Journal of Inequalities in Pure and
Applied Mathematics
IMPROVEMENT OF AN OSTROWSKI TYPE INEQUALITY FOR
MONOTONIC MAPPINGS AND ITS APPLICATION
FOR SOME SPECIAL MEANS
volume 2, issue 3, article 31,
2001.
S.S. DRAGOMIR AND M.L. FANG
School of Communications and Informatics
Victoria University of Technology
PO Box 14428, Melbourne City MC
8001 Victoria, AUSTRALIA.
EMail: sever.dragomir@vu.edu.au
URL: http://rgmia.vu.edu.au/SSDragomirWeb.html
Department of Mathematics,
Nanjing Normal University,
Nanjing 210097, P. R. China
EMail: mlfang@pine.njnu.edu.cn
Received 10 January, 2001;
accepted 23 April, 2001.
Communicated by: B. Mond
Abstract
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2000
School of Communications and Informatics,Victoria University of Technology
ISSN (electronic): 1443-5756
006-01
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Abstract
We first improve two Ostrowski type inequalities for monotonic functions, then
provide its application for special means.
2000 Mathematics Subject Classification: 26D15, 26D10.
Key words: Ostrowski’s inequality, Trapezoid inequality, Special means.
Project supported by the National Natural Science Foundation of China (Grant
No.10071038)
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
S.S. Dragomir and M.L. Fang
Contents
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1
2
3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Application for Special Means . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1
Mapping f (x) = xp . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2
Mapping f (x) = −1/x . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3
Mapping f (x) = ln x . . . . . . . . . . . . . . . . . . . . . . . . . . 10
References
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J. Ineq. Pure and Appl. Math. 2(3) Art. 31, 2001
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1.
Introduction
In [1], Dragomir established the following Ostrowski’s inequality for monotonic
mappings.
Theorem 1.1. Let f : [a, b] → R be a monotonic nondecreasing mapping on
[a, b]. Then for all x ∈ [a, b], we have the following inequality
Z b
1
f (x) −
f
(t)dt
b−a a
Z b
1
[2x − (a + b)]f (x) +
sgn(t − x)f (t)dt
≤
b−a
a
1
≤
[(x − a)(f (x) − f (a)) + (b − x)(f (b) − f (x))]
"b − a #
1 x − a+b
2
(1.1)
(f (b) − f (a)).
≤
+
b−a
2
The constant
1
2
is the best possible one.
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
S.S. Dragomir and M.L. Fang
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In [2], Dragomir, Pečarić and Wang generalized Theorem 1.1 and proved
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Theorem 1.2. Let f : [a, b] → R be a monotonic nondecreasing mapping on
[a, b] and t1 , t2 , t3 ∈ (a, b) be such that t1 ≤ t2 ≤ t3 . Then
Z b
f
(x)dx
−
[(t
−
a)f
(a)
+
(t
−
t
)f
(t
)
+
(b
−
t
)f
(b)]
1
3
1
2
3
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≤ (b − t3 )f (b) + (2t2 − t1 − t3 )f (t2 ) − (t1 − a)f (a)
Z b
+
T (x)f (x)dx
a
(1.2)
≤ (b − t3 )(f (b) − f (t3 )) + (t3 − t2 )(f (t3 ) − f (t2 ))
+ (t2 − t1 )(f (t2 ) − f (t1 )) + (t1 − a)(f (t1 ) − f (a))
≤ max{t1 − a, t2 − t1 , t3 − t2 , b − t3 }(f (b) − f (a)),
where T (x) = sgn(t1 − x), for x ∈ [a, t2 ], and T (x) = sgn(t3 − x), for
x ∈ [t2 , b].
In the present paper, we firstly improve the above results, and then provide
its application for some special means.
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
S.S. Dragomir and M.L. Fang
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2.
Main Result
We shall start with the following result.
Theorem 2.1. Let f : [a, b] → R be a monotonic nondecreasing mapping on
[a, b] and let t1 , t2 , t3 ∈ [a, b] be such that t1 ≤ t2 ≤ t3 . Then
Z b
f
(x)dx
−
[(t
−
a)f
(a)
+
(t
−
t
)f
(t
)
+
(b
−
t
)f
(b)]
1
3
1
2
3
a
(2.1)
(2.2)
≤ max{(b − t3 )(f (b) − f (t3 )) + (t2 − t1 )(f (t2 ) − f (t1 )),
(t3 − t2 )(f (t3 ) − f (t2 )) + (t1 − a)(f (t1 ) − f (a))}
≤ max{t1 − a, t2 − t1 , t3 − t2 , b − t3 }(f (b) − f (a)).
Proof. Since f (x) is a monotonic nondecreasing mapping on [a, b], we have
Z b
f (x)dx − [(t1 − a)f (a) + (t3 − t1 )f (t2 ) + (b − t3 )f (b)]
a
Z t1
Z t3
Z b
= (f (x) − f (a))dx +
(f (x) − f (t2 ))dx +
(f (x) − f (b))dx
a
t
t
Z
3
Z1 t3
t
1
= (f (x) − f (a))dx +
(f (x) − f (t2 ))dx
a
t2
Z t2
Z b
−
(f (t2 ) − f (x))dx +
(f (b) − f (x))dx t1
t3
≤ max{(b − t3 )(f (b) − f (t3 )) + (t2 − t1 )(f (t2 ) − f (t1 )),
(t3 − t2 )(f (t3 ) − f (t2 )) + (t1 − a)(f (t1 ) − f (a))}
≤ max{t1 − a, t2 − t1 , t3 − t2 , b − t3 }(f (b) − f (a)).
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
S.S. Dragomir and M.L. Fang
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Thus (2.1) and (2.2) is proved.
For t1 = t2 = t3 = x, Theorem 2.1 becomes the following corollary.
Corollary 2.2. Let f be defined as in Theorem 2.1. Then
Z b
f
(x)dx
−
[(x
−
a)f
(a)
+
(b
−
x)f
(b)]
a
≤ max{(b − x)(f (b) − f (x)), (x − a)(f (x) − f (a))}
≤ max{x − a, b − x} · max{(f (x) − f (a)), (f (b) − f (x))}
a + b 1
(f (b) − f (a)).
≤ (b − a) + x −
2
2 For x =
a+b
,
2
we get trapezoid inequality.
Corollary 2.3. Let f be defined as in Theorem 2.1. Then
Z b
f (a) + f (b)
(b − a)
f (x)dx −
2
a
a+b
a+b
b−a
(2.3)
≤
max
f
− f (a) , f (b) − f
2
2
2
1
≤ (b − a)(f (b) − f (a)).
2
For t1 = a, t2 = x, t3 = b, we get Theorem 1.1.
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
S.S. Dragomir and M.L. Fang
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3.
Application for Special Means
In this section, we shall give application of Corollary 2.3. Let us recall the
following means.
1. The arithmetic mean:
A = A(a, b) :=
a+b
,
2
a, b ≥ 0.
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
2. The geometric mean:
√
G = G(a, b) :=
ab,
a, b ≥ 0.
3. The harmonic mean:
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H = H(a, b) :=
2
,
1/a + 1/b
a, b ≥ 0.
4. The logarithmic mean:
L = L(a, b) :=
a, b ≥ 0, a 6= b; If a = b, then L(a, b) = a.
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5. The identric mean:
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b−a
,
ln b − ln a
1
I = I(a, b) :=
e
S.S. Dragomir and M.L. Fang
bb
aa
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1
b−a
,
a, b ≥ 0, a 6= b; If a = b, then I(a, b) = a.
J. Ineq. Pure and Appl. Math. 2(3) Art. 31, 2001
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6. The p-logarithmic mean:
bp+1 − ap+1
Lp = Lp (a, b) :=
(p + 1)(b − a)
p1
,
a 6= b; If a = b, then Lp (a, b) = a,
where p 6= −1, 0 and a, b > 0.
The following simple relationships are known in the literature
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
H ≤ G ≤ L ≤ I ≤ A.
We are going to use inequality (2.3) in the following equivalent version:
Z b
1
f
(a)
+
f
(b)
f
(t)dt
−
b − a
2
a
1
a+b
a+b
(3.1)
≤ max
f
− f (a) , f (b) − f
2
2
2
1
≤ (f (b) − f (a)),
2
where f : [a, b] → R is monotonic nondecreasing on [a, b].
S.S. Dragomir and M.L. Fang
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3.1.
Mapping f (x) = x
p
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p
Consider the mapping f : [a, b] ⊂ (0, ∞) → R, f (x) = x , p > 0. Then
Z b
1
f (t)dt = Lpp (a, b),
b−a a
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f (a) + f (b)
= A(ap , bp ),
2
f (b) − f (a) = p(b − a)Lp−1
p−1 .
Then by (3.1), we get
p
p p
a+b
a+b
p p
Lp (a, b) − A(ap , bp ) ≤ 1 max
− a ,b −
2
2
2
p 1 p
a+b
=
b −
2
2
p
a+b
1 p
1
p
p
−a
= (b − a ) −
2
2
2
1
p(b − a)ap−1
(3.2)
≤ p(b − a)Lp−1
.
p−1 −
2
4
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
S.S. Dragomir and M.L. Fang
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Remark 3.1. The following result was proved in [2].
p
Lp (a, b) − A(ap , bp ) ≤ 1 p(b − a)Lp−1
p−1 .
2
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3.2.
Mapping f (x) = −1/x
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Consider the mapping f : [a, b] ⊂ (0, ∞) → R, f (x) = − x1 . Then
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1
b−a
Z
−1
f (t)dt = −L (a, b),
a
J. Ineq. Pure and Appl. Math. 2(3) Art. 31, 2001
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f (a) + f (b)
A(a, b)
=− 2
,
2
G (a, b)
b−a
f (b) − f (a) = 2
.
G (a, b)
Then by (3.1), we get
A(a, b)
1
1
2
2
1
−1
G2 (a, b) − L (a, b) ≤ 2 max a − a + b , a + b − b
1
b−a
1 b−a 1
b−a
= ·
= ·
− ·
2 a(a + b)
2
ab
2 b(a + b)
1
b−a
1 b−a
≤ · 2
− ·
.
2 G (a, b) 2 b(a + b)
S.S. Dragomir and M.L. Fang
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Thus we get
(3.3)
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
0 ≤ AL − G2 ≤
1 b
(b − a)L.
2a+b
Remark 3.2. The following result was proved in [2].
1
0 ≤ AG − G2 ≤ (b − a)L.
2
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3.3.
Mapping f (x) = ln x
Consider the mapping f : [a, b] ⊂ (0, ∞) → R, f (x) = ln x. Then
Z b
1
f (t)dt = ln I(a, b),
b−a a
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f (a) + f (b)
= ln G(a, b),
2
b−a
f (b) − f (a) =
.
L(a, b)
Then by (3.1), we get
1
a+b
a+b
|ln I(a, b) − ln G(a, b)| ≤ max ln
− ln a, ln b − ln
2
2
2
1 a+b
1 b−a
1
2b
= ln
=
− ln
.
2
2a
2 L(a, b) 2 a + b
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
S.S. Dragomir and M.L. Fang
Thus we get
(3.4)
I
1≤
≤
G
r
b−a
a + b 12 · L(a,b)
e
.
2b
Remark 3.3. The following result was proved in [2].
1≤
1 b−a
I
≤ e 2 · L(a,b) .
G
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References
[1] S.S. DRAGOMIR, Ostrowski’s inequality for monotonic mapping and applications, J. KSIAM, 3(1) (1999), 129–135.
[2] S.S. DRAGOMIR, J. PEČARIĆ AND S. WANG, The unified treatment of
trapezoid, Simpson, and Ostrowski type inequalities for monotonic mappings and applications, Math. Comput. Modelling, 31 (2000), 61–70.
[3] 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 Math. Applic., 33(11)
(1997), 15–20.
[4] S.S. DRAGOMIR AND S. WANG, Applications of Ostrowski inequality to
the estimation of error bounds for some special means and some numerical
quadrature rules, Appl. Math. Lett., 11(1) (1998), 105–109.
[5] M. MATIĆ, J. PEČARIĆ AND N. UJEVIĆ, Improvement and further generalization of inequalities of Ostrowski-Grüss type, Computers Math. Applic.,
39(3/4) (2000), 161–175.
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
S.S. Dragomir and M.L. Fang
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[6] D.S. MITRINOVIĆ, J. PEČARIĆ AND A.M. FINK, Classical and New Inequalities in Analysis, Kluwer Academic, Dordrecht, 1993.
[7] D.S. MITRINOVIĆ, J. PEČARIĆ AND A.M. FINK, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic, Dordrecht, 1991.
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[8] G.S. YANG AND K.L. TSENG, On certain integral inequalities related to
Hermite-Hadamard inequalities, J. Math. Anal. Appl., 239 (1999), 180–187.
Improvement of An Ostrowski
Type Inequality for Monotonic
Mappings and Its Application
for Some Special Means
S.S. Dragomir and M.L. Fang
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