Journal of Inequalities in Pure and
Applied Mathematics
A GENERALIZATION FOR OSTROWSKI’S INEQUALITY IN R2
BAOGUO JIA
School of Mathematics and Scientific Computer
Zhongshan University
Guangzhou 510275, China.
volume 7, issue 5, article 182,
2006.
Received 27 February, 2006;
accepted 28 April, 2006.
Communicated by: G.A. Anastassiou
EMail: mcsjbg@zsu.edu.cn
Abstract
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Abstract
We establish a new Ostrowski’s inequality in R2 by using an idea of B.G. Pachpatte.
2000 Mathematics Subject Classification: 33B15, 33C05.
Key words: Ostrowski’s inequality, Mean value theorem.
This project was supported in part by the Foundations of the Natural Science Committee and Zhongshan University Advanced Research Centre, China.
A Generalization for Ostrowski’s
Inequality in R2
Baoguo Jia
A.M. Ostrowski proved the following inequality (see [1, p. 226–227]):
#
"
Z b
a+b 2
x
−
1
1
2
f (x) −
(b − a)||f 0 ||∞ ,
(1)
f (t)dt ≤
+
2
(b − a)
b−a a
4
for all x ∈ [a, b], where f : [a, b] ⊆ R → R is continuous on [a, b] and differentiable on (a, b), whose derivative f 0 : (a, b) → R is bounded on (a, b), i.e.,
||f 0 ||∞ = sup |f 0 (x)| < ∞.
x∈(a,b)
Recently, by using a fairly elementary analysis, B.G. Pachpatte [2] established the following inequality of type (1) involving two functions and their
derivatives.
Theorem 1. Let f, g : [a, b] ⊆ R → R be continuous functions on [a, b] and
differentiable on (a, b), whose derivative f 0 , g 0 : (a, b) → R are bounded on
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(a, b), i.e., ||f 0 ||∞ = sup |f 0 (x)| < ∞, ||g 0 ||∞ = sup |g 0 (x)| < ∞. Then
x∈(a,b)
x∈(a,b)
Z b
Z b
1
f (x)g(x) −
g(x)
f (y)dy + f (x)
g(y)dy 2(b − a)
a
" a
#
a+b 2
x
−
1
1
2
(b − a),
+
≤ {|g(x)| ||f 0 ||∞ + |f (x)| ||g 0 ||∞ }
2
4
(b − a)2
A Generalization for Ostrowski’s
Inequality in R2
for all x ∈ [a, b].
In this paper, by means of B.G. Pachpatte’s idea, we prove the following
Baoguo Jia
Theorem 2. Let D = [a, b] × [a, b], intD = (a, b) × (a, b), f, g : D → R be
continuous functions on D and differentiable on intD, whose partial derivatives
fx , fy , gx , gy : intD → R are bounded on intD, i.e.,
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||fx ||∞ =
sup
|fx (x, y)| < ∞,
||fy ||∞ =
(x,y)∈intD
||gx ||∞ =
sup
sup
|fy (x, y)| < ∞,
(x,y)∈intD
|gx (x, y)| < ∞,
(x,y)∈intD
||gy ||∞ =
sup
|gy (x, y)| < ∞.
(x,y)∈intD
Then
f (u1 , v1 )g(u1 , v1 ) −
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ZZ
1
g(u1 , v1 )
f (u2 , v2 )du2 dv2
2(b − a)2
D
ZZ
+ f (u1 , v1 )
g(u2 , v2 )du2 dv2 D
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J. Ineq. Pure and Appl. Math. 7(5) Art. 182, 2006
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"
2 #
u1 − a+b
1
1
2
≤ [|g(u1 , v1 )| ||fx ||∞ + |f (u1 , v1 )| ||gx ||∞ ]
+
(b − a)
2
4
(b − a)2
"
2 #
v1 − a+b
1
1
2
+ [|g(u1 , v1 )| ||fy ||∞ + |f (u1 , v1 )| ||gy ||∞ ]
+
(b − a)
2
4
(b − a)2
for all (u1 , v1 ) ∈ D.
By taking g(x, y) = 1 in Theorem 2, we get the following Ostrowski like
inequality in R2 ,
Corollary 3.
f (u1 , v1 ) −
Baoguo Jia
1
f (u2 , v2 )du2 dv2 2
(b − a)
D
"
2 #
u1 − a+b
1
2
(b − a)
≤ ||fx ||∞
+
4
(b − a)2
"
2 #
v1 − a+b
1
2
+ ||fy ||∞
+
(b − a).
4
(b − a)2
ZZ
Proof of Theorem 2. By the mean value theorem, there exist ξ1 , η1 , ξ2 , η2 ∈
(a, b) such that
(2)
A Generalization for Ostrowski’s
Inequality in R2
f (u1 , v1 ) − f (u2 , v2 ) = fx (ξ1 , v1 )(u1 − u2 ) + fy (u2 , η1 )(v1 − v2 ),
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and
J. Ineq. Pure and Appl. Math. 7(5) Art. 182, 2006
(3)
g(u1 , v1 ) − g(u2 , v2 ) = gx (ξ2 , v1 )(u1 − u2 ) + gy (u2 , η2 )(v1 − v2 ).
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Multiplying both sides of (2) and (3) by g(u1 , v1 ) and f (u1 , v1 ) respectively and
adding we get
2f (u1 , v1 )g(u1 , v1 ) − [f (u2 , v2 )g(u1 , v1 ) + f (u1 , v1 )g(u2 , v2 )]
= g(u1 , v1 )[fx (ξ1 , v1 )(u1 − u2 ) + fy (u2 , η1 )(v1 − v2 )]
+ f (u1 , v1 )[gx (ξ2 , v1 )(u1 − u2 ) + gy (u2 , η2 )(v1 − v2 )].
Integrate both sides with respect to u2 , v2 over D. Note that by the proof of
the mean value theorem, we know that fx (ξ1 , v1 ), fy (u2 , η1 ), gx (ξ2 , v1 ) and
gy (u2 , η2 ) are Riemann-integrable for (u2 , v2 ) ∈ D. Rewriting we get
ZZ
1
f (u1 , v1 )g(u1 , v1 ) −
g(u1 , v1 )
f (u2 , v2 )du2 dv2
2(b − a)2
D
ZZ
+f (u1 , v1 )
g(u2 , v2 )du2 dv2
D
ZZ
1
=
g(u1 , v1 )
[fx (ξ1 , v1 )(u1 − u2 ) + fy (u2 , η1 )(v1 − v2 )]du2 dv2
2(b − a)2
D
ZZ
1
+
f (u1 , v1 )
[gx (ξ2 , v1 )(u1 − u2 ) + gy (u2 , η2 )(v1 − v2 )]du2 dv2 .
2(b − a)2
D
A Generalization for Ostrowski’s
Inequality in R2
Baoguo Jia
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So
f (u1 , v1 )g(u1 , v1 ) −
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ZZ
1
g(u1 , v1 )
f (u2 , v2 )du2 dv2
2(b − a)2
D
ZZ
+f (u1 , v1 )
g(u2 , v2 )du2 dv2 D
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J. Ineq. Pure and Appl. Math. 7(5) Art. 182, 2006
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≤
1
|g(u1 , v1 )|
2(b − a)2
ZZ
ZZ
× ||fx ||∞
|u1 − u2 |du2 dv2 + ||fy ||∞
|v1 − v2 |du2 dv2
D
D
1
+
|f (u1 , v1 )|
2(b − a)2
ZZ
ZZ
× ||gx ||∞
|u1 − u2 |du2 dv2 + ||gy ||∞
|v1 − v2 |du2 dv2 .
D
D
A Generalization for Ostrowski’s
Inequality in R2
Note that
Baoguo Jia
(u1 − a)2 + (u1 − b)2
|u1 − u2 |du2 dv2 = (b − a)
2
D
#
"
2
u1 − a+b
1
2
(b − a)
+
=
(b − a)2
4
ZZ
and
2 #
v1 − a+b
1
2
|v1 − v2 |du2 dv2 =
+
(b − a).
2
4
(b
−
a)
D
"
ZZ
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We obtain that
f (u1 , v1 )g(u1 , v1 ) −
ZZ
1
g(u1 , v1 )
f (u2 , v2 )du2 dv2
2(b − a)2
D
ZZ
+f (u1 , v1 )
g(u2 , v2 )du2 dv2 D
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J. Ineq. Pure and Appl. Math. 7(5) Art. 182, 2006
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"
2 #
u1 − a+b
1
1
2
≤ [|g(u1 , v1 )| ||fx ||∞ + |f (u1 , v1 )| ||gx ||∞ ]
+
(b − a)
2
4
(b − a)2
"
2 #
v1 − a+b
1
1
2
+ [|g(u1 , v1 )| ||fy ||∞ + |f (u1 , v1 )| ||gy ||∞ ]
+
(b − a).
2
4
(b − a)2
Remark 1. Let f (x, y) = f (x), g(x, y) = g(x) in Theorem 2. Then fy =
0, gy = 0. We obtain Theorem 1.
Remark 2. Let f (x, y) = f (x), g(x, y) = 1 in Theorem 2, we recapture the
well known Ostrowski inequality.
A Generalization for Ostrowski’s
Inequality in R2
Baoguo Jia
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References
[1] A.M. OSTROWSKI, Uber die Absolutabweichung einer differentiebaren
funktion von ihrem integralmitelwert, Comment. Math. Helv., 10 (1938),
226–227.
[2] B.G. PACHPATTE, A note on Ostrowski like inequalities, J. Ineq. Pure
Appl. Math., 6(4) (2005), Art. 114. [ONLINE: http://jipam.vu.
edu.au/article.php?sid=588].
A Generalization for Ostrowski’s
Inequality in R2
Baoguo Jia
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