Journal of Inequalities in Pure and
Applied Mathematics
OSTROWSKI-GRÜSS TYPE INEQUALITIES IN TWO DIMENSIONS
NENAD UJEVIĆ
Department of Mathematics
University of Split
Teslina 12/III, 21000 Split
CROATIA.
EMail: ujevic@pmfst.hr
volume 4, issue 5, article 101,
2003.
Received 08 January, 2003;
accepted 07 August, 2003.
Communicated by: B.G. Pachpatte
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
003-03
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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.
Contents
Ostrowski-Grüss type
Inequalities in Two Dimensions
Nenad Ujević
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
A General Ostrowski-Grüss Inequality . . . . . . . . . . . . . . . . . . . 4
3
A Particular Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
References
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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 interior Int I of I, and let a, b ∈ Int I, a < b. If |f 0 (t)| ≤
M , ∀t ∈ [a,
#
b], then we have
"
Z b
a+b 2
(x
−
)
1
1
2
f (x) −
(1.1)
f (t)dt ≤
+
(b − a)M,
b−a a
4
(b − a)2
for x ∈ [a, b].
The first (direct) generalization of Ostrowski’s inequality was given by G.V.
Milovanović and J. Pečarić 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 OstrowskiGrüss 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 versions 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 Section 3
we use the mentioned general inequality to obtain a particular two-dimensional
Ostrowski-Grüss type inequality.
Ostrowski-Grüss type
Inequalities in Two Dimensions
Nenad Ujević
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2.
A General Ostrowski-Grüss Inequality
Let Ω = [a, b] × [a, b] and let f : Ω → R be a given function. Here we suppose
that f ∈ C 2n (Ω). Let Pk (s) and Qk (t) be harmonic or Appell-like polynomials,
i.e.
(2.1)
Pk0 (s) = Pk−1 (s) and Q0k (t) = Qk−1 (t), k = 1, 2, . . . , n + 1,
with
P0 (s) = Q0 (t) = 1.
(2.2)
We also define
(2.3)
Rk (s, t) = Pk (s)Qk (t), k = 0, 1, 2, . . . , n + 1.
Lemma 2.1. Let Rk (s, t) be defined by (2.3). Then we have
(2.4)
∂ 2 Rk (s, t)
= Rk−1 (s, t)
∂s∂t
for k = 1, 2, . . . , n + 1.
Proof. From (2.1) – (2.3) it follows that
∂ 2 Rk (s, t)
∂ ∂Rk (s, t)
=
∂s∂t
∂t
∂s
∂
=
(P 0 (s)Qk (t))
∂t k
= Pk−1 (s)Q0k (t)
= Pk−1 (s)Qk−1 (t) = Rk−1 (s, t).
Ostrowski-Grüss type
Inequalities in Two Dimensions
Nenad Ujević
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We now define
Z b
∂ 2k−1 f (b, t)
∂ 2k−1 f (a, t)
(2.5)
Jk =
Rk (b, t)
− Rk (a, t)
dt,
∂sk−1 ∂tk
∂sk−1 ∂tk
a
∂ k−1 f (b, t)
∂ k−1 f (a, t)
,
v
(t)
=
,
k−1
∂sk−1
∂sk−1
for k = 1, 2, . . . , n. We also define
Z b
Z b
∂ 2k−1 f (b, t)
(k)
(2.7)
Jk,1 =
Rk (b, t)
dt = Pk (b)
Qk (t)uk−1 (t)dt
k−1 ∂tk
∂s
a
a
(2.6)
uk−1 (t) =
Ostrowski-Grüss type
Inequalities in Two Dimensions
Nenad Ujević
and
Z
(2.8)
b
Jk,2 =
a
∂ 2k−1 f (a, t)
dt = Pk (a)
Rk (a, t)
∂sk−1 ∂tk
b
Z
(k)
Qk (t)vk−1 (t)dt
a
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such that
JJ
J
Jk = Jk,1 − Jk,2 .
(2.9)
Lemma 2.2. Let Jk,1 be defined by (2.7). Then we have
(2.10) Jk,1 = Pk (b)
k
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h
i
(j−1)
(j−1)
(−1)k−j Qj (b)uk−1 (b) − Qj (a)uk−1 (a)
j=1
k
Z
+ (−1) Pk (b)
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b
uk−1 (t)dt,
a
for k = 1, 2, . . . , n.
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Proof. We introduce the notation
b
Z
Uk (uk−1 ) =
(k)
Qk (t)uk−1 (t)dt.
a
Then we have
k
k
Z
b
(−1) Uk (uk−1 ) = (−1)
(k)
Qk (t)uk−1 (t)dt
ha
i
(k−1)
(k−1)
= (−1) Qk (b)uk−1 (b) − Qk (a)uk−1 (a)
Z b
(k−1)
k−1
Qk−1 (t)uk−1 (t)dt.
+ (−1)
k
Ostrowski-Grüss type
Inequalities in Two Dimensions
Nenad Ujević
a
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We can write the above relation in the form
(−1)k Uk (uk−1 )
i
h
(k−1)
(k−1)
k
= (−1) Qk (b)uk−1 (b) − Qk (a)uk−1 (a) + (−1)k−1 Uk−1 (uk−1 ).
In a similar way we get
k−1
(−1)
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k−1
Uk−1 (uk−1 ) = (−1)
Z
b
(k−1)
Qk−1 (t)uk−1 (t)dt
ha
i
(k−2)
(k−2)
= (−1)k−1 Qk−1 (b)uk−1 (b) − Qk−1 (a)uk−1 (a)
Z b
(k−2)
k−2
+ (−1)
Qk−2 (t)uk−1 (t)dt
a
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or
(−1)k−1 Uk−1 (uk−1 )
h
i
(k−2)
(k−2)
= (−1)k−1 Qk−1 (b)uk−1 (b) − Qk−1 (a)uk−1 (a)
+ (−1)k−2 Uk−2 (uk−1 ).
If we continue the above procedure then we obtain
(−1)k Uk (uk−1 )
=
k
X
Ostrowski-Grüss type
Inequalities in Two Dimensions
i
h
(j−1)
(j−1)
(−1)j Qj (b)uk−1 (b) − Qj (a)uk−1 (a) + U0 (uk−1 )
Nenad Ujević
j=1
=
k
X
j
(−1)
h
(j−1)
Qj (b)uk−1 (b)
−
i
(j−1)
Qj (a)uk−1 (a)
b
Z
+
uk−1 (t)dt.
a
j=1
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Note now that
Jk,1 = Pk (b)Uk (uk−1 )
such that (2.10) holds.
(2.11) Jk,2 = Pk (a)
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Lemma 2.3. Let Jk,2 be defined by (2.8). Then we have
k
X
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k−j
(−1)
h
(j−1)
Qj (b)vk−1 (b)
−
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i
(j−1)
Qj (a)vk−1 (a)
j=1
+ (−1)k Pk (a)
Z
Page 7 of 26
b
vk−1 (t)dt,
a
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for k = 1, 2, . . . , n.
Proof. The proof is almost identical to that of Lemma 2.2.
We now define
Z b
∂Rk (s, b) ∂ 2k−2 f (s, b) ∂Rk (s, a) ∂ 2k−2 f (s, a)
(2.12) Kk =
−
ds,
∂s
∂sk−1 ∂tk−1
∂s
∂sk−1 ∂tk−1
a
for k = 2, . . . , n,
(2.13)
xk−1 (s) =
∂ k−1 f (s, b)
∂ k−1 f (s, a)
,
y
(s)
=
k−1
∂tk−1
∂tk−1
Nenad Ujević
and
b
Z
(2.14)
Z
x0 (s)ds − Q1 (a)
K1 = Q1 (b)
a
b
y0 (s)ds.
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b
∂Rk (s, b) ∂ 2k−2 f (s, b)
ds
∂s
∂sk−1 ∂tk−1
a
Z b
(k−1)
= Qk (b)
Pk−1 (s)xk−1 (s)ds
Z
Kk,1 =
a
and
Kk,2
b
∂Rk (s, a) ∂ 2k−2 f (s, a)
=
ds
∂s
∂sk−1 ∂tk−1
a
Z b
(k−1)
= Qk (a)
Pk−1 (s)yk−1 (s)ds
Z
(2.16)
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a
We also define
(2.15)
Ostrowski-Grüss type
Inequalities in Two Dimensions
a
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such that
(2.17)
Kk = Kk,1 − Kk,2 , k = 1, 2, . . . , n.
Lemma 2.4. Let Kk,1 be defined by (2.15). Then we have
(2.18) Kk,1 = Qk (b)
k
X
h
i
(j−2)
(j−2)
(−1)k−j+1 Pj−1 (b)xk−1 (b) − Pj−1 (a)xk−1 (a)
j=2
k−1
+ (−1)
Z
Qk (b)
b
xk−1 (s)ds,
a
Ostrowski-Grüss type
Inequalities in Two Dimensions
Nenad Ujević
for k = 2, . . . , n.
Title Page
Proof. We introduce the notation
Contents
b
Z
Uk−1 (xk−1 ) =
(k−1)
JJ
J
Pk−1 (s)xk−1 (s)ds.
a
Then we have
II
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(−1)k−1 Uk−1 (xk−1 ) = (−1)k−1
Z
b
(k−1)
Close
Pk−1 (s)xk−1 (s)ds
a
k−1
= (−1)
h
(k−2)
Pk−1 (b)xk−1 (b)
+ (−1)k−2
Z
b
−
i
(k−2)
Pk−1 (a)xk−1 (a)
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(k−2)
Pk−2 (s)xk−1 (s)ds.
a
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We can write the above relation in the form
(−1)k−1 Uk−1 (xk−1 )
h
i
(k−2)
(k−2)
= (−1)k−1 Pk−1 (b)xk−1 (b) − Pk−1 (a)xk−1 (a)
+ (−1)k−2 Uk−2 (xk−1 ).
In a similar way we get
k−2
(−1)
k−2
Uk−2 (xk−1 ) = (−1)
Z
b
(k−2)
Pk−2 (s)xk−1 (s)ds
ha
i
(k−3)
(k−3)
= (−1)
Pk−2 (b)xk−1 (b) − Pk−2 (a)xk−1 (a)
Z b
(k−3)
k−3
Pk−3 (s)xk−1 (s)ds
+ (−1)
k−2
a
(−1)k−2 Uk−2 (xk−1 )
i
h
(k−3)
(k−3)
= (−1)k−2 Pk−2 (b)xk−1 (b) − Pk−2 (a)xk−1 (a)
+ (−1)k−3 Uk−3 (xk−1 ).
If we continue the above procedure then we get
(−1)k−1 Uk−1 (xk−1 )
k
X
j=2
Nenad Ujević
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or
=
Ostrowski-Grüss type
Inequalities in Two Dimensions
h
i
(j−2)
(j−2)
(−1)j−1 Pj−1 (b)xk−1 (b) − Pj−1 (a)xk−1 (a) + U0 (xk−1 )
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=
k
X
j−1
(−1)
h
(j−2)
Pj−1 (b)xk−1 (b)
−
i
(j−2)
Pj−1 (a)xk−1 (a)
Z
+
b
xk−1 (t)dt.
a
j=2
Note now that
Kk,1 = Qk (b)Uk−1 (xk−1 )
such that (2.18) holds.
Lemma 2.5. Let Kk,2 be defined by (2.16). Then we have
(2.19) Kk,2 = Qk (a)
k
X
Ostrowski-Grüss type
Inequalities in Two Dimensions
k−j+1
(−1)
h
(j−2)
Pj−1 (b)yk−1 (b)
−
i
(j−2)
Pj−1 (a)yk−1 (a)
Nenad Ujević
j=2
k−1
+ (−1)
Z
Qk (a)
b
yk−1 (s)ds,
a
for k = 2, . . . , n.
Proof. The proof is almost identical to that of Lemma 2.4.
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Let (X, h·, ·i) be a real inner product space and e ∈ X, kek = 1. Let
γ, ϕ, Γ, Φ be real numbers and x, y ∈ X such that the conditions
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hΦe − x, x − ϕei ≥ 0 and hΓe − y, y − γei ≥ 0
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(2.20)
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hold. In [5] we can find the inequality
(2.21)
|hx, yi − hx, ei hy, ei| ≤
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1
|Φ − ϕ| |Γ − γ| .
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We also have
(2.22)
|hx, yi − hx, ei hy, ei| ≤ kxk2 − hx, ei2
12
kyk2 − he, yi2
12
.
Let X = L2 (Ω) and e = 1/(b − a). If we define
Z bZ b
1
(2.23) T (f, g) =
f (t, s)g(t, s)dtds
(b − a)2 a a
Z bZ b
Z bZ b
1
−
f (t, s)dtds
g(t, s)dtds,
(b − a)4 a a
a
a
then from (2.20) and (2.21) we obtain the Grüss inequality in L2 (Ω),
1
|T (f, g)| ≤ (Γ − γ)(Φ − ϕ),
4
(2.24)
γ ≤ f (x, y) ≤ Γ, ϕ ≤ g(x, y) ≤ Φ, (x, y) ∈ Ω.
T (f, g)2 ≤ T (f, f )T (g, g).
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I
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We now define
Z bZ
b
In =
a
a
∂ 2n f (s, t)
dsdt
Rn (s, t)
∂sn ∂tn
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and
(2.27)
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From (2.22), we have the pre-Grüss inequality
(2.26)
Nenad Ujević
Title Page
if
(2.25)
Ostrowski-Grüss type
Inequalities in Two Dimensions
Sn =
1
(b − a)2
Z bZ
Z bZ
b
Rn (s, t)dsdt
a
a
a
a
b
∂ 2n f (s, t)
dsdt.
∂sn ∂tn
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Lemma 2.6. Let In and Sn be defined by (2.26) and (2.27), respectively. Then
we have the inequality
|In − Sn | ≤
(2.28)
where
M2n − m2n
C(b − a)2 ,
2
∂ 2n f (s, t)
∂ 2n f (s, t)
,
m
=
min
2n
(s,t)∈Ω ∂sn ∂tn
(s,t)∈Ω ∂sn ∂tn
M2n = max
and
(2.29) C =
1
(b − a)2
Z
a
b
Pn (s)2 ds
Z
Ostrowski-Grüss type
Inequalities in Two Dimensions
b
Qn (t)2 dt
Nenad Ujević
a
1
−
(b − a)4
Z
b
Z
Pn (s)ds
a
b
2 ) 12
.
Qn (t)dt
Contents
Proof. From (2.23), (2.26) and (2.27) we see that
∂ 2n f (s, t)
2
.
In − Sn = (b − a) T Rn (s, t),
∂sn ∂tn
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Then from (2.25) we get
2
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a
1
2
|In − Sn | ≤ (b − a) T (Rn (s, t), Rn (s, t)) T
∂ 2n f (s, t) ∂ 2n f (s, t)
,
∂sn ∂tn
∂sn ∂tn
From (2.24) we have
2n
1
∂ f (s, t) ∂ 2n f (s, t) 2
M2n − m2n
T
,
≤
.
n
n
n
n
∂s ∂t
∂s ∂t
2
12
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.
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We also have
1
T (Rn (s, t), Rn (s, t)) =
(b − a)2
Z
b
2
Z
b
Qn (t)2 dt
Pn (s) ds
a
1
−
(b − a)4
Z
a
b
Z
Pn (s)ds
a
b
2
Qn (t)dt .
a
From the last three relations we see that (2.28) holds.
Theorem 2.7. Let Ω = [a, b]×[a, b] and let f : Ω → R be a given function such
that f ∈ C 2n (Ω). Let the conditions of Lemma 2.6 hold. If Jk , Kk are given by
(2.9), (2.17), where Jk,1 , Jk,2 , Kk,1 , Kk,2 are given by Lemmas 2.2 – 2.5, then
we have the inequality
Ostrowski-Grüss type
Inequalities in Two Dimensions
Nenad Ujević
Title Page
(2.30)
Z Z
n
n
b b
X
X
f (s, t)dsdt +
Jk −
Kk − S n a a
k=1
k=1
M2n − m2n
C(b − a)2 ,
≤
2
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JJ
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where
Close
1
(2.31) Sn =
[Pn+1 (b) − Pn+1 (a)] [Qn+1 (b) − Qn+1 (a)]
(b − a)2
× [v(b, b) − v(b, a) − v(a, b) + v(a, a)] ,
and v(s, t) =
∂ 2n−2 t(s,t)
.
∂sn−1 ∂tn−1
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Proof. We have
Z bZ
(2.32)
b
∂ 2n f (s, t)
In =
Rn (s, t)
dsdt
∂sn ∂tn
a
a
Z b Z b
∂ ∂ 2n−1 f (s, t)
=
dt
Rn (s, t)
ds
∂s ∂sn−1 ∂tn
a
a
Z b
∂ 2n−1 f (a, t)
∂ 2n−1 f (b, t)
=
Rn (b, t)
− Rn (a, t)
dt
∂sn−1 ∂tn
∂sn−1 ∂tn
a
Z bZ b
∂Rn (s, t) ∂ 2n−1 f (s, t)
dsdt
−
∂s
∂sn−1 ∂tn
a
a
= Jn − Ln ,
Ostrowski-Grüss type
Inequalities in Two Dimensions
Nenad Ujević
Title Page
where
Z bZ
Ln =
a
a
b
∂Rn (s, t) ∂ 2n−1 f (s, t)
dsdt.
∂s
∂sn−1 ∂tn
We also have
Z b Z b
∂Rn (s, t) ∂ ∂ 2n−2 f (s, t)
Ln =
ds
dt
∂s
∂t ∂sn−1 ∂tn−1
a
a
Z b
∂Rn (s, b) ∂ 2n−2 f (s, b) ∂Rn (s, a) ∂ 2n−2 f (s, a)
=
−
ds
∂s
∂sn−1 ∂tn−1
∂s
∂sn−1 ∂tn−1
a
Z bZ b
∂ 2n−2 f (s, t)
−
Rn−1 (s, t) n−1 n−1 dsdt
∂s ∂t
a
a
= Kn − 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
In =
(2.33)
n
X
Jk −
k=1
n
X
Ostrowski-Grüss type
Inequalities in Two Dimensions
Kk + I0 ,
k=1
Nenad Ujević
where
Z bZ
I0 =
(2.34)
b
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f (s, t)dsdt.
a
Contents
a
We now consider the term
1
Sn =
(b − a)2
(2.35)
We have
Z bZ
Z bZ
a
Z bZ
Rn (s, t)dsdt
a
a
a
a
b
∂ 2n f (s, t)
dsdt.
∂sn ∂tn
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b
Z
Rn (s, t)dsdt =
a
b
JJ
J
b
Z
Pn (s)ds
a
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b
Qn (t)dt
a
Page 16 of 26
= [Pn+1 (b) − Pn+1 (a)] [Qn+1 (b) − Qn+1 (a)]
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and
Z bZ
a
a
b
∂ 2n f (s, t)
dsdt
∂sn ∂tn
Z b Z b
∂ ∂ 2n−1 f (s, t)
=
dt
ds
∂sn−1 ∂tn
a
a ∂s
Z b 2n−1
∂
f (b, t) ∂ 2n−1 f (a, t)
=
−
dt
∂sn−1 ∂tn
∂sn−1 ∂tn
a
∂ 2n−2 f (b, b) ∂ 2n−2 f (b, a) ∂ 2n−1 f (a, b) ∂ 2n−1 f (a, a)
=
−
−
+
∂sn−1 ∂tn−1
∂sn−1 ∂tn−1
∂sn−1 ∂tn−1
∂sn−1 ∂tn−1
= [v(b, b) − v(b, a) − v(a, b) + v(a, a)] ,
Thus (2.31) holds. From (2.33) – (2.35) we see that
Z bZ
In − Sn =
b
f (s, t)dsdt +
a
a
n
X
Jk −
k=1
Then from Lemma 2.6 we conclude that (2.30) holds.
Ostrowski-Grüss type
Inequalities in Two Dimensions
Nenad Ujević
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n
X
k=1
Contents
Kk − S n .
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3.
A Particular Inequality
Here we use the notations introduced in Section 2. In Theorem 2.7 we proved
a general inequality of Ostrowski-Grüss type. Many particular inequalities can
be obtained if we choose specific harmonic or Appell-like polynomials Pk (s),
Qk (t) in (2.30). For example, in [8] we can find the following harmonic polynomials
1
Pk (s) = (s − a)k ,
k!
k
1
a+b
Pk (s) =
s−
,
k!
2
(b − a)k
s−a
Pk (s) =
Bk
,
k!
b−a
(b − a)k
s−a
Pk (s) =
Ek
,
k!
b−a
where Bk (s) and Ek (s) are Bernoulli and Euler polynomials, respectively. We
shall not consider all possible combinations of these polynomials. Here we
choose the following combination
s−a
(b − a)k
(3.1)
Pk (s) =
Bk
,
k!
b−a
(b − a)k
t−a
Qk (t) =
Bk
.
k!
b−a
We now substitute the above polynomials in (2.10), (2.11), (2.18), (2.19) to
obtain
Ostrowski-Grüss type
Inequalities in Two Dimensions
Nenad Ujević
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(3.2)
Jk,1 = J¯k,1
k
j
X
(b − a)k
k−j (b − a)
=
Bk (1)
(−1)
k!
j!
j=1
h
i
(j−1)
(j−1)
× Bj (1)uk−1 (b) − Bj (0)uk−1 (a)
Z
(b − a)k b
k
+ (−1) Bk (1)
uk−1 (t)dt,
k!
a
(3.3)
Ostrowski-Grüss type
Inequalities in Two Dimensions
Jk,2 = J¯k,2
k
X
(b − a)k
(b − a)j
=
Bk (0)
(−1)k−j
k!
j!
j=1
h
i
(j−1)
(j−1)
× Bj (1)vk−1 (b) − Bj (0)vk−1 (a)
Z
(b − a)k b
k
vk−1 (t)dt,
+ (−1) Bk (0)
k!
a
(3.4)
Kk,1 = K̄k,1
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k
=
Nenad Ujević
k
X
j−1
(b − a)
(b − a)
Bk (1)
(−1)k−j+1
k!
(j − 1)!
j=2
h
i
(j−2)
(j−2)
× Bj−1 (1)xk−1 (b) − Bj−1 (0)xk−1 (a)
Z b
k
k−1 (b − a)
Bk (1)
xk−1 (s)ds,
+ (−1)
k!
a
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and
(3.5)
Kk,2 = K̄k,2
=
k
X
(b − a)k
(b − a)j−1
Bk (0)
(−1)k−j+1
k!
(j − 1)!
j=2
h
i
(j−2)
(j−2)
× Bj−1 (1)yk−1 (b) − Bj−1 (0)yk−1 (a)
Z b
k
k−1 (b − a)
+ (−1)
Bk (0)
yk−1 (s)ds.
k!
a
We have
(3.6)
Ostrowski-Grüss type
Inequalities in Two Dimensions
Nenad Ujević
Jk = J¯k = J¯k,1 − J¯k,2 , k = 1, 2, . . . , n,
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(3.7)
Kk = K̄k = K̄k,1 − K̄k,2 , k = 2, . . . , n
and
(3.8)
b−a
K̄1 =
2
Z
b
Z
x0 (s)ds +
a
b
y0 (s)ds ,
a
where J¯k,1 , J¯k,2 , K̄k,1 , K̄k,2 are defined by (3.2) – (3.5), respectively.
Basic properties of Bernoulli polynomials can be found in [1]. Here we
emphasize the following properties:
Z 1
(3.9)
Bk (s)ds = 0, k = 1, 2, . . .
0
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and
Z
(3.10)
1
Bk (s)Bj (s)ds = (−1)k−1
0
k!j!
Bk+j , k, j = 1, 2, . . . ,
(k + j)!
where
(3.11)
Bk = Bk (0), k = 0, 1, 2, . . .
are Bernoulli numbers. We also have
(3.12)
B2i+1 = 0, i = 1, 2, . . . ,
Ostrowski-Grüss type
Inequalities in Two Dimensions
Nenad Ujević
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(3.13)
Bk (0) = Bk (1) = Bk , k = 0, 2, 3, 4, . . . ,
and, in particular,
(3.14)
1
1
B1 (0) = − , B1 (1) = .
2
2
From (3.2) – (3.8) and (3.12) we see that
(3.15)
J¯2i+1 = K̄2i+1 = 0, i = 1, 2, . . . , n.
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Note also that sums in (3.2) – (3.5) have only even-indexed terms and the term
for j = 1 (j = 2) is non-zero.
J. Ineq. Pure and Appl. Math. 4(5) Art. 101, 2003
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Theorem 3.1. Under the assumptions of Theorem 2.7 we have
Z Z
n
n
b b
X
X
¯
(3.16)
f (s, t)dsdt +
Jk −
K̄k a a
k=1
k=1
M2n − m2n |B2n |
·
(b − a)2n+2 ,
2
(2n)!
≤
where Bk are Bernoulli numbers and J¯k , K̄k are given by (3.6), (3.7), respectively.
Proof. The proof follows from the proof of Theorem 2.7, since the following is
valid. Let Pn and Qn be defined by (3.1), for k = n.
Firstly, we have
Z bZ b
Z b Z b 2n
1
∂ f (s, t)
Sn =
Rn (s, t)dsdt
dsdt = 0,
2
(b − a) a a
∂sn ∂tn
a
a
since
Z bZ
b
Z
b
(s, t)dsdt =
a
an
Pn (s)ds
a
a
=
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2
b
Nenad Ujević
b
Z
Z
Ostrowski-Grüss type
Inequalities in Two Dimensions
Pn (s)ds
Quit
a

n+1
(b − a)
=
n!
Z1
2
Bn (s)ds = 0,
0
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J. Ineq. Pure and Appl. Math. 4(5) Art. 101, 2003
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because of (3.9).
Secondly, we have
Z b
Z b
1
2
Pn (s) ds
C=
Qn (t)2 dt
2
(b − a) a
a
Z b
2 ) 12
Z b
1
Pn (s)ds
Qn (t)dt
−
(b − a)4
a
a
12
Z b
Z b
1
=
Pn (s)2 ds
Qn (t)2 dt
(b − a)2 a
a
Z b
1
=
Pn (s)2 ds
b−a a
Z
1
(b − a)2n+1 1
·
Bn (s)2 ds
=
2
b−a
(n!)
0
2
2n
(b − a) (n!)
|B2n |
=
|B2n | =
(b − a)2n ,
2
(n!) (2n)!
(2n)!
since (3.10) holds.
Ostrowski-Grüss type
Inequalities in Two Dimensions
Nenad Ujević
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References
[1] M. ABRAMOWITZ AND I.A. STEGUN (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National
Bureau of Standards, Applied Mathematics Series 55, 4th printing, Washington, 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. DRAGOMIR AND S. 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.,
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[7] G. HANNA, P. CERONE AND J. ROUMELIOTIS, An Ostrowski type
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(2000), 671–689.
Ostrowski-Grüss type
Inequalities in Two Dimensions
Nenad Ujević
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[8] M. MATIĆ, J. PEČARIĆ AND N. UJEVIĆ, On new estimation of the
remainder in generalized Taylor’s formula, Math. Inequal. Appl., 2(3),
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[10] M. MATIĆ, J. PEČARIĆ AND N. UJEVIĆ, Generalizations of weighted
version of Ostrowski’s inequality and some related results, J. Inequal.
Appl., 5 (2000), 639–666.
[11] M. MATIĆ, J. PEČARIĆ AND N. UJEVIĆ, Weighted version of multivariate Ostrowski type inequalities, Rocky Mountain J. Math, 31(2) (2001),
511–538.
[12] G.V. MILOVANOVIĆ, On some integral inequalities, Univ. Beograd Publ.
Elektrotehn. Fak. Ser. Mat. Fiz., No 498-541, (1975), 119–124.
[13] G.V. MILOVANOVIĆ, O nekim funkcionalnim nejednakostima, Univ.
Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No 599, (1977), 1–59.
[14] G.V. MILOVANOVIĆ AND J.E. PEČARIĆ, On generalization of the inequality of A. Ostrowski and some related applications, Univ. Beograd
Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No 544-576, (1976), 155-158.
[15] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and New
Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
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Inequalities in Two Dimensions
Nenad Ujević
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[16] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Acad. Publ.,
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[17] A. OSTROWSKI, Über die Absolutabweichung einer Differentiebaren
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[18] B.G. PACHPATTE, On multidimensional Grüss type inequalities, J. Inequal. Pure Appl. Math., 3(2) (2002), Article 27. [ONLINE http://
jipam.vu.edu.au/v3n2/063_01.html].
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Nenad Ujević
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