Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 960672, 8 pages
doi:10.1155/2010/960672
Research Article
On Ostrowski-Type Inequalities for
Higher-Order Partial Derivatives
Zhao Changjian1 and Wing-Sum Cheung2
1
Department of Information and Mathematics Sciences, College of Science, China Jiliang University,
Hangzhou 310018, China
2
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Correspondence should be addressed to Zhao Changjian, chjzhao@163.com
Received 4 November 2009; Accepted 14 January 2010
Academic Editor: S. S. Dragomir
Copyright q 2010 Z. Changjian and W.-S. Cheung. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We establish some new Ostrowski-type integral inequalities involving higher-order partial
derivatives. As applications, we get some interrelated results. Our results provide new estimates
on inequalities of this type.
1. Introduction
The following inequality is well known in the literature as Ostrowski’s integral inequality
see 1, page 468.
Theorem 1.1. Let f : a, b → R be a differentiable mapping on a, b whose derivative f :
a, b → R is bounded on a, b; that is, f ∞ supt∈a,b |f t| < ∞, then
b
1 x − a b/22
1
ftdt ≤
b − af ∞ ,
fx −
2
b−a a
4
b − a
1.1
for all x ∈ a, b.
Many generalizations, extensions and variations of this inequality have appeared
in the literature; see 1–10 and the references given therein. In particular, in 2009, Wang
2
Journal of Inequalities and Applications
and Zhao 11 established a new Ostrowski-type inequality for higher-order derivatives as
follows see 11 for definitions and notations:
b
f n 1
a b b − a n b − an
∞
x
−
.
fxdx ≤
fx −
b−a a
n!
2 2
n 1
1.2
The main purpose of the present paper is to establish the following Ostrowski-type
inequality involving higher-order partial derivatives see next section for definitions and
notations:
bd
1
f x, y dxdy
f x, y −
b − ad − c a c
n
K a b b − a m c d d − c n−m
≤
y
−
Cnm x −
n! m0
2 2
2 2
n
Cnm b − am1 d − cn−m1
1
,
b − ad − c m0
m 1n − m 1
1.3
where K > 0 is a constant, Cnm n!/m!n − m!, m, n ∈ N, and 0 ≤ m ≤ n.
This is a generalization of inequality 1.2.
Moreover, as applications, we get some interrelated results. Our results provide new
estimates on such type of inequalities.
2. Main Results
Theorem 2.1. Suppose that
1 fx, y : a, b × c, d → R is continuous on a, b × c, d;
2 fx, y : a, b × c, d → R is differentiable in a, b × c, d up to order n, with bounded
nth-order mixed partial derivatives ∂n fx, y/∂xm ∂yn−m (m, n are natural numbers, and
0 ≤ m ≤ n), that is,
sup
∂n
∂sm ∂tn−m fs, t K < ∞;
s,t∈a,b×c,d
0≤m≤n
2.1
3 there exists x0 , y0 ∈ a, b × c, d such that ∂k fs, t/∂sm ∂tk−m |x0 ,y0 0, k 1, 2, . . . , n − 1; m 0, 1, . . . , n,
Journal of Inequalities and Applications
3
then for any x, y ∈ a, b × c, d,
bd
1
fs, tdsdt
f x, y −
b − ad − c a c
n
K a b b − a m c d d − c n−m
≤
y
−
Cnm x −
n! m0
2 2
2 2
n
Cnm b − am1 d − cn−m1
1
,
b − ad − c m0
m 1n − m 1
2.2
where Cnm n!/m!n − m!.
Proof. From the hypotheses and using the n-dimensional Taylor expansion of fx, y at
x0 , y0 , we have, for some 0 < θ < 1,
n
1
n−m
f x, y f x0 , y0 Cm x − x0 m y − y0
n! m0 n
∂n f
×
.
m
n−m
∂x ∂y
x0 θx−x0 ,y0 θy−y0 2.3
Dividing both sides of 2.3 by b − ad − c, then integrating over y from c to d first, and
then integrating the resulting inequality over x from a to b, we observe that
1
b − ad − c
bd
fs, tdsdt
a
f x0 , y0 ×
bd
n
a
c
1
n!b − ad − c
2.4
n−m
Cnm s − x0 m t − y0
·
c m0
∂n f
dsdt.
m
n−m
∂x ∂y
x0 θs−x0 ,y0 θt−y 0
Note that we have replaced the dummy variables x, y by s, t, respectively. From 2.3 and
2.4, we have
1
f x, y −
b − ad − c
bd
fs, tdsdt
a
c
n
n−m
∂n f
1
1
Cnm x − x0 m y − y0
·
−
m
n−m
n! m0
∂x ∂y
n!b
−
ad
− c
x0 θx−x0 ,y0 θy−y0 ×
bd
n
a
Cnm s
c m0
− x0 m
t − y0
n−m
∂n f
·
dsdt.
∂xm ∂yn−m x0 θs−x0 ,y0 θt−y0 2.5
4
Journal of Inequalities and Applications
Hence
bd
1
fs, tdsdt
f x, y −
b − ad − c a c
n
1
n−m
∂n f
1
m
m
≤
Cn x − x0 y − y0
·
m
n−m
n! m0
∂x ∂y
n!b
−
ad
− c
x0 θx−x0 ,y0 θy−y0 n
b d
n−m
∂n f
m
m
×
Cn s − x0 t − y0
·
dsdt
a c m0
∂xm ∂yn−m x0 θs−x0 ,y0 θt−y0 n n−m K m
≤
Cn x − x0 m y − y0
n! m0
1
b − ad − c
bd
n n−m m
m
Cn s − x0 t − y0
dsdt
a
c m0
n
n−m
K Cm |x − x0 |m y − y0 n! m0 n
bd
n
n−m
1
m
C
|s − x0 |m t − y0 dsdt
b − ad − c m0 n a c
n
n
n−m
Cnm
1
K Cnm |x − x0 |m y − y0 n! m0
b − ad − c m0 m 1n − m 1
× b − x0 m1
x0 − a
m1
d − y0
n−m1
y0 − c
n−m1 .
2.6
On the other hand, by applying the following two elementary inequalities 11:
x − t2 ≤
2
x − a b b − a ,
2 2
n
n
n
b − t t − a ≤ b − a ,
a ≤ x ≤ b, a ≤ t ≤ b,
a ≤ t ≤ b, n ≥ 1
2.7
Journal of Inequalities and Applications
5
to the right-hand side of 2.6, we obtain
bd
1
fs, tdsdt
f x, y −
b − ad − c a c
n
K a b b − a m c d d − c n−m
m ≤
x−
C
y − 2 2
n! m0 n 2 2
n
Cnm b − am1 d − cn−m1
1
.
b − ad − c m0
m 1n − m 1
2.8
This completes the proof.
Remark 2.2. With suitable modifications, it is easy to see that 2.2 reduces to the following
inequality in the 1-dimensional situation:
b
n
f n 1
b − an
∞
x − a b b − a
,
fxdx ≤
fx −
b−a a
n!
2 2
n1
2.9
where fx : a, b → R is continuous on a, b, with nth-order derivative f n : a, b → R
bounded on a, b, that is, f n ∞ supt∈a,b |f n t| < ∞, and f k x0 0, k 1, 2, . . . , n − 1.
Observe that this is a recent result of Wang and Zhao 11.
Theorem 2.3. Suppose that
1 fx, y : a, b × c, d → R is continuous on a, b × c, d;
2 fx, y : a, b × c, d → R is twice differentiable in a, b × c, d with bounded secondorder partial derivatives ∂2 fx, y/∂xm ∂y2−m m 0, 1, 2, that is,
∂2
sup
m 2−m fs, t L < ∞;
∂s
∂t
s,t∈a,b×c,d
2.10
m0,1,2
3 there exists x0 , y0 ∈ a, b × c, d such that ∂fs, t/∂s|x0 ,y0 ∂fs, t/∂t|x0 ,y0 0.
Then, one has
bd
1
fs, tdsdt
f x, y −
b − ad − c a c
L
≤
2
2
x − a b b − a y − c d d − c
2 2
2 2
b − a2 d − c2 b − ad − c
.
3
3
2
2.11
6
Journal of Inequalities and Applications
Proof. From the hypotheses and in view of the 2-dimensional Taylor expansion, it easily
follows that for some 0 < θ < 1,
f x, y −
1
b − ad − c
2 1
2 ∂ f
x − x0 2
∂x2 bd
fs, tdsdt
a
c
∂2 f x − x0 y − y0
∂x∂y x0 θx−x0 ,y0 θy−y0 2 ∂2 f 1
y − y0
2
∂y2 1
−
2b − ad − c
x0 θx−x0 ,y0 θy−y0 x0 θx−x0 ,y0 θy−y0 a
⎛
2 ∂
f
⎝s − x0 ∂x2 x
c
bd
2
0 θs−x0 ,y0 θt−y0 ∂2 f 2s − x0 t − y0
∂x∂y 2 ∂2 f t − y0
∂y2 x
x0 θs−x0 ,y0 θt−y0 ⎞
⎠dsdt.
0 θs−x0 ,y0 θt−y0 2.12
Hence,
bd
1
fs, tdsdt
f x, y −
b − ad − c a c
≤L
1 2
1
x − x0 2 x − x0 y − y0 y − y0
2
2
1
2b − ad − c
L
b d
2 2
s − x0 2s − x0 t − y0 t − y0 dsdt
a
c
1 2
1
x − x0 2 x − x0 y − y0 y − y0
2
2
3 3 1
1
1 d − y0 y0 − c b − a
b − x0 3 x0 − a3 d − c 2b − ad − c 3
3
2 2 1 2
2 d − y0 y0 − c
b − x0 x0 − a
2
Journal of Inequalities and Applications
c d d − c 2
a b b − a L
≤
x − 2 2 y − 2 2
2
7
b − a2 d − c2 b − ad − c
.
3
3
2
2.13
This proves Theorem 2.3.
Let fx, y and ∂2 fx, y/∂xm ∂y2−m m 0, 1, 2 change to fx and f x,
respectively, and with suitable modifications, Theorem 2.3 reduces to the following.
Theorem 2.4. Suppose that
1 fx : a, b → R is continuous on a, b;
2 fx : a, b → R is twice differentiable in a, b with bounded second -order derivative,
that is, f ∞ supt∈a,b |ft| < ∞;
3 there exists x0 ∈ a, b such that f x0 0 (or fa fb),
then, one has
b
2
1 1
7
|x − a b/2|
2 x − a b/2
f ∞ b − a
. 2.14
ftdt ≤
fx −
2
b−a a
b−a
12
b − a2
This is another recent result of Wang and Zhao in 11.
Acknowledgments
The research is supported by the National Natural Sciences Foundation of China 10971205.
It is partially supported by the Research Grants Council of the Hong Kong SAR, China
Project no. HKU7016/07P and an HKU Seed Grant for Basic Research.
References
1 D. S. Mtrinović, J. E. Pečarić, and A. M. Fink, Inequalities for Functions and Their Integrals and Dervatives,
Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994.
2 B. G. Pachpatte, “On an inequality of Ostrowski type in three independent variables,” Journal of
Mathematical Analysis and Applications, vol. 249, no. 2, pp. 583–591, 2000.
3 B. G. Pachpatte, “On a new Ostrowski type inequality in two independent variables,” Tamkang Journal
of Mathematics, vol. 32, no. 1, pp. 45–49, 2001.
4 G. A. Anastassiou, “Multivariate Ostrowski type inequalities,” Acta Mathematica Hungarica, vol. 76,
no. 4, pp. 267–278, 1997.
5 S. S. Dragomir and S. Wang, “A new inequality of Ostrowski’s type in L1 norm and applications to
some special means and to some numerical quadrature rules,” Tamkang Journal of Mathematics, vol. 28,
no. 3, pp. 239–244, 1997.
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,”
Computers & Mathematics with Applications, vol. 33, no. 11, pp. 15–20, 1997.
8
Journal of Inequalities and Applications
7 S. S. Dragomir and S. Wang, “Applications of Ostrowski’s inequality to the estimation of error bounds
for some special means and for some numerical quadrature rules,” Applied Mathematics Letters, vol.
11, no. 1, pp. 105–109, 1998.
8 N. S. Barnett and S. S. Dragomir, “An Ostrowski type inequality for double integrals and applications
for cubature formulae,” RGMIA Research Report Collection, vol. 1, pp. 13–23, 1998.
9 D. Baı̆nov and P. Simeonov, Integral Inequalities and Applications, vol. 57 of Mathematics and Its
Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.
10 D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Inequalities Involving Functions and Their Integrals and
Derivatives, vol. 53 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The
Netherlands, 1991.
11 M. Wang and X. Zhao, “Ostrowski type inequalities for higher-order derivatives,” Journal of
Inequalities and Applications, vol. 2009, Article ID 162689, 8 pages, 2009.