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volume 4, issue 5, article 101, 2003.

Received 08 January, 2003;

accepted 07 August, 2003.

Communicated by:B.G. Pachpatte

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Journal of Inequalities in Pure and Applied Mathematics

OSTROWSKI-GRÜSS TYPE INEQUALITIES IN TWO DIMENSIONS

NENAD UJEVI ´C

Department of Mathematics University of Split

Teslina 12/III, 21000 Split CROATIA.

EMail:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 003-03

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Ostrowski-Grüss type Inequalities in Two Dimensions

Nenad Ujevi´c

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J. Ineq. Pure and Appl. Math. 4(5) Art. 101, 2003

http://jipam.vu.edu.au

Abstract

A general Ostrowski-Grüss type inequality in two dimensions is established. A particular inequality of the same type is also given.

2000 Mathematics Subject Classification:26D10, 26D15.

Key words: Ostrowski’s inequality, 2-dimensional generalization, Ostrowski-Grüss inequality.

1. Introduction

In 1938 A. Ostrowski proved the following integral inequality ([17] or [16, p.

468]).

Theorem 1.1. Let f : I → R, where I ⊂ R is an interval, be a mapping differentiable in the interiorInt I ofI, and leta, b∈ Int I, a < b. If|f⁰(t)| ≤ M,∀t ∈[a, b], then we have

(1.1)

f(x)− 1 b−a

Z b a

f(t)dt

≤

"

1

4+ (x−^a+b₂ )² (b−a)²

#

(b−a)M, forx∈[a, b].

The first (direct) generalization of Ostrowski’s inequality was given by G.V.

Milovanović and J. Peˇcarić in [14]. In recent years a number of authors have written about generalizations of Ostrowski’s inequality. For example, this topic is considered in [2], [4], [6], [9] and [14]. In this way, some new types of inequalities have been formed, such as inequalities of Ostrowski-Grüss type, inequalities of Ostrowski-Chebyshev type, etc. The first inequality of Ostrowski- Grüss type was given by S.S. Dragomir and S. Wang in [6]. It was generalized and improved in [9]. X.L. Cheng gave a sharp version of the mentioned inequality in [4]. The first multivariate version of Ostrowski’s inequality was given by G.V. Milovanović in [12] (see also [13] and [16, p. 468]). Multivariate ver- sions of Ostrowski’s inequality were also considered in [3], [7] and [11]. In this paper we give a general two-dimensional Ostrowski-Grüss inequality. For that purpose, we introduce specially defined polynomials, which can be considered as harmonic or Appell-like polynomials in two dimensions. In Section3 we use the mentioned general inequality to obtain a particular two-dimensional Ostrowski-Grüss type inequality.

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2. A General Ostrowski-Grüss Inequality

LetΩ = [a, b]×[a, b]and letf : Ω→Rbe a given function. Here we suppose thatf ∈C²ⁿ(Ω). LetP_k(s)andQ_k(t)be harmonic or Appell-like polynomials, i.e.

(2.1) P_k⁰(s) =Pk−1(s)andQ⁰_k(t) =Qk−1(t), k= 1,2, . . . , n+ 1, with

(2.2) P₀(s) = Q₀(t) = 1.

We also define

(2.3) R_k(s, t) =P_k(s)Q_k(t), k = 0,1,2, . . . , n+ 1.

Lemma 2.1. LetR_k(s, t)be defined by (2.3). Then we have

(2.4) ∂²R_k(s, t)

∂s∂t =Rk−1(s, t) fork = 1,2, . . . , n+ 1.

Proof. From (2.1) – (2.3) it follows that

∂²R_k(s, t)

∂s∂t = ∂

∂t

∂R_k(s, t)

∂s

= ∂

∂t(P_k⁰(s)Q_k(t))

=Pk−1(s)Q⁰_k(t)

=Pk−1(s)Qk−1(t) =Rk−1(s, t).

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We now define (2.5) J_k =

Z b a

R_k(b, t)∂^2k−1f(b, t)

∂s^k−1∂t^k −R_k(a, t)∂^2k−1f(a, t)

∂s^k−1∂t^k

dt,

(2.6) uk−1(t) = ∂^k−1f(b, t)

∂s^k−1 , vk−1(t) = ∂^k−1f(a, t)

∂s^k−1 , fork = 1,2, . . . , n. We also define

(2.7) J_k,1 = Z b

a

R_k(b, t)∂^2k−1f(b, t)

∂s^k−1∂t^k dt =P_k(b) Z b

a

Q_k(t)u^(k)_k−1(t)dt and

(2.8) Jk,2 = Z b

a

Rk(a, t)∂^2k−1f(a, t)

∂s^k−1∂t^k dt=Pk(a) Z b

a

Qk(t)v_k−1^(k) (t)dt such that

(2.9) J_k =J_k,1−J_k,2.

Lemma 2.2. LetJ_k,1be defined by (2.7). Then we have (2.10) J_k,1 =P_k(b)

k

X

j=1

(−1)^k−jh

Q_j(b)u^(j−1)_k−1 (b)−Q_j(a)u^(j−1)_k−1 (a)i + (−1)^kPk(b)

Z b a

uk−1(t)dt, fork = 1,2, . . . , n.

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Proof. We introduce the notation

U_k(uk−1) = Z b

a

Q_k(t)u^(k)_k−1(t)dt.

Then we have

(−1)^kU_k(uk−1) = (−1)^k Z b

a

Q_k(t)u^(k)_k−1(t)dt

= (−1)^kh

Q_k(b)u^(k−1)_k−1 (b)−Q_k(a)u^(k−1)_k−1 (a)i + (−1)^k−1

Z b a

Q_k−1(t)u^(k−1)_k−1 (t)dt.

We can write the above relation in the form (−1)^kU_k(uk−1)

= (−1)^kh

Q_k(b)u^(k−1)_k−1 (b)−Q_k(a)u^(k−1)_k−1 (a)i

+ (−1)^k−1Uk−1(uk−1).

In a similar way we get

(−1)^k−1Uk−1(uk−1) = (−1)^k−1 Z b

a

Qk−1(t)u^(k−1)_k−1 (t)dt

= (−1)^k−1h

Qk−1(b)u^(k−2)_k−1 (b)−Qk−1(a)u^(k−2)_k−1 (a)i + (−1)^k−2

Z b a

Q_k−2(t)u^(k−2)_k−1 (t)dt

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or

(−1)^k−1U_k−1(u_k−1)

= (−1)^k−1h

Qk−1(b)u^(k−2)_k−1 (b)−Qk−1(a)u^(k−2)_k−1 (a) i

+ (−1)^k−2Uk−2(uk−1).

If we continue the above procedure then we obtain (−1)^kU_k(uk−1)

=

k

X

j=1

(−1)^jh

Q_j(b)u^(j−1)_k−1 (b)−Q_j(a)u^(j−1)_k−1 (a)i

+U₀(uk−1)

=

k

X

j=1

(−1)^jh

Q_j(b)u^(j−1)_k−1 (b)−Q_j(a)u^(j−1)_k−1 (a)i +

Z b a

uk−1(t)dt.

Note now that

J_k,1 =P_k(b)U_k(uk−1) such that (2.10) holds.

Lemma 2.3. LetJk,2be defined by (2.8). Then we have (2.11) Jk,2 =Pk(a)

k

X

j=1

(−1)^k−jh

Qj(b)v^(j−1)_k−1 (b)−Qj(a)v^(j−1)_k−1 (a) i

+ (−1)^kP_k(a) Z b

a

v_k−1(t)dt,

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fork = 1,2, . . . , n.

Proof. The proof is almost identical to that of Lemma2.2.

We now define (2.12) K_k =

Z b a

∂R_k(s, b)

∂s

∂^2k−2f(s, b)

∂s^k−1∂t^k−1 − ∂R_k(s, a)

∂s

∂^2k−2f(s, a)

∂s^k−1∂t^k−1

ds, fork = 2, . . . , n,

(2.13) xk−1(s) = ∂^k−1f(s, b)

∂t^k−1 , yk−1(s) = ∂^k−1f(s, a)

∂t^k−1 and

(2.14) K₁ =Q₁(b) Z b

a

x₀(s)ds−Q₁(a) Z b

a

y₀(s)ds.

We also define

K_k,1 = Z b

a

∂R_k(s, b)

∂s

∂^2k−2f(s, b)

∂s^k−1∂t^k−1 ds (2.15)

=Q_k(b) Z b

a

Pk−1(s)x^(k−1)_k−1 (s)ds and

K_k,2 = Z b

a

∂R_k(s, a)

∂s

∂^2k−2f(s, a)

∂s^k−1∂t^k−1 ds (2.16)

=Q_k(a) Z b

a

P_k−1(s)y_k−1^(k−1)(s)ds

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such that

(2.17) K_k =K_k,1−K_k,2,k = 1,2, . . . , n.

Lemma 2.4. LetK_k,1be defined by (2.15). Then we have

(2.18) Kk,1 =Qk(b)

k

X

j=2

(−1)^k−j+1h

Pj−1(b)x^(j−2)_k−1 (b)−Pj−1(a)x^(j−2)_k−1 (a) i

+ (−1)^k−1Q_k(b) Z b

a

x_k−1(s)ds, fork = 2, . . . , n.

Proof. We introduce the notation

Uk−1(xk−1) = Z b

a

Pk−1(s)x^(k−1)_k−1 (s)ds.

Then we have

(−1)^k−1Uk−1(xk−1) = (−1)^k−1 Z b

a

Pk−1(s)x^(k−1)_k−1 (s)ds

= (−1)^k−1h

Pk−1(b)x^(k−2)_k−1 (b)−Pk−1(a)x^(k−2)_k−1 (a)i + (−1)^k−2

Z b a

P_k−2(s)x^(k−2)_k−1 (s)ds.

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We can write the above relation in the form (−1)^k−1U_k−1(x_k−1)

= (−1)^k−1h

P_k−1(b)x^(k−2)_k−1 (b)−P_k−1(a)x^(k−2)_k−1 (a)i

+ (−1)^k−2Uk−2(xk−1).

In a similar way we get

(−1)^k−2Uk−2(xk−1) = (−1)^k−2 Z b

a

Pk−2(s)x^(k−2)_k−1 (s)ds

= (−1)^k−2h

Pk−2(b)x^(k−3)_k−1 (b)−Pk−2(a)x^(k−3)_k−1 (a)i + (−1)^k−3

Z b a

Pk−3(s)x^(k−3)_k−1 (s)ds or

(−1)^k−2Uk−2(xk−1)

= (−1)^k−2h

Pk−2(b)x^(k−3)_k−1 (b)−Pk−2(a)x^(k−3)_k−1 (a)i

+ (−1)^k−3Uk−3(xk−1).

If we continue the above procedure then we get (−1)^k−1Uk−1(xk−1)

=

k

X

j=2

(−1)^j−1h

Pj−1(b)x^(j−2)_k−1 (b)−Pj−1(a)x^(j−2)_k−1 (a)i

+U₀(xk−1)

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=

k

X

j=2

(−1)^j−1h

P_j−1(b)x^(j−2)_k−1 (b)−P_j−1(a)x^(j−2)_k−1 (a)i +

Z b a

x_k−1(t)dt.

Note now that

K_k,1 =Q_k(b)Uk−1(xk−1) such that (2.18) holds.

Lemma 2.5. LetK_k,2be defined by (2.16). Then we have (2.19) K_k,2 =Q_k(a)

k

X

j=2

(−1)^k−j+1h

Pj−1(b)y_k−1^(j−2)(b)−Pj−1(a)y_k−1^(j−2)(a)i + (−1)^k−1Q_k(a)

Z b a

yk−1(s)ds, fork = 2, . . . , n.

Proof. The proof is almost identical to that of Lemma2.4.

Let (X,h·,·i) be a real inner product space and e ∈ X, kek = 1. Let γ, ϕ,Γ,Φbe real numbers andx, y ∈X such that the conditions

(2.20) hΦe−x, x−ϕei ≥0and hΓe−y, y−γei ≥0 hold. In [5] we can find the inequality

(2.21) |hx, yi − hx, ei hy, ei| ≤ 1

4|Φ−ϕ| |Γ−γ|.

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We also have

(2.22) |hx, yi − hx, ei hy, ei| ≤ kxk²− hx, ei²¹₂

kyk²− he, yi²¹₂ . LetX =L₂(Ω)ande= 1/(b−a). If we define

(2.23) T(f, g) = 1 (b−a)²

Z b a

f(t, s)g(t, s)dtds

− 1 (b−a)⁴

Z b a

f(t, s)dtds Z b

a

Z b a

g(t, s)dtds, then from (2.20) and (2.21) we obtain the Grüss inequality inL₂(Ω),

(2.24) |T(f, g)| ≤ 1

4(Γ−γ)(Φ−ϕ), if

γ ≤f(x, y)≤Γ, ϕ ≤g(x, y)≤Φ,(x, y)∈Ω.

From (2.22), we have the pre-Grüss inequality (2.25) T(f, g)² ≤T(f, f)T(g, g).

We now define

(2.26) In=

Z b a

Rn(s, t)∂²ⁿf(s, t)

∂sⁿ∂tⁿ dsdt and

(2.27) S_n= 1

(b−a)² Z b

a

Z b a

R_n(s, t)dsdt Z b

a

Z b a

∂²ⁿf(s, t)

∂sⁿ∂tⁿ dsdt.

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Lemma 2.6. LetI_nandS_nbe defined by (2.26) and (2.27), respectively. Then we have the inequality

(2.28) |I_n−S_n| ≤ M_2n−m_2n

2 C(b−a)², where

M2n= max

(s,t)∈Ω

∂²ⁿf(s, t)

∂sⁿ∂tⁿ , m2n = min

(s,t)∈Ω

∂²ⁿf(s, t)

∂sⁿ∂tⁿ and

(2.29) C =

1 (b−a)²

Z b a

Pn(s)²ds Z b

a

Qn(t)²dt

− 1 (b−a)⁴

Z b a

P_n(s)ds Z b

a

Q_n(t)dt ²)¹₂

. Proof. From (2.23), (2.26) and (2.27) we see that

I_n−S_n = (b−a)²T

R_n(s, t),∂²ⁿf(s, t)

∂sⁿ∂tⁿ

. Then from (2.25) we get

|I_n−S_n| ≤(b−a)²T (R_n(s, t), R_n(s, t))¹² T

∂²ⁿf(s, t)

∂sⁿ∂tⁿ ,∂²ⁿf(s, t)

∂sⁿ∂tⁿ ¹₂

. From (2.24) we have

T

∂²ⁿf(s, t)

∂sⁿ∂tⁿ ,∂²ⁿf(s, t)

∂sⁿ∂tⁿ ¹₂

≤ M_2n−m_2n

2 .

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We also have

T (R_n(s, t), R_n(s, t)) = 1 (b−a)²

Z b a

P_n(s)²ds Z b

a

Q_n(t)²dt

− 1 (b−a)⁴

Z b a

P_n(s)ds Z b

a

Q_n(t)dt ²

. From the last three relations we see that (2.28) holds.

Theorem 2.7. LetΩ = [a, b]×[a, b]and letf : Ω→Rbe a given function such thatf ∈C²ⁿ(Ω). Let the conditions of Lemma2.6hold. IfJk,Kkare given by (2.9), (2.17), where J_k,1, J_k,2, K_k,1, K_k,2 are given by Lemmas2.2 –2.5, then we have the inequality

(2.30)

Z b a

f(s, t)dsdt+

n

X

k=1

J_k−

n

X

k=1

K_k−S_n

≤ M_2n−m_2n

2 C(b−a)², where

(2.31) S_n = 1

(b−a)²[P_n+1(b)−P_n+1(a)] [Q_n+1(b)−Q_n+1(a)]

×[v(b, b)−v(b, a)−v(a, b) +v(a, a)], andv(s, t) = _∂s^∂²ⁿ⁻²n−1∂t^t(s,t)ⁿ⁻¹.

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Proof. We have

I_n= Z b

a

Z b a

R_n(s, t)∂²ⁿf(s, t)

∂sⁿ∂tⁿ dsdt (2.32)

= Z b

a

dt Z b

a

R_n(s, t) ∂

∂s

∂²ⁿ⁻¹f(s, t)

∂sⁿ⁻¹∂tⁿ

ds

= Z b

a

R_n(b, t)∂²ⁿ⁻¹f(b, t)

∂sⁿ⁻¹∂tⁿ −R_n(a, t)∂²ⁿ⁻¹f(a, t)

∂sⁿ⁻¹∂tⁿ

dt

− Z b

a

Z b a

∂R_n(s, t)

∂s

∂²ⁿ⁻¹f(s, t)

∂sⁿ⁻¹∂tⁿ dsdt

=J_n−L_n, where

L_n = Z b

a

Z b a

∂Rn(s, t)

∂s

∂²ⁿ⁻¹f(s, t)

∂sⁿ⁻¹∂tⁿ dsdt.

We also have L_n =

Z b a

ds Z b

a

∂R_n(s, t)

∂s

∂

∂t

∂²ⁿ⁻²f(s, t)

∂sⁿ⁻¹∂tⁿ⁻¹

dt

= Z b

a

∂R_n(s, b)

∂s

∂²ⁿ⁻²f(s, b)

∂sⁿ⁻¹∂tⁿ⁻¹ −∂R_n(s, a)

∂s

∂²ⁿ⁻²f(s, a)

ds

− Z b

a

Z b a

Rn−1(s, t)∂²ⁿ⁻²f(s, t)

∂sⁿ⁻¹∂tⁿ⁻¹dsdt

=K_n−In−1.

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Hence, we have

In=Jn−Kn+In−1. In a similar way we obtain

In−1 =Jn−1−Kn−1+In−2. If we continue this procedure then we get

(2.33) I_n =

n

X

k=1

J_k−

n

X

k=1

K_k+I₀, where

(2.34) I₀ =

Z b a

f(s, t)dsdt.

We now consider the term

(2.35) S_n = 1

(b−a)² Z b

a

Z b a

R_n(s, t)dsdt Z b

a

Z b a

∂²ⁿf(s, t)

∂sⁿ∂tⁿ dsdt.

We have Z b

a

Z b a

R_n(s, t)dsdt= Z b

a

P_n(s)ds Z b

a

Q_n(t)dt

= [Pn+1(b)−Pn+1(a)] [Qn+1(b)−Qn+1(a)]

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and Z b

a

Z b a

∂²ⁿf(s, t)

∂sⁿ∂tⁿ dsdt

= Z b

a

dt Z b

a

∂

∂s

∂²ⁿ⁻¹f(s, t)

∂sⁿ⁻¹∂tⁿ

ds

= Z b

a

∂²ⁿ⁻¹f(b, t)

∂sⁿ⁻¹∂tⁿ − ∂²ⁿ⁻¹f(a, t)

∂sⁿ⁻¹∂tⁿ

dt

= ∂²ⁿ⁻²f(b, b)

∂sⁿ⁻¹∂tⁿ⁻¹ − ∂²ⁿ⁻²f(b, a)

∂sⁿ⁻¹∂tⁿ⁻¹ − ∂²ⁿ⁻¹f(a, b)

∂sⁿ⁻¹∂tⁿ⁻¹ +∂²ⁿ⁻¹f(a, a)

= [v(b, b)−v(b, a)−v(a, b) +v(a, a)], Thus (2.31) holds. From (2.33) – (2.35) we see that

I_n−S_n= Z b

a

Z b a

f(s, t)dsdt+

n

X

k=1

J_k−

n

X

k=1

K_k−S_n. Then from Lemma2.6we conclude that (2.30) holds.

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3. A Particular Inequality

Here we use the notations introduced in Section2. In Theorem2.7 we proved a general inequality of Ostrowski-Grüss type. Many particular inequalities can be obtained if we choose specific harmonic or Appell-like polynomials P_k(s), Q_k(t)in (2.30). For example, in [8] we can find the following harmonic polynomials

Pk(s) = 1

k!(s−a)^k, P_k(s) = 1

k!

s− a+b 2

k

, P_k(s) = (b−a)^k

k! B_k

s−a b−a

, P_k(s) = (b−a)^k

k! E_k

s−a b−a

,

whereB_k(s)andE_k(s)are Bernoulli and Euler polynomials, respectively. We shall not consider all possible combinations of these polynomials. Here we choose the following combination

P_k(s) = (b−a)^k k! B_k

s−a b−a

(3.1) ,

Q_k(t) = (b−a)^k k! B_k

t−a b−a

.

We now substitute the above polynomials in (2.10), (2.11), (2.18), (2.19) to obtain

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J_k,1 = ¯J_k,1 (3.2)

= (b−a)^k k! B_k(1)

k

X

j=1

(−1)^k−j(b−a)^j j!

×h

Bj(1)u^(j−1)_k−1 (b)−Bj(0)u^(j−1)_k−1 (a) i

+ (−1)^kB_k(1)(b−a)^k k!

Z b a

uk−1(t)dt, J_k,2 = ¯J_k,2

(3.3)

= (b−a)^k k! B_k(0)

k

X

j=1

(−1)^k−j(b−a)^j j!

×h

Bj(1)v_k−1^(j−1)(b)−Bj(0)v_k−1^(j−1)(a) i

+ (−1)^kB_k(0)(b−a)^k k!

Z b a

vk−1(t)dt, K_k,1 = ¯K_k,1

(3.4)

= (b−a)^k k! B_k(1)

k

X

j=2

(−1)^k−j+1(b−a)^j−1 (j−1)!

×h

Bj−1(1)x^(j−2)_k−1 (b)−Bj−1(0)x^(j−2)_k−1 (a)i + (−1)^k−1(b−a)^k

k! B_k(1) Z b

a

x_k−1(s)ds,

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and

K_k,2 = ¯K_k,2 (3.5)

= (b−a)^k k! Bk(0)

k

X

j=2

(−1)^k−j+1(b−a)^j−1 (j −1)!

×h

Bj−1(1)y^(j−2)_k−1 (b)−Bj−1(0)y_k−1^(j−2)(a)i + (−1)^k−1(b−a)^k

k! B_k(0) Z b

a

yk−1(s)ds.

We have

(3.6) J_k = ¯J_k = ¯J_k,1−J¯_k,2,k = 1,2, . . . , n,

(3.7) K_k = ¯K_k= ¯K_k,1−K¯_k,2, k = 2, . . . , n and

(3.8) K¯1 = b−a 2

Z b a

x0(s)ds+ Z b

a

y0(s)ds

,

whereJ¯_k,1,J¯_k,2,K¯_k,1,K¯_k,2are defined by (3.2) – (3.5), respectively.

Basic properties of Bernoulli polynomials can be found in [1]. Here we emphasize the following properties:

(3.9)

Z 1 0

B_k(s)ds = 0, k = 1,2, . . .

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and (3.10)

Z 1 0

B_k(s)B_j(s)ds = (−1)^k−1 k!j!

(k+j)!B_k+j, k, j = 1,2, . . . , where

(3.11) B_k=B_k(0),k = 0,1,2, . . . are Bernoulli numbers. We also have

(3.12) B_2i+1 = 0,i= 1,2, . . . ,

(3.13) B_k(0) =B_k(1) =B_k,k= 0,2,3,4, . . . , and, in particular,

(3.14) B₁(0) =−1

2, B₁(1) = 1 2. From (3.2) – (3.8) and (3.12) we see that

(3.15) J¯_2i+1 = ¯K_2i+1 = 0, i= 1,2, . . . , n.

Note also that sums in (3.2) – (3.5) have only even-indexed terms and the term forj = 1 (j = 2)is non-zero.

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Theorem 3.1. Under the assumptions of Theorem2.7we have

(3.16)

Z b a

f(s, t)dsdt+

n

X

k=1

J¯_k−

n

X

k=1

K¯_k

≤ M_2n−m_2n

2 ·|B_2n|

(2n)!(b−a)²ⁿ⁺², where B_k are Bernoulli numbers and J¯_k, K¯_k are given by (3.6), (3.7), respec- tively.

Proof. The proof follows from the proof of Theorem2.7, since the following is valid. LetP_nandQ_nbe defined by (3.1), fork=n.

Firstly, we have S_n = 1

(b−a)² Z b

a

Z b a

R_n(s, t)dsdt Z b

a

Z b a

∂²ⁿf(s, t)

∂sⁿ∂tⁿ dsdt= 0, since

Z b a

Z b an

(s, t)dsdt= Z b

a

P_n(s)ds Z b

a

Q_n(t)dt

= Z b

a

P_n(s)ds ²

=





(b−a)ⁿ⁺¹ n!

1

Z

0

Bn(s)ds





2

= 0,

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because of (3.9).

Secondly, we have C =

1 (b−a)²

Z b a

P_n(s)²ds Z b

a

Q_n(t)²dt

− 1 (b−a)⁴

Z b a

P_n(s)ds Z b

a

Q_n(t)dt ²)¹₂

= 1

(b−a)² Z b

a

P_n(s)²ds Z b

a

Q_n(t)²dt ¹₂

= 1

b−a Z b

a

P_n(s)²ds

= 1

b−a ·(b−a)²ⁿ⁺¹ (n!)²

Z 1 0

Bn(s)²ds

= (b−a)²ⁿ (n!)²

(n!)²

(2n)!|B2n|= |B_2n|

(2n)!(b−a)²ⁿ, since (3.10) holds.

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References

[1] M. ABRAMOWITZ ANDI.A. STEGUN (Eds), Handbook of Mathemati- cal Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 4th printing, Wash- ington, 1965

[2] G.A. ANASTASSIOU, Multivariate Ostrowski type inequalities, Acta Math. Hungar., 76 (1997), 267–278.

[3] G.A. ANASTASSIOU, Ostrowski type inequalities, Proc. Amer. Math.

Soc., 123(12) (1995), 3775–3781.

[4] X.L. CHENG, Improvement of some Ostrowski-Grüss type inequalities, Comput. Math. Appl., 42 (2001), 109–114.

[5] S.S. DRAGOMIR, A generalization of Grüss inequality in inner product spaces and applications, J. Math. Anal. Appl., 237 (1999), 74–82.

[6] S.S. DRAGOMIRANDS. WANG, An inequality of Ostrowski–Grüss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, Comput. Math. Appl., 33(11) (1997), 16–20.

[7] G. HANNA, P. CERONE AND J. ROUMELIOTIS, An Ostrowski type inequality in two dimensions using the three point rule, ANZIAM J., 42(E) (2000), 671–689.

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[8] M. MATI Ć, J. PE ˇCARI Ć AND N. UJEVI Ć, On new estimation of the remainder in generalized Taylor’s formula, Math. Inequal. Appl., 2(3), (1999), 343–361.

[9] M. MATI Ć, J. PE ˇCARI ĆANDN. UJEVI Ć, Improvement and further gen- eralization of some inequalities of Ostrowski-Grüss type, Comput. Math.

Appl., 39 (2000), 161–175.

[10] M. MATI Ć, J. PE ˇCARI Ć AND N. UJEVI Ć, Generalizations of weighted version of Ostrowski’s inequality and some related results, J. Inequal.

Appl., 5 (2000), 639–666.

[11] M. MATI Ć, J. PE ˇCARI ĆANDN. UJEVI Ć, Weighted version of multivari- ate Ostrowski type inequalities, Rocky Mountain J. Math, 31(2) (2001), 511–538.

[12] G.V. MILOVANOVI ´C, On some integral inequalities, Univ. Beograd Publ.

Elektrotehn. Fak. Ser. Mat. Fiz., No 498-541, (1975), 119–124.

[13] G.V. MILOVANOVI ´C, O nekim funkcionalnim nejednakostima, Univ.

Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No 599, (1977), 1–59.

[14] G.V. MILOVANOVI ´C AND J.E. PE ˇCARI ´C, On generalization of the in- equality of A. Ostrowski and some related applications, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No 544-576, (1976), 155-158.

[15] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´CANDA.M. FINK, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.

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[16] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´CANDA.M. FINK, Inequalities Involv- ing Functions and Their Integrals and Derivatives, Kluwer Acad. Publ., Dordrecht, Boston/London, 1991.

[17] A. OSTROWSKI, Über die Absolutabweichung einer Differentiebaren Funktion von ihren Integralmittelwert, Comment. Math. Helv., 10 (1938), 226–227.

[18] B.G. PACHPATTE, On multidimensional Grüss type inequalities, J. In- equal. Pure Appl. Math., 3(2) (2002), Article 27. [ONLINE http://

jipam.vu.edu.au/v3n2/063_01.html].