Volume 2010, Article ID 960672,8pages doi:10.1155/2010/960672
Research Article
On Ostrowski-Type Inequalities for
Higher-Order Partial Derivatives
Zhao Changjian
1and Wing-Sum Cheung
21Department of Information and Mathematics Sciences, College of Science, China Jiliang University,
Hangzhou 310018, China
2Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Correspondence should be addressed to Zhao Changjian,[email protected]
Received 4 November 2009; Accepted 14 January 2010
Academic Editor: S. S. Dragomir
Copyrightq2010 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
see1, page 468.
Theorem 1.1. Let f : a, b → R be a differentiable mapping on a, b whose derivative f :
a, b → Ris bounded ona, b;that is,f∞supt∈a,b|ft|<∞,then
fx−
1 b−a
b
a
ftdt
≤
1 4
x−ab/22
b−a2
b−af∞, 1.1
for allx∈a, b.
and Zhao11established a new Ostrowski-type inequality for higher-order derivatives as followssee11for definitions and notations:
fx−
1 b−a
b
a
fxdx
≤
fn
∞
n!
x−ab 2
b−a
2
n
b−an
n1
. 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:
f x, y
− 1
b−ad−c
b
a d
c
f x, ydxdy
≤ K
n!
n
m0
Cmn
x−ab 2
b−a
2
m
y−cd 2
d−c
2
n−m
1
b−ad−c
n
m0
Cmnb−am1d−cn−m1
m1n−m1
,
1.3
whereK >0 is a constant,Cm
n n!/m!n−m!,m, n∈N,and 0≤m≤n.
This is a generalization of inequality1.2.
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
1fx, y:a, b×c, d → Ris continuous ona, b×c, d;
2fx, y:a, b×c, d → Ris differentiable ina, b×c, dup to ordern, with bounded
nth-order mixed partial derivatives∂nfx, y/∂xm∂yn−m(m, nare natural numbers, and
0≤m≤n), that is,
sup
s,t∈a,b×c,d 0≤m≤n
∂sm∂∂tnn−mfs, t
K <∞; 2.1
3there exists x0, y0 ∈ a, b× c, d such that ∂kfs, t/∂sm∂tk−m|x0,y0 0, k
then for anyx, y∈a, b×c, d,
f x, y
− 1
b−ad−c
b
a d
c
fs, tdsdt
≤ K
n!
n
m0
Cm n
x−ab 2
b−a
2
m
y−cd 2
d−c
2
n−m
1
b−ad−c
n
m0
Cm
nb−am1d−cn−m1
m1n−m1
,
2.2
whereCm
n 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,
f x, yf x0, y0
1
n!
n
m0
Cnmx−x0m y−y0 n−m
× ∂nf
∂xm∂yn−m
x0θx−x0,y0θy−y0
.
2.3
Dividing both sides of2.3byb−ad−c, then integrating overyfromctodfirst, and then integrating the resulting inequality overxfromatob, we observe that
1
b−ad−c
b
a d
c
fs, tdsdt
f x0, y0
1
n!b−ad−c
× b
a d
c n
m0
Cnms−x0m t−y0
n−m· ∂nf
∂xm∂yn−m
x0θs−x0,y0θt−y0
dsdt.
2.4
Note that we have replaced the dummy variablesx, y bys, t, respectively. From2.3and
2.4, we have
f x, y− 1 b−ad−c
b
a d
c
fs, tdsdt
1
n!
n
m0
Cmnx−x0m y−y0
n−m· ∂nf
∂xm∂yn−m
x0θx−x0,y0θy−y0
− 1
n!b−ad−c
× b
a d
c n
m0
Cm
ns−x0m t−y0
n−m· ∂nf
∂xm∂yn−m
x0θs−x0,y0θt−y0
dsdt.
Hence
f x, y
− 1
b−ad−c
b
a d
c
fs, tdsdt
≤1
n!
n
m0
Cmnx−x0m y−y0
n−m· ∂nf
∂xm∂yn−m
x0θx−x0,y0θy−y0
1 n!b−ad−c
× b
a d
c n
m0
Cmns−x0m t−y0
n−m· ∂nf
∂xm∂yn−m
x0θs−x0,y0θt−y0
dsdt
≤ K
n!
n
m0 Cm
nx−x0m y−y0 n−m
1
b−ad−c
b
a d
c n
m0 Cm
ns−x0m t−y0 n−m
dsdt
K
n!
n
m0
Cm
n|x−x0|my−y0n−m
1
b−ad−c
n
m0
Cm n
b
a d
c
|s−x0|mt−y0n−mdsdt
K
n!
n
m0
Cm
n|x−x0|my−y0n−m
1
b−ad−c
n
m0
Cm n
m1n−m1
×b−x0m1 x0−am1 d−y0
n−m1
y0−c
n−m1
.
2.6
On the other hand, by applying the following two elementary inequalities11:
x−t2 ≤x−ab
2
b−a
2
2
, a≤x≤b, a≤t≤b,
b−tn t−an≤b−an, a≤t≤b, n≥1
to the right-hand side of2.6, we obtain
f x, y
− 1
b−ad−c
b
a d
c
fs, tdsdt
≤ K
n!
n
m0
Cmn
x−ab 2
b−a
2
m
y−cd 2
d−c
2
n−m
1
b−ad−c
n
m0
Cmnb−am1d−cn−m1
m1n−m1
.
2.8
This completes the proof.
Remark 2.2. With suitable modifications, it is easy to see that2.2reduces to the following inequality in the 1-dimensional situation:
fx−
1 b−a
b
a
fxdx
≤
fn
∞
n!
x−ab 2
b−a
2
n
b−an
n1
, 2.9
wherefx:a, b → Ris continuous ona, b, withnth-order derivativefn :a, b → R
bounded ona, b, that is,fn
∞supt∈a,b|fnt|<∞, andfkx0 0, k1,2, . . . , n−1.
Observe that this is a recent result of Wang and Zhao11.
Theorem 2.3. Suppose that
1fx, y:a, b×c, d → Ris continuous ona, b×c, d;
2fx, y:a, b×c, d → Ris twice differentiable ina, b×c, dwith bounded second-order partial derivatives∂2fx, y/∂xm∂y2−mm0,1,2,that is,
sup
s,t∈a,b×c,d m0,1,2
∂
2
∂sm∂t2−mfs, t
L <∞; 2.10
3there existsx0, y0∈a, b×c, dsuch that∂fs, t/∂s|x0,y0 ∂fs, t/∂t|x0,y0 0. Then, one has
f x, y
− 1
b−ad−c
b
a d
c
fs, tdsdt
≤ L
2
x−ab 2
b−a
2
y−cd 2
d−c
2
2
b−a2
3
d−c2 3
b−ad−c
2
.
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
b
a d
c
fs, tdsdt
1
2x−x0
2∂2f
∂x2
x0θx−x0,y0θy−y0
x−x0 y−y0 ∂2f
∂x∂y
x0θx−x0,y0θy−y0
1
2 y−y0
2∂2f
∂y2
x0θx−x0,y0θy−y0
− 1
2b−ad−c
b
a d
c ⎛
⎝s−x02∂2f
∂x2
x0θs−x0,y0θt−y0
2s−x0 t−y0 ∂2f
∂x∂y
x0θs−x0,y0θt−y0
t−y0
2∂2f
∂y2
x0θs−x0,y0θt−y0 ⎞ ⎠dsdt.
2.12
Hence,
f x, y
− 1
b−ad−c
b
a d
c
fs, tdsdt
≤L
1
2x−x0
2x−x
0 y−y0
1 2 y−y0
2
1
2b−ad−c
b
a d
c
s−x022s−x0 t−y0 t−y0 2
dsdt
L
1
2x−x0
2
x−x0 y−y01
2 y−y0
2
1
2b−ad−c
1 3
b−x03 x0−a3
d−c 1
3 d−y0
3
y0−c 3
b−a
1
2
b−x02 x0−a2 d−y0 2
y0−c
≤ L
2
x− ab 2
b−a
2
y− cd 2
d−c
2
2
b−a2
3
d−c2 3
b−ad−c
2
.
2.13
This provesTheorem 2.3.
Let fx, y and ∂2fx, y/∂xm∂y2−mm 0,1,2 change to fx and fx, respectively, and with suitable modifications,Theorem 2.3reduces to the following.
Theorem 2.4. Suppose that
1fx:a, b → Ris continuous ona, b;
2fx:a, b → Ris twice differentiable ina, bwith bounded second -order derivative, that is,f∞supt∈a,b|ft|<∞;
3there existsx0∈a, bsuch thatfx0 0(orfa fb), then, one has
fx−
1 b−a
b
a
ftdt≤ 1 2f
∞b−a2
x−ab/22
b−a2
|x−ab/2| b−a
7 12
. 2.14
This is another recent result of Wang and Zhao in11.
Acknowledgments
The research is supported by the National Natural Sciences Foundation of China10971205. It is partially supported by the Research Grants Council of the Hong Kong SAR, China
Project no. HKU7016/07Pand an HKU Seed Grant for Basic Research.
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