a, b], n≥1.

The functionµˇ_nis differentiable,µˇ⁰_n(t) = ˇµn−1(t)andµˇ_n(a) = 0,for every n ≥ 2,while for n= 1

ˇ µ₁(t) =

Z

[a,t]

dµ(s) =µ([a, t]),

which means that µˇ₁(t) is equal to the distribution function of µ. A sequence of functions P_n : [a, b]→R, n≥1,is called aµ-harmonic sequence of functions on[a, b]if

P_n⁰(t) =Pn−1(t), n≥2; P1(t) =c+ ˇµ1(t), t∈[a, b],

for somec∈R.The sequence(ˇµ_n, n ≥1)is an example of aµ-harmonic sequence of functions on[a, b]. The notion of aµ-harmonic sequence of functions has been introduced in [2]. See also [1].

Remark 2.1. Letw₀ : [a, b] → Rbe an absolutely integrable function and letµ ∈ M[a, b]be defined by

dµ(t) = w₀(t)dt.

If (P_n, n ≥ 1)is a µ-harmonic sequence of functions on [a, b], thenw₀, P₁, . . . , P_n is a w₀- Appell type sequence of functions on[a, b].

Forµ∈M[a, b]letµ=µ₊−µ−be the Jordan-Hahn decomposition ofµ,whereµ₊andµ−

are orthogonal and positive measures. Then we have|µ|=µ++µ−and kµk=|µ|([a, b]) = kµ₊k+kµ−k=µ₊([a, b]) +µ−([a, b]).

The measureµ∈M[a, b]is said to be balanced ifµ([a, b]) = 0.This is equivalent to kµ+k=kµ−k= 1

2kµk.

Measureµ ∈ M[a, b]is calledn-balanced ifµˇ_n(b) = 0.We see that a1-balanced measure is the same as a balanced measure. We also write

m_k(µ) = Z

[a,b]

t^kdµ(t), k≥0 for thek-th moment ofµ.

Lemma 2.2. For everyf ∈C[a, b]andµ∈M[a, b]we have Z

[a,b]

f(t)dµˇ₁(t) = Z

[a,b]

f(t)dµ(t)−µ({a})f(a).

Proof. DefineI, J :C[a, b]×M[a, b]→Rby I(f, µ) =

Z

[a,b]

f(t)dˇµ₁(t) and

J(f, µ) = Z

[a,b]

f(t)dµ(t)−µ({a})f(a).

ThenI andJ are continuous bilinear functionals, since

|I(f, µ)| ≤ kfk kµk, |J(f, µ)| ≤2kfk kµk.

Let us prove that I(f, µ) = J(f, µ)for every f ∈ C[a, b]and every discrete measure µ ∈ M[a, b].

Forx∈[a, b]letµ=δ_xbe the Dirac measure atx,i.e. the measure defined by R

[a,b]

f(t)dδ_x(t) = f(x).

If a < x≤b, then ˇ

µ1(t) =δx([a, t]) =

( 0, a≤t < x 1, x≤t≤b and by a simple calculation we have

I(f, δ_x) = Z

[a,b]

f(t)dˇµ₁(t) =f(x) = R

[a,b]

f(t)dδ_x(t)−0

= R

[a,b]

f(t)dδx(t)−δx({a})f(a) = J(f, δx).

Similarly, ifx=a, then ˇ

µ₁(t) = δ_a([a, t]) = 1, a≤t≤b and by a similar calculation we have

I(f, δ_a) = Z

[a,b]

f(t)dˇµ₁(t) = 0 =f(a)−f(a)

= R

[a,b]

f(t)dδa(t)−δa({a})f(a) = J(f, δx).

Therefore, for every f ∈ C[a, b] and every x ∈ [a, b] we haveI(f, δ_x) = J(f, δ_x). Every discrete measureµ∈M[a, b]has the form

µ=X

k≥1

c_kδ_xk,

where(ck, k ≥1)is a sequence inRsuch that X

k≥1

|c_k|<∞, and{x_k;k ≥1}is a subset of[a, b].

By using the continuity ofI and J, for every f ∈ C[a, b]and every discrete measure µ ∈ M[a, b]we have

I(f, µ) = I f,X

k≥1

c_kδ_xk

!

=X

k≥1

c_kI(f, δ_xk)

=X

k≥1

c_kJ(f, δ_xk) =J f,X

k≥1

c_kδ_xk

!

=J(f, µ).

Since the Banach subspace M[a, b]_d of all discrete measures is weakly^∗ dense in M[a, b]

and the functionalsI(f,·)andJ(f,·)are also weakly^∗continuous we conclude thatI(f, µ) =

J(f, µ)for everyf ∈C[a, b]andµ∈M[a, b].

Theorem 2.3. Let f : [a, b] → Rbe such thatf⁽ⁿ⁻¹⁾ has bounded variation for somen ≥ 1.

Then for everyµ-harmonic sequence(P_n, n ≥1)we have (2.1)

Z

[a,b]

f(t)dµ(t) =µ({a})f(a) +S_n+R_n, where

(2.2) S_n =

n

X

k=1

(−1)^k−1

P_k(b)f^(k−1)(b)−P_k(a)f^(k−1)(a) and

(2.3) Rn= (−1)ⁿ

Z

[a,b]

Pn(t)df⁽ⁿ⁻¹⁾(t).

Proof. By partial integration, forn ≥2,we have Rn = (−1)ⁿ

Z

[a,b]

Pn(t)df⁽ⁿ⁻¹⁾(t)

= (−1)ⁿ

P_n(b)f⁽ⁿ⁻¹⁾(b)−P_n(a)f⁽ⁿ⁻¹⁾(a)

−(−1)ⁿ Z

[a,b]

Pn−1(t)f⁽ⁿ⁻¹⁾(t)dt

= (−1)ⁿ

P_n(b)f⁽ⁿ⁻¹⁾(b)−P_n(a)f⁽ⁿ⁻¹⁾(a)

+Rn−1. By Lemma 2.2 we have

R₁ =− Z

[a,b]

P₁(t)df(t)

=−[P₁(b)f(b)−P_n(a)f(a)] + Z

[a,b]

f(t)dP₁(t)

=−[P₁(b)f(b)−P_n(a)f(a)] + Z

[a,b]

f(t)dˇµ₁(t)

=−[P₁(b)f(b)−P_n(a)f(a)] + Z

[a,b]

f(t)dµ(t)−µ({a})f(a).

Therefore, by iteration, we have Rn=

n

X

k=1

(−1)^k

Pk(b)f^(k−1)(b)−Pk(a)f^(k−1)(a) +

Z

[a,b]

f(t)dµ(t)−µ({a})f(a),

which proves our assertion.

Remark 2.4. By Remark 2.1 we see that identity (2.1) is a generalization of the integration-by- parts formula (1.1).

Corollary 2.5. Letf : [a, b] → Rbe such thatf⁽ⁿ⁻¹⁾ has bounded variation for somen ≥ 1.

Then for everyµ∈M[a, b]we have Z

[a,b]

f(t)dµ(t) = ˇS_n+ ˇR_n, where

Sˇn=

n

X

k=1

(−1)^k−1µˇk(b)f^(k−1)(b) and

Rˇ_n= (−1)ⁿ Z

[a,b]

ˇ

µ_n(t)df⁽ⁿ⁻¹⁾(t).

Proof. Apply the theorem above for theµ-harmonic sequence(ˇµ_n, n ≥1)and note thatµˇ_n(a) =

0,forn≥2.

Corollary 2.6. Letf : [a, b] → Rbe such thatf⁽ⁿ⁻¹⁾ has bounded variation for somen ≥ 1.

Then for everyx∈[a, b]we have f(x) =

n

X

k=1

(x−b)^k−1

(k−1)! f^(k−1)(b) +R_n(x),

where

R_n(x) = (−1)ⁿ (n−1)!

Z

[x,b]

(t−x)ⁿ⁻¹df⁽ⁿ⁻¹⁾(t).

Proof. Apply Corollary 2.5 forµ=δ_xand note that in this case ˇ

µk(t) = (t−x)^k−1

(k−1)! , x≤t ≤b, and µˇk(t) = 0, a≤t < x,

fork ≥1.

Corollary 2.7. Letf : [a, b] → Rbe such thatf⁽ⁿ⁻¹⁾ has bounded variation for somen ≥ 1.

Further, let(c_m, m≥1)be a sequence inRsuch that X

m≥1

|c_m|<∞

and let{x_m;m≥1} ⊂[a, b]. Then X

m≥1

cmf(xm) = X

m≥1 n

X

k=1

cm

(x_m−b)^k−1

(k−1)! f^(k−1)(b) +X

m≥1

cmRn(xm), whereR_n(x_m)is from Corollary 2.6.

Proof. Apply Corollary 2.5 for the discrete measureµ=P

m≥1c_mδ_xm.

3. SOME OSTROWSKI-TYPEINEQUALITIES

In this section we shall use the same notations as above.

Theorem 3.1. Let f : [a, b] → Rbe such thatf⁽ⁿ⁻¹⁾ is L-Lipschitzian for somen ≥ 1.Then for everyµ-harmonic sequence(P_n, n≥1)we have

(3.1)

Z

[a,b]

f(t)dµ(t)−µ({a})f(a)−S_n

≤L Z b

a

|P_n(t)|dt, whereS_nis defined by (2.2).

Proof. By Theorem 2.3 we have

|R_n|= Z

[a,b]

P_n(t)df⁽ⁿ⁻¹⁾(t)

≤L Z b

a

|P_n(t)|dt,

which proves our assertion.

Corollary 3.2. Iff isL-Lipschitzian, then for everyc∈Randµ∈M[a, b]we have

Z

[a,b]

f(t)dµ(t)−µ([a, b])f(b)−c[f(b)−f(a)]

≤L Z b

a

|c+ ˇµ1(t)|dt.

Proof. Putn = 1in the theorem above and note thatP₁(t) = c+ ˇµ₁(t),for somec∈R. Corollary 3.3. Iff isL-Lipschitzian, then for everyc≥0andµ≥0we have

Z

[a,b]

f(t)dµ(t)−µ([a, b])f(b)−c[f(b)−f(a)]

≤L[c(b−a) + ˇµ2(b)]

≤L(b−a)(c+kµk).

Proof. Apply Corollary 3.2 and note that in this case Z b

a

|c+ ˇµ₁(t)|dt = Z b

a

[c+ ˇµ₁(t)]dt

=c(b−a) + ˇµ2(b)

≤c(b−a) + (b−a)kµk

= (b−a)(c+kµk).

Corollary 3.4. Letf beL-Lipschitzian,(c_m, m≥1)a sequence in[0,∞)such that

X

m≥1

c_m <∞,

and let{x_m;m≥1} ⊂[a, b]. Then for everyc≥0we have

X

m≥1

c_m[f(b)−f(x_m)] +c[f(b)−f(a)]

≤L

"

c(b−a) +X

m≥1

c_m(b−x_m)

#

≤L(b−a)

"

c+X

m≥1

c_m

# .

Proof. Apply Corollary 3.3 for the discrete measureµ=P

m≥1c_mδ_xm.

Corollary 3.5. Iff isL-Lipschitzian andµ≥0, then

Z

[a,b]

f(t)dµ(t)−µ([a, x])f(a)−µ((x, b])f(b)

≤L[(2x−a−b)ˇµ₁(x)−2ˇµ₂(x) + ˇµ₂(b)], for everyx∈[a, b].

Proof. Apply Corollary 3.2 forc=−ˇµ1(x).Then

c+ ˇµ₁(b) =µ((x, b]), µˇ₁(x) = µ([a, x]) and

Z b

a

|−ˇµ₁(x) + ˇµ₁(t)|dt = Z x

a

(ˇµ₁(x)−µˇ₁(t))dt+ Z b

x

(ˇµ₁(t)−µˇ₁(x))dt

= (2x−a−b)ˇµ₁(x)−2ˇµ₂(x) + ˇµ₂(b).

Corollary 3.6. Letf : [a, b] → Rbe such thatf⁽ⁿ⁻¹⁾ isL-Lipschitzian for somen ≥ 1.Then for everyµ∈M[a, b]we have

Z

[a,b]

f(t)dµ(t)−Sˇ_n

≤L Z b

a

|ˇµ_n(t)|dt≤ (b−a)ⁿ

n! Lkµk, whereSˇ_nis from Corollary 2.5.

Proof. Apply the theorem above for theµ-harmonic sequence(ˇµ_n, n≥1).

Corollary 3.7. Letf : [a, b] → Rbe such thatf⁽ⁿ⁻¹⁾ isL-Lipschitzian for somen ≥ 1.Then for everyx∈[a, b]we have

f(x)−

n

X

k=1

(x−b)^k−1

(k−1)! f^(k−1)(b)

≤ (b−x)ⁿ n! L.

Proof. Apply Corollary 3.6 forµ=δ_xand note that in this case ˇ

µ_k(t) = (t−x)^k−1

(k−1)! , x≤t ≤b, and µˇ_k(t) = 0, a≤t < x,

fork ≥1.

Corollary 3.8. Let f : [a, b] → R be such that f⁽ⁿ⁻¹⁾ is L-Lipschitzian, for some n ≥ 1.

Further, let(c_m, m≥1)be a sequence inRsuch that X

m≥1

|c_m|<∞

and let{xm;m≥1} ⊂[a, b]. Then

X

m≥1

c_mf(x_m)−X

m≥1 n

X

k=1

c_m(x_m−b)^k−1

(k−1)! f^(k−1)(b)

≤ L n!

X

m≥1

|c_m|(b−x_m)ⁿ

≤ L

n!(b−a)ⁿX

m≥1

|c_m|.

Proof. Apply Corollary 3.6 for the discrete measureµ=P

m≥1c_mδ_xm.

Theorem 3.9. Let f : [a, b] → Rbe such thatf⁽ⁿ⁻¹⁾ has bounded variation for somen ≥ 1.

Then for everyµ-harmonic sequence(P_n, n ≥1)we have

Z

[a,b]

f(t)dµ(t)−µ({a})f(a)−S_n

≤ max

t∈[a,b]|P_n(t)|

b

_

a

(f⁽ⁿ⁻¹⁾), whereWb

a(f⁽ⁿ⁻¹⁾)is the total variation off⁽ⁿ⁻¹⁾ on[a, b].

Proof. By Theorem 2.3 we have

|R_n|= Z

[a,b]

P_n(t)df⁽ⁿ⁻¹⁾(t)

≤ max

t∈[a,b]|P_n(t)|

b

_

a

(f⁽ⁿ⁻¹⁾),

which proves our assertion.

Corollary 3.10. Iff is a function of bounded variation, then for everyc∈ Randµ ∈M[a, b]

we have Z

[a,b]

f(t)dµ(t)−µ([a, b])f(b)−c[f(b)−f(a)]

≤ max

t∈[a,b]|c+ ˇµ₁(t)|

b

_

a

(f).

Proof. Putn = 1in the theorem above.

Corollary 3.11. If f is a function of bounded variation, then for everyc ≥ 0 andµ ≥ 0we have

Z

[a,b]

f(t)dµ(t)−µ([a, b])f(b)−c[f(b)−f(a)]

≤[c+kµk]

b

_

a

(f).

Proof. In this case we have max

t∈[a,b]|c+ ˇµ₁(t)|=c+ ˇµ₁(b) = c+kµk.

Corollary 3.12. Letf be a function of bounded variation,(c_m, m ≥ 1)a sequence in [0,∞) such that

X

m≥1

cm <∞ and let{x_m;m≥1} ⊂[a, b]. Then for everyc≥0we have

X

m≥1

c_m[f(b)−f(x_m)] +c[f(b)−f(a)]

≤

"

c+X

m≥1

c_m

# _b _

a

(f).

Proof. Apply Corollary 3.11 for the discrete measureµ=P

m≥1c_mδ_xm.

Corollary 3.13. Iff is a function of bounded variation andµ≥0,then we have

Z

[a,b]

f(t)dµ(t)−µ([a, x])f(a)−µ((x, b])f(b)

≤ 1

2[ˇµ1(b)−µˇ1(a) +|ˇµ1(a) + ˇµ1(b)−2ˇµ1(x)|]

b

_

a

(f).

Proof. Apply Corollary 3.11 forc=−ˇµ₁(x).Then max

t∈[a,b]|c+ ˇµ₁(t)|= max

t∈[a,b]|µˇ₁(t)−µˇ₁(x)|

= max{ˇµ₁(x)−µˇ₁(a),µˇ₁(b)−µˇ₁(x)}

= 1

2[ˇµ₁(b)−µˇ₁(a) +|ˇµ₁(a) + ˇµ₁(b)−2ˇµ₁(x)|].

Corollary 3.14. Letf : [a, b]→Rbe such thatf⁽ⁿ⁻¹⁾ has bounded variation for somen≥ 1.

Then for everyµ∈M[a, b]we have

Z

[a,b]

f(t)dµ(t)−Sˇ_n

≤ max

t∈[a,b]|ˇµ_n(t)|

b

_

a

(f⁽ⁿ⁻¹⁾)

≤ (b−a)ⁿ⁻¹ (n−1)! kµk

b

_

a

(f⁽ⁿ⁻¹⁾), whereSˇ_nis from Corollary 2.5.

Proof. Apply the theorem above for theµ-harmonic sequence(ˇµ_n, n≥1).

Corollary 3.15. Letf : [a, b]→Rbe such thatf⁽ⁿ⁻¹⁾ has bounded variation for somen≥ 1.

Then for everyx∈[a, b]we have

f(x)−

n

X

k=1

(x−b)^k−1

(k−1)! f^(k−1)(b)

≤ (b−x)ⁿ⁻¹ (n−1)!

b

_

a

(f⁽ⁿ⁻¹⁾).

Proof. Apply Corollary 3.14 forµ=δ_x and note that in this case max

t∈[a,b]|µˇ_n(t)|= (b−x)ⁿ⁻¹ (n−1)! .

Corollary 3.16. Letf : [a, b]→Rbe such thatf⁽ⁿ⁻¹⁾ has bounded variation for somen≥ 1.

Further, let(c_m, m≥1)be a sequence inRsuch that X

m≥1

|c_m|<∞ and let{x_m;m≥1} ⊂[a, b]. Then

X

m≥1

c_mf(x_m)−X

m≥1 n

X

k=1

c_m(x_m−b)^k−1

(k−1)! f^(k−1)(b)

≤ 1 (n−1)!

b

_

a

(f⁽ⁿ⁻¹⁾)X

m≥1

|c_m|(b−x_m)ⁿ⁻¹

≤ (b−a)ⁿ⁻¹ (n−1)!

b

_

a

(f⁽ⁿ⁻¹⁾)X

m≥1

|c_m| Proof. Apply Corollary 3.14 for the discrete measureµ=P

m≥1c_mδ_xm.

Theorem 3.17. Letf : [a, b] →Rbe such thatf⁽ⁿ⁾ ∈L_p[a, b]for somen ≥1.Then for every µ-harmonic sequence(P_n, n≥1)we have

Z

[a,b]

f(t)dµ(t)−µ({a})f(a)−S_n

≤ kP_nk_qkf⁽ⁿ⁾k_p, wherep, q ∈[1,∞]and1/p+ 1/q= 1.

Proof. By Theorem 2.3 and the Hölder inequality we have

|R_n|= Z

[a,b]

P_n(t)df⁽ⁿ⁻¹⁾(t)

= Z

[a,b]

P_n(t)f⁽ⁿ⁾(t)dt

≤ Z b

a

|Pn(t)|^qdt

¹q Z b

a

f⁽ⁿ⁾(t)

pdt p¹

=kP_nk_qkf⁽ⁿ⁾k_p.

Remark 3.18. We see that the inequality of the theorem above is a generalization of inequality (1.2).

Corollary 3.19. Let f : [a, b] → R be such that f⁽ⁿ⁾ ∈ L_p[a, b] for some n ≥ 1, and µ ∈ M[a, b].Then

Z

[a,b]

f(t)dµ(t)−Sˇ_n

≤ kˇµ_nk_qkf⁽ⁿ⁾k_p

≤ (b−a)^n−1+1/q

(n−1)! [(n−1)q+ 1]^1/q kµk kf⁽ⁿ⁾k_p, wherep, q ∈[1,∞]and1/p+ 1/q= 1.

Proof. Apply the theorem above for theµ-harmonic sequence(ˇµ_n, n≥1).

Corollary 3.20. Let f : [a, b] → Rbe such that f⁽ⁿ⁾ ∈ L_p[a, b], for somen ≥ 1. Further, let (c_m, m ≥1)be a sequence inRsuch that

X

m≥1

|c_m|<∞ and let{x_m;m≥1} ⊂[a, b]. Then

X

m≥1

c_mf(x_m)−X

m≥1 n

X

k=1

c_m(x_m−b)^k−1

(k−1)! f^(k−1)(b)

≤ kf⁽ⁿ⁾k_p

(n−1)! [(n−1)q+ 1]^1/q X

m≥1

|c_m|(b−x_m)^n−1+1/q

≤ (b−a)^n−1+1/qkf⁽ⁿ⁾k_p (n−1)! [(n−1)q+ 1]^1/q

X

m≥1

|c_m|, wherep, q ∈[1,∞]and1/p+ 1/q= 1.

Proof. Apply the theorem above for the discrete measureµ=P

m≥1c_mδ_xm.

4. SOMEGRÜSS-TYPE INEQUALITIES

Letf : [a, b]→Rbe such thatf⁽ⁿ⁾∈L∞[a, b],for somen ≥1.Then m_n ≤f⁽ⁿ⁾(t)≤M_n, t ∈[a, b], a.e.

for some real constantsm_nandM_n.

Theorem 4.1. Letf : [a, b] → R be such thatf⁽ⁿ⁾ ∈ L∞[a, b], for some n ≥ 1.Further, let (P_k, k≥1)be aµ-harmonic sequence such that

P_n+1(a) = P_n+1(b), for that particularn.Then

Z

[a,b]

f(t)dµ(t)−µ({a})f(a)−S_n

≤ Mn−mn

2

Z b

a

|P_n(t)|dt.

Proof. Apply Theorem 2.3 for the special case when f⁽ⁿ⁻¹⁾ is absolutely continuous and its derivativef⁽ⁿ⁾,existinga.e.,is boundeda.e.Define the measureν_nby

dν_n(t) =−P_n(t)dt.

Then

νn([a, b]) =− Z b

a

Pn(t)dt =Pn+1(a)−Pn+1(b) = 0,

which means thatν_nis balanced. Further, kν_nk=

Z b

a

|P_n(t)|dt and by [1, Theorem 2]

|R_n|=

Z b

a

P_n(t)f⁽ⁿ⁾(t)dt

≤ M_n−m_n 2 kν_nk

= M_n−m_n 2

Z b

a

|P_n(t)|dt,

which proves our assertion.

Corollary 4.2. Letf : [a, b]→Rbe such thatf⁽ⁿ⁾ ∈L∞[a, b], for somen ≥1.Then for every (n+ 1)-balanced measureµ∈M[a, b]we have

Z

[a,b]

f(t)dµ(t)−Sˇn

≤ M_n−m_n 2

Z b

a

|ˇµn(t)|dt

≤ M_n−m_n 2

(b−a)ⁿ n! kµk, whereSˇ_nis from Corollary 2.5.

Proof. Apply Theorem 4.1 for theµ-harmonic sequence(ˇµ_k, k≥1)and note that the condition P_n+1(a) =P_n+1(b)reduces toµˇ_n+1(b) = 0,which means thatµis(n+ 1)-balanced.

Corollary 4.3. Letf : [a, b] → R be such thatf⁽ⁿ⁾ ∈ L∞[a, b] for some n ≥ 1.Further, let (cm, m ≥1)be a sequence inRsuch that

X

m≥1

|cm|<∞

and let{x_m;m≥1} ⊂[a, b]satisfy the condition X

m≥1

c_m(b−x_m)ⁿ= 0.

Then

X

m≥1

cmf(xm)−X

m≥1 n

X

k=1

cm

(x_m−b)^k−1

(k−1)! f^(k−1)(b)

≤ M_n−m_n 2n!

X

m≥1

|c_m|(b−x_m)ⁿ

≤ M_n−m_n

2n! (b−a)ⁿX

m≥1

|c_m|. Proof. Apply Corollary 4.2 for the discrete measureµ=P

m≥1cmδxm.

Corollary 4.4. Letf : [a, b]→Rbe such thatf⁽ⁿ⁾ ∈ L∞[a, b]for somen ≥1.Then for every µ∈M[a, b],such that allk-moments ofµare zero fork = 0, . . . , n,we have

Z

[a,b]

f(t)dµ(t)

≤ M_n−m_n 2

Z b

a

|ˇµ_n(t)|dt

≤ M_n−m_n 2

(b−a)ⁿ n! kµk.

Proof. By [1, Theorem 5], the conditionm_k(µ) = 0, k = 0, . . . , nis equivalent toµˇ_k(b) = 0, k = 1, . . . , n+ 1.Apply Corollary 4.2 and note that in this caseSˇ_n = 0.

Corollary 4.5. Letf : [a, b] → R be such thatf⁽ⁿ⁾ ∈ L_∞[a, b] for some n ≥ 1.Further, let (c_m, m ≥1)be a sequence inRsuch that

X

m≥1

|c_m|<∞ and let{x_m;m≥1} ⊂[a, b]. If

X

m≥1

c_m =X

m≥1

c_mx_m =· · ·=X

m≥1

c_mxⁿ_m = 0, then

X

m≥1

c_mf(x_m)

≤ M_n−m_n 2n!

X

m≥1

|c_m|(b−x_m)ⁿ

≤ M_n−m_n

2n! (b−a)ⁿX

m≥1

|c_m|. Proof. Apply Corollary 4.4 for the discrete measureµ=P

m≥1c_mδ_xm.

REFERENCES

[1] A. ˇCIVLJAK, LJ. DEDI ´CANDM. MATI ´C, Euler-Grüss type inequalities involving measures, sub- mitted.

[2] A. ˇCIVLJAK, LJ. DEDI ´CANDM. MATI ´C, Euler harmonic identities for measures, Nonlinear Func- tional Anal. & Applics., 12(1) (2007).

[3] Lj. DEDI ´C, M. MATI ´C, J. PE ˇCARI ´CANDA. AGLI ´C ALJINOVI ´C, On weighted Euler harmonic identities with applications, Math. Inequal. & Appl., 8(2), (2005), 237–257.

[4] S.S. DRAGOMIR, The generalised integration by parts formula for Appell sequences and related results, RGMIA Res. Rep. Coll., 5(E) (2002), Art. 18. [ONLINE:http://rgmia.vu.edu.au/

v5(E).html].

[5] P. CERONE, Generalised Taylor’s formula with estimates of the remainder, RGMIA Res. Rep. Coll., 5(2) (2002), Art. 8. [ONLINE:http://rgmia.vu.edu.au/v5n2.html].

[6] P. CERONE, Perturbated generalised Taylor’s formula with sharp bounds, RGMIA Res. Rep. Coll., 5(2) (2002), Art. 6. [ONLINE:http://rgmia.vu.edu.au/v5n2.html].

[7] S.S. DRAGOMIR ANDA. SOFO, A perturbed version of the generalised Taylor’s formula and ap- plications, RGMIA Res. Rep. Coll., 5(2) (2002), Art. 16. [ONLINE:http://rgmia.vu.edu.

au/v5n2.html].