Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 4, Issue 5, Article 101, 2003
OSTROWSKI-GRÜSS TYPE INEQUALITIES IN TWO DIMENSIONS
NENAD UJEVIĆ
D EPARTMENT OF M ATHEMATICS
U NIVERSITY OF S PLIT
T ESLINA 12/III, 21000 S PLIT
CROATIA.
ujevic@pmfst.hr
Received 08 January, 2003; accepted 07 August, 2003
Communicated by B.G. Pachpatte
A BSTRACT. A general Ostrowski-Grüss type inequality in two dimensions is established. A
particular inequality of the same type is also given.
Key words and phrases: Ostrowski’s inequality, 2-dimensional generalization, Ostrowski-Grüss inequality.
2000 Mathematics Subject Classification. 26D10, 26D15.
1. I NTRODUCTION
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 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 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
ISSN (electronic): 1443-5756
c 2003 Victoria University. All rights reserved.
003-03
2
N ENAD U JEVI Ć
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.
2. A G ENERAL O STROWSKI -G RÜSS I NEQUALITY
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
Rk (s, t) = Pk (s)Qk (t), k = 0, 1, 2, . . . , n + 1.
(2.3)
Lemma 2.1. Let Rk (s, t) be defined by (2.3). Then we have
∂ 2 Rk (s, t)
= Rk−1 (s, t)
∂s∂t
(2.4)
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).
We now define
(2.5)
Z b
∂ 2k−1 f (b, t)
∂ 2k−1 f (a, t)
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
k
∂s ∂t
a
a
uk−1 (t) =
(2.6)
and
Z
(2.8)
Jk,2 =
a
b
∂ 2k−1 f (a, t)
Rk (a, t)
dt = Pk (a)
∂sk−1 ∂tk
Z
b
(k)
Qk (t)vk−1 (t)dt
a
such that
(2.9)
J. Inequal. Pure and Appl. Math., 4(5) Art. 101, 2003
Jk = Jk,1 − Jk,2 .
http://jipam.vu.edu.au/
O STROWSKI -G RÜSS TYPE I NEQUALITIES IN T WO D IMENSIONS
3
Lemma 2.2. Let Jk,1 be defined by (2.7). Then we have
(2.10)
Jk,1 = Pk (b)
k
X
h
i
(j−1)
(j−1)
(−1)k−j Qj (b)uk−1 (b) − Qj (a)uk−1 (a)
j=1
Z
k
+ (−1) Pk (b)
b
uk−1 (t)dt,
a
for k = 1, 2, . . . , n.
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)
k
= (−1) Qk (b)uk−1 (b) − Qk (a)uk−1 (a)
Z b
(k−1)
k−1
+ (−1)
Qk−1 (t)uk−1 (t)dt.
a
We can write the above relation in the form
h
i
(k−1)
(k−1)
(−1)k Uk (uk−1 ) = (−1)k 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)
k−1
Z
b
Uk−1 (uk−1 ) = (−1)
(k−1)
Qk−1 (t)uk−1 (t)dt
ha
i
(k−2)
(k−2)
= (−1)
Qk−1 (b)uk−1 (b) − Qk−1 (a)uk−1 (a)
Z b
(k−2)
k−2
Qk−2 (t)uk−1 (t)dt
+ (−1)
k−1
a
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
j
h
(j−1)
Qj (b)uk−1 (b)
j
h
(j−1)
Qj (b)uk−1 (b)
(−1)
i
−
(j−1)
Qj (a)uk−1 (a)
−
(j−1)
Qj (a)uk−1 (a)
+ U0 (uk−1 )
j=1
=
k
X
(−1)
i
Z
b
uk−1 (t)dt.
+
a
j=1
Note now that
Jk,1 = Pk (b)Uk (uk−1 )
such that (2.10) holds.
J. Inequal. Pure and Appl. Math., 4(5) Art. 101, 2003
http://jipam.vu.edu.au/
4
N ENAD U JEVI Ć
Lemma 2.3. Let Jk,2 be defined by (2.8). Then we have
(2.11)
Jk,2 = Pk (a)
k
X
h
i
(j−1)
(j−1)
(−1)k−j Qj (b)vk−1 (b) − Qj (a)vk−1 (a)
j=1
b
Z
k
+ (−1) Pk (a)
vk−1 (t)dt,
a
for k = 1, 2, . . . , n.
Proof. The proof is almost identical to that of Lemma 2.2.
We now define
Z b
(2.12)
Kk =
a
∂Rk (s, b) ∂ 2k−2 f (s, b) ∂Rk (s, a) ∂ 2k−2 f (s, a)
−
ds,
∂s
∂sk−1 ∂tk−1
∂s
∂sk−1 ∂tk−1
for k = 2, . . . , n,
xk−1 (s) =
(2.13)
∂ k−1 f (s, b)
∂ k−1 f (s, a)
,
y
(s)
=
k−1
∂tk−1
∂tk−1
and
Z
b
Z
x0 (s)ds − Q1 (a)
K1 = Q1 (b)
(2.14)
b
y0 (s)ds.
a
a
We also define
b
Z
(2.15)
Kk,1 =
a
∂Rk (s, b) ∂ 2k−2 f (s, b)
ds = Qk (b)
∂s
∂sk−1 ∂tk−1
b
Z
(k−1)
Pk−1 (s)xk−1 (s)ds
a
and
Z
(2.16)
Kk,2 =
a
b
∂Rk (s, a) ∂ 2k−2 f (s, a)
ds = Qk (a)
∂s
∂sk−1 ∂tk−1
Z
b
(k−1)
Pk−1 (s)yk−1 (s)ds
a
such that
Kk = Kk,1 − Kk,2 , k = 1, 2, . . . , n.
(2.17)
Lemma 2.4. Let Kk,1 be defined by (2.15). Then we have
(2.18)
Kk,1 = Qk (b)
k
X
k−j+1
(−1)
h
(j−2)
Pj−1 (b)xk−1 (b)
−
i
(j−2)
Pj−1 (a)xk−1 (a)
j=2
k−1
+ (−1)
Z
Qk (b)
b
xk−1 (s)ds,
a
for k = 2, . . . , n.
Proof. We introduce the notation
Z
Uk−1 (xk−1 ) =
b
(k−1)
Pk−1 (s)xk−1 (s)ds.
a
J. Inequal. Pure and Appl. Math., 4(5) Art. 101, 2003
http://jipam.vu.edu.au/
O STROWSKI -G RÜSS TYPE I NEQUALITIES IN T WO D IMENSIONS
5
Then we have
k−1
(−1)
k−1
Z
b
(k−1)
Pk−1 (s)xk−1 (s)ds
Uk−1 (xk−1 ) = (−1)
ha
i
(k−2)
(k−2)
= (−1)k−1 Pk−1 (b)xk−1 (b) − Pk−1 (a)xk−1 (a)
Z b
(k−2)
k−2
Pk−2 (s)xk−1 (s)ds.
+ (−1)
a
We can write the above relation in the form
(−1)k−1 Uk−1 (xk−1 )
i
h
(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
Z
b
Uk−2 (xk−1 ) = (−1)
(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
or
(−1)k−2 Uk−2 (xk−1 )
h
i
(k−3)
(k−3)
k−2
= (−1)
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
h
i
(j−2)
(j−2)
(−1)j−1 Pj−1 (b)xk−1 (b) − Pj−1 (a)xk−1 (a) + U0 (xk−1 )
j=2
=
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
k−j+1
(−1)
h
(j−2)
Pj−1 (b)yk−1 (b)
−
i
(j−2)
Pj−1 (a)yk−1 (a)
j=2
k−1
+ (−1)
Z
b
yk−1 (s)ds,
Qk (a)
a
for k = 2, . . . , n.
Proof. The proof is almost identical to that of Lemma 2.4.
J. Inequal. Pure and Appl. Math., 4(5) Art. 101, 2003
http://jipam.vu.edu.au/
6
N ENAD U JEVI Ć
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
hΦe − x, x − ϕei ≥ 0 and hΓe − y, y − γei ≥ 0
(2.20)
hold. In [5] we can find the inequality
|hx, yi − hx, ei hy, ei| ≤
(2.21)
1
|Φ − ϕ| |Γ − γ| .
4
We also have
|hx, yi − hx, ei hy, ei| ≤ kxk2 − hx, ei2
(2.22)
21
kyk2 − he, yi2
12
.
Let X = L2 (Ω) and e = 1/(b − a). If we define
Z bZ b
1
f (t, s)g(t, s)dtds
(2.23)
T (f, g) =
(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)
if
γ ≤ f (x, y) ≤ Γ, ϕ ≤ g(x, y) ≤ Φ, (x, y) ∈ Ω.
From (2.22), we have the pre-Grüss inequality
T (f, g)2 ≤ T (f, f )T (g, g).
(2.25)
We now define
Z bZ
b
In =
(2.26)
Rn (s, t)
a
a
∂ 2n f (s, t)
dsdt
∂sn ∂tn
and
1
Sn =
(b − a)2
(2.27)
Z bZ
b
Z bZ
Rn (s, t)dsdt
a
a
a
a
b
∂ 2n f (s, t)
dsdt.
∂sn ∂tn
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
b
2
Z
Pn (s) ds
a
b
Qn (t)2 dt
a
1
−
(b − a)4
J. Inequal. Pure and Appl. Math., 4(5) Art. 101, 2003
Z
b
Z
Pn (s)ds
a
b
2 ) 12
Qn (t)dt
.
a
http://jipam.vu.edu.au/
O STROWSKI -G RÜSS TYPE I NEQUALITIES IN T WO D IMENSIONS
7
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
Then from (2.25) we get
1
2
|In − Sn | ≤ (b − a)2 T (Rn (s, t), Rn (s, t)) T
∂ 2n f (s, t) ∂ 2n f (s, t)
,
∂sn ∂tn
∂sn ∂tn
12
.
From (2.24) we have
T
∂ 2n f (s, t) ∂ 2n f (s, t)
,
∂sn ∂tn
∂sn ∂tn
12
≤
M2n − m2n
.
2
We also have
T (Rn (s, t), Rn (s, t))
Z b
2
Z b
Z b
Z b
1
1
2
2
Pn (s) ds
Qn (t) dt −
Pn (s)ds
Qn (t)dt .
=
(b − a)2 a
(b − a)4
a
a
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
Z Z
n
n
b b
M −m
X
X
2n
2n
C(b − a)2 ,
(2.30)
f (s, t)dsdt +
Jk −
Kk − S n ≤
a a
2
k=1
k=1
where
(2.31)
Sn =
and v(s, t) =
1
[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)] ,
∂ 2n−2 t(s,t)
.
∂sn−1 ∂tn−1
Proof. We have
Z bZ
(2.32)
b
∂ 2n f (s, t)
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 (b, t)
∂ 2n−1 f (a, 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
= J n − Ln ,
In =
where
Z bZ
Ln =
a
J. Inequal. Pure and Appl. Math., 4(5) Art. 101, 2003
a
b
∂Rn (s, t) ∂ 2n−1 f (s, t)
dsdt.
∂s
∂sn−1 ∂tn
http://jipam.vu.edu.au/
8
N ENAD U JEVI Ć
We also have
Z
b
Ln =
Z
b
∂Rn (s, t) ∂ ∂ 2n−2 f (s, t)
dt
∂s
∂t ∂sn−1 ∂tn−1
ds
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 .
a
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
Kk + I0 ,
k=1
where
Z bZ
I0 =
(2.34)
b
f (s, t)dsdt.
a
a
We now consider the term
1
Sn =
(b − a)2
(2.35)
Z bZ
b
Z bZ
Rn (s, t)dsdt
a
a
a
a
b
∂ 2n f (s, t)
dsdt.
∂sn ∂tn
We have
Z bZ
b
Z
Rn (s, t)dsdt =
a
a
b
b
Z
Pn (s)ds
a
Qn (t)dt
a
= [Pn+1 (b) − Pn+1 (a)] [Qn+1 (b) − Qn+1 (a)]
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 b
n
n
X
X
In − Sn =
f (s, t)dsdt +
Jk −
Kk − S n .
a
a
k=1
Then from Lemma 2.6 we conclude that (2.30) holds.
J. Inequal. Pure and Appl. Math., 4(5) Art. 101, 2003
k=1
http://jipam.vu.edu.au/
O STROWSKI -G RÜSS TYPE I NEQUALITIES IN T WO D IMENSIONS
9
3. A PARTICULAR I NEQUALITY
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
(s − a)k ,
k!
k
1
a+b
s−
,
Pk (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
Pk (s) =
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
(b − a)k
s−a
(b − a)k
t−a
(3.1)
Pk (s) =
Bk
,
Qk (t) =
Bk
.
k!
b−a
k!
b−a
We now substitute the above polynomials in (2.10), (2.11), (2.18), (2.19) to obtain
(3.2) Jk,1 = J¯k,1
k
X
(b − a)k
(b − a)j
=
Bk (1)
(−1)k−j
k!
j!
j=1
×
(3.3)
h
(j−1)
Bj (1)uk−1 (b)
−
i
(j−1)
Bj (0)uk−1 (a)
(b − a)k
+ (−1) Bk (1)
k!
k
Z
b
uk−1 (t)dt,
a
Jk,2 = J¯k,2
k
i
j h
X
(b − a)k
(j−1)
(j−1)
k−j (b − a)
=
Bk (0)
(−1)
Bj (1)vk−1 (b) − Bj (0)vk−1 (a)
k!
j!
j=1
Z
(b − a)k b
k
+ (−1) Bk (0)
vk−1 (t)dt,
k!
a
(3.4)
Kk,1 = K̄k,1
k
j−1
X
(b − a)k
k−j+1 (b − a)
=
Bk (1)
(−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)
+ (−1)
Bk (1)
xk−1 (s)ds,
k!
a
J. Inequal. Pure and Appl. Math., 4(5) Art. 101, 2003
http://jipam.vu.edu.au/
10
N ENAD U JEVI Ć
and
Kk,2 = K̄k,2
(3.5)
=
k
X
(b − a)j−1
(b − a)k
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)
Jk = J¯k = J¯k,1 − J¯k,2 , k = 1, 2, . . . , n,
(3.7)
Kk = K̄k = K̄k,1 − K̄k,2 , k = 2, . . . , n
and
b−a
K̄1 =
2
(3.8)
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
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, . . . ,
(3.13)
Bk (0) = Bk (1) = Bk , k = 0, 2, 3, 4, . . . ,
and, in particular,
1
1
B1 (0) = − , B1 (1) = .
2
2
From (3.2) – (3.8) and (3.12) we see that
(3.14)
(3.15)
J¯2i+1 = K̄2i+1 = 0, i = 1, 2, . . . , n.
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. Inequal. Pure and Appl. Math., 4(5) Art. 101, 2003
http://jipam.vu.edu.au/
O STROWSKI -G RÜSS TYPE I NEQUALITIES IN T WO D IMENSIONS
11
Theorem 3.1. Under the assumptions of Theorem 2.7 we have
Z Z
n
n
b b
M −m
X
X
2n
2n |B2n |
(3.16)
f (s, t)dsdt +
J¯k −
K̄k ≤
·
(b − a)2n+2 ,
a a
2
(2n)!
k=1
k=1
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
b
Z
(s, t)dsdt =
a
an
b
Z
Pn (s)ds
Qn (t)dt
a
a
Z
=
2
b
Pn (s)ds
a

=
n+1
(b − a)
n!
Z1
2
Bn (s)ds = 0,
0
because of (3.9).
Secondly, we have
(
Z b
2 ) 12
Z b
Z b
Z b
1
1
C=
Pn (s)2 ds
Qn (t)2 dt −
Pn (s)ds
Qn (t)dt
(b − a)2 a
(b − a)4
a
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
b−a
(n!)2
0
(b − a)2n (n!)2
|B2n |
=
|B2n | =
(b − a)2n ,
2
(n!) (2n)!
(2n)!
since (3.10) holds.
R EFERENCES
[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.
J. Inequal. Pure and Appl. Math., 4(5) Art. 101, 2003
http://jipam.vu.edu.au/
12
N ENAD U JEVI Ć
[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., 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.
[8] M. MATIĆ, J. PEČARIĆ AND N. UJEVIĆ, On new estimation of the remainder in generalized
Taylor’s formula, Math. Inequal. Appl., 2(3), (1999), 343–361.
[9] M. MATIĆ, J. PEČARIĆ AND N. UJEVIĆ, Improvement and further generalization of some inequalities of Ostrowski-Grüss type, Comput. Math. Appl., 39 (2000), 161–175.
[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.
[16] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequalities Involving 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. Inequal. Pure Appl. Math.,
3(2) (2002), Article 27. [ONLINE http://jipam.vu.edu.au/v3n2/063_01.html].
J. Inequal. Pure and Appl. Math., 4(5) Art. 101, 2003
http://jipam.vu.edu.au/