Journal of Inequalities in Pure and
Applied Mathematics
A GENERALIZED OSTROWSKI-GRÜSS TYPE INEQUALITY FOR
TWICE DIFFERENTIABLE MAPPINGS AND APPLICATIONS
volume 7, issue 4, article 124,
2006.
A. RAFIQ, N.A. MIR AND FIZA ZAFAR
COMSATS Institute of Information Technology
Islamabad, Pakistan
EMail: arafiq@comsats.edu.pk
EMail: namir@comsats.edu.pk
CASPAM, B. Z. University
Multan, Pakistan
EMail: fizazafar@gmail.com
Received 15 September, 2005;
accepted 10 May, 2006.
Communicated by: N.S. Barnett
Abstract
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c
2000
Victoria University
ISSN (electronic): 1443-5756
275-05
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Abstract
A generalized Ostrowski type inequality for twice differentiable mappings in
terms of the upper and lower bounds of the second derivative is established.
The inequality is applied to numerical integration.
2000 Mathematics Subject Classification: 26D15.
Key words: Ostrowski inequality, Grüss inequality.
The authors express thanks to Prof. N.S. Barnett for giving valuable suggestions
during the preparation of this manuscript.
A Generalized Ostrowski-Grüss
Type Inequality for Twice
Differentiable Mappings and
Applications
A. Rafiq, N.A. Mir and Fiza Zafar
Contents
Title Page
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Applications in Numerical Integration . . . . . . . . . . . . . . . . . . . 11
References
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1.
Introduction
The integral inequality that establishes a connection between the integral of
the product of two functions and the product of the integrals is known in the
literature as the Grüss inequality. The inequality is as follows:
Theorem 1.1. Let f, g : [a, b] −→ R be integrable functions such that Ψ ≤
f (x) ≤ ϕ and γ ≤ g(x) ≤ Γ for all x ∈ [a, b], where Ψ, ϕ, γ and Γ are
constants. It follows that,
Z b
Z b
Z b
1
1
1
(1.1) f (x)g(x)dx −
f (x)dx
g(x)dx
b−a a
b−a a
b−a a
1
≤ (ϕ − Ψ) (Γ − γ) ,
4
where the constant 41 is sharp.
In [2], S.S. Dragomir and S. Wang proved the following Ostrowski type
inequality in terms of lower and upper bounds of the first derivative.
Theorem 1.2. Let f : [a, b] −→ R be continuous on [a, b] and differentiable on
(a, b) and where the first derivative satisfies the condition,
γ ≤ f 0 (x) ≤ Γ
for all x ∈ [a, b],
then,
(1.2)
A Generalized Ostrowski-Grüss
Type Inequality for Twice
Differentiable Mappings and
Applications
A. Rafiq, N.A. Mir and Fiza Zafar
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Z b
1
f
(b)
−
f
(a)
a
+
b
f (x) −
f (t)dt −
x−
b−a a
b−a
2
1
≤ (b − a) (Γ − γ)
4
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for all x ∈ [a, b].
In [1], S.S. Dragomir and N.S. Barnett, proved the following inequality.
Theorem 1.3. Let f : [a, b] −→ R be continuous on [a, b] and twice differentiable on (a, b), where the second derivative f 00 : (a, b) −→ R satisfies the
condition,
ϕ ≤ f 00 (x) ≤ Φ for all x ∈ (a, b) ,
then,
(1.3)
"
2 # 0
2
(b
−
a)
1
a
+
b
f (b) − f 0 (a)
+
x−
f (x) +
24
2
2
b−a
Z b
1
a+b
0
− x−
f (x) −
f (t)dt
2
b−a a
2
1
a + b 1
≤ (Φ − ϕ)
(b − a) + x −
8
2
2 for all x ∈ [a, b].
In this paper we establish a more general form of (1.3) and apply the result
to numerical integration.
A Generalized Ostrowski-Grüss
Type Inequality for Twice
Differentiable Mappings and
Applications
A. Rafiq, N.A. Mir and Fiza Zafar
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2.
Main Results
Theorem 2.1. Let f : [a, b] −→ R be a continuous mapping on [a, b], and
twice differentiable on (a, b) with second derivative f 00 : (a, b) −→ R satisfying
the condition:
b−a
b−a
00
ϕ ≤ f (x) ≤ Φ,
for all x ∈ a + h
.
,b − h
2
2
It follows that,
f (a) + f (b)
a+b
0
(2.1) (1 − h) f (x) − x −
f (x) + h
2
2
#
"
2
2 0
a+b
(3h − 1) (b − a)
f (b) − f 0 (a)
1
+
(1 − h) x −
−
2
2
24
b−a
2
Z b
1
1
1
a
+
b
,
f (t)dt ≤ (Φ − ϕ)
(b − a) (1 − h) + x −
−
b−a a
8
2
2 for all x ∈ [a +
h b−a
,b
2
−
h b−a
]
2
and h ∈ [0, 1] .
Proof. The proof uses the following identity,
Z b
(2.2)
f (t)dt = (b − a) (1 − h) f (x)
a
a+b
b−a
− (b − a) (1 − h) x −
f 0 (x) + h
(f (a) + f (b))
2
2
Z b
h2 (b − a)2 0
0
(f (b) − f (a)) +
K (x, t) f 00 (t) dt.
−
8
a
A Generalized Ostrowski-Grüss
Type Inequality for Twice
Differentiable Mappings and
Applications
A. Rafiq, N.A. Mir and Fiza Zafar
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for all x ∈ [a + h b−a
, b − h b−a
], where the kernel K : [a, b]2 → R is defined by
2
2
K(x, t) =
(2.3)


1
2
2
t − a + h b−a
if t ∈ [a, x]
2

1
2
2
t − b − h b−a
if t ∈ (x, b].
2
This is a particular form of the identity given in [3, p. 59; Corollary 2.3].
Observe that the kernel K satisfies the estimation
 1
2
b−a a+b
−
x
,
x
∈
a
+
h
,
 2 b − h b−a
2
2
2
(2.4)
0 ≤ K(x, t) ≤
a+b
 1
b−a 2
b−a
x
−
a
+
h
,
x
∈
,
b
−
h
.
2
2
2
2
A Generalized Ostrowski-Grüss
Type Inequality for Twice
Differentiable Mappings and
Applications
A. Rafiq, N.A. Mir and Fiza Zafar
Applying the Grüss inequality for the mappings f 00 (·) and K (x, ·) we get,
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(2.5)
Z b
Z b
Z b
1
1
1
00
00
K (x, t) f (t) dt −
K (x, t) dt
f (t) dt
b − a
b−a a
b−a a
a
 1
2
b − h b−a
− x , x ∈ a + h b−a
, a+b

2
2
2
2
1
≤ (Φ − ϕ) ×
2
 1
4
x − a + h b−a
, x ∈ a+b
, b − h b−a
.
2
2
2
2
b
Z
K(x, t)dt =
(2.6)
a
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Observe that,
Z
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a
x
2
2
Z b
t − a + h b−a
t − b − h b−a
2
2
dt +
dt
2
2
x
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#
3 3
b−a
h3 (b − a)3
b−a
b−h
+
x− a+h
−x +
2
2
4
#
"
2
(b − a)2 (1 − h)2 1
a+b
h3 (b − a)3
= (b − a) (1 − h)
+
x−
+
.
24
2
2
24
1
=
6
"
Using (2.6) in (2.5) , we get
"
2
1 Z b
a+b
(b − a)2 (1 − h)3 1
00
K (x, t) f (t) dt −
+ (1 − h) x −
b − a a
24
2
2
#
h3 (b − a)2
f 0 (b) − f 0 (a) +
24
b−a
 1
2
b−a
b−a a+b
b
−
h
−
x
,
x
∈
a
+
h
,

2
2
2
2
1
≤ (Φ − ϕ) ×
2
 1
4
x − a + h b−a
, x ∈ a+b
, b − h b−a
.
2
2
2
2
Also, by using identity (2.2) , the above inequality reduces to,
(1 − h) f (x) − x − a + b f 0 (x) + h f (a) + f (b)
2
2
"
#
2
1
a+b
(3h − 1) (b − a)2
f 0 (b) − f 0 (a)
+
(1 − h) x −
−
2
2
24
b−a
Z b
1
−
f (t)dt
b−a a
A Generalized Ostrowski-Grüss
Type Inequality for Twice
Differentiable Mappings and
Applications
A. Rafiq, N.A. Mir and Fiza Zafar
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≤


1
(Φ − ϕ) ×

4
2
b−a a+b
b − h b−a
−
x
,
x
∈
a
+
h
;
,
2
2
2
1
2
1
2
2
x − a + h b−a
, x ∈ a+b
, b − h b−a
.
2
2
2
Since,
(
max
2 )
b−a 2
−
x
x
−
a
+
h
b − h b−a
2
2
,
2
2
 1
2
− x , x ∈ a + h b−a
, a+b
 2 b − h b−a
2
2
2
=
2
 1
x − a + h b−a
, x ∈ a+b
, b − h b−a
,
2
2
2
2
A Generalized Ostrowski-Grüss
Type Inequality for Twice
Differentiable Mappings and
Applications
A. Rafiq, N.A. Mir and Fiza Zafar
but on the other hand,
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(
max
b−
h b−a
2
2
2 2 )
−x
x − a + h b−a
2
,
2
2
1 1
a
+
b
,
=
(b − a) (1 − h) + x −
2 2
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inequality (2.1) is proved.
Remark 1. For h = 0 in (2.1), we obtain (1.3).
Corollary 2.2. If f is as in Theorem 2.1, then we have the following perturbed
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midpoint inequality:
(2.7)
(1 − h) f a + b + h f (a) + f (b)
2
2
Z b
(3h − 1) (b − a) 0
1
0
−
(f (b) − f (a)) −
f (t)dt
24
b−a a
1
(Φ − ϕ) (b − a)2 (1 − h)2 ,
≤
32
giving,
(2.8)
Z b
a
+
b
(b
−
a)
1
0
0
f
+
(f
(b)
−
f
(a))
−
f
(t)dt
2
24
b−a a
1
(Φ − ϕ) (b − a)2 ,
≤
32
for h = 0.
Remark 2. The classical midpoint inequality states that
Z b
a
+
b
1
f
≤ 1 (b − a)2 kf 00 k .
(2.9)
−
f
(t)
dt
∞
24
2
b−a a
A Generalized Ostrowski-Grüss
Type Inequality for Twice
Differentiable Mappings and
Applications
A. Rafiq, N.A. Mir and Fiza Zafar
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If Φ − ϕ ≤ 43 kf 00 k∞ , then the estimation provided by (2.7) is better than the
estimation in the classical midpoint inequality (2.9). A sufficient condition for
Φ − ϕ ≤ 43 kf 00 k∞ to be true is 0 ≤ ϕ ≤ Φ.Indeed, if 0 ≤ ϕ ≤ Φ, then
Φ − ϕ ≤ kf 00 k∞ < 34 kf 00 k∞ .
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Corollary 2.3. Let f be as in Theorem 2.1, then,
(2.10)
Z b
f (a) + f (b) (b − a) 0
1
0
−
(f (b) − f (a)) −
f (t)dt
2
12
b−a a
1
≤
(Φ − ϕ) (2 − h)2 (b − a)2 .
32
Proof. Put x = a and x = b in turn in (2.1) and use the triangle inequality.
Corollary 2.4. Let f be as in Theorem 2.1, then we have the following perturbed Trapezoid inequality:
(2.11)
Z b
f (a) + f (b) (b − a) 0
1
0
−
(f
(b)
−
f
(a))
−
f
(t)dt
2
12
b−a a
1
≤
(Φ − ϕ) (b − a)2 .
32
Proof. Put h = 1 in (2.10).
Remark 3. The classical Trapezoid inequality states that
Z b
f (a) + f (b)
1
1
(2.12)
−
f (t) dt ≤
(b − a)2 kf 00 k∞ .
2
b−a a
12
If we assume that Φ − ϕ ≤ 23 kf 00 k∞ , then the estimation provided by (2.10) is
better than the estimation in the classical Trapezoid inequality (2.12).
A Generalized Ostrowski-Grüss
Type Inequality for Twice
Differentiable Mappings and
Applications
A. Rafiq, N.A. Mir and Fiza Zafar
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3.
Applications in Numerical Integration
Let In : a = x0 < x1 < · · · < xn−1 < xn = b be a division of the interval
[a, b], ξi ∈ [xi , xi+1 ], (i = 0, 1, . . . , n − 1) a sequence of intermediate points
and hi := xi+1 − xi (i = 0, 1, . . . , n − 1) . Following the approach taken in [1]
we have the following:
Theorem 3.1. Let f : [a, b] −→ R be continuous on [a, b] and a twice differentiable function on (a, b), whose second derivative, f 00 : (a, b) −→ R satisfies:
ϕ ≤ f 00 (x) ≤ Φ,
for all x ∈ (a, b),
A Generalized Ostrowski-Grüss
Type Inequality for Twice
Differentiable Mappings and
Applications
then,
A. Rafiq, N.A. Mir and Fiza Zafar
Z
(3.1)
b
f (t)dt = A (f, f 0 , In , ξ, δ) + R (f, f 0 , In , ξ, δ) ,
a
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where
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(3.2) A (f, f 0 , In , ξ, δ)
n−1
X
n−1
X
xi + xi−1
= (1 − δ)
hi f (ξi ) − (1 − δ)
hi ξi −
f 0 (ξi )
2
i=0
i=0
"
X
2
n−1
n−1
X
f (xi ) + f (xi+1 )
1
xi + xi+1
+δ
hi
+
(1 − δ) ξi −
2
2
2
i=0
i=0
(3δ − 1) h2i
−
(f 0 (xi+1 ) − f 0 (xi ))
24
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and the remainder R (f, f 0 , In , ξ, δ) satisfies the estimation:
|R (f, f 0 , In , ξ, δ)|
2
n−1
X
1
(1 − δ)
xi + xi+1 ≤ (Φ − ϕ)
hi
hi + ξi −
8
2
2
i=0
n−1
(3.3)
X
1
≤
(Φ − ϕ) (1 − δ)2
h3i ,
32
i=0
where δ ∈ [0, 1] and xi + δ h2i ≤ ξi ≤ xi+1 − δ h2i .
Proof. Applying Theorem 2.1 on the interval [xi , xi+1 ] (i = 0, . . . , n − 1) gives:
f
(x
)
+
f
(x
)
x
+
x
i
i+1
i
i+1
0
(1 − δ) hi f (ξi ) − hi ξi −
f (ξi ) + δhi
2
2
"
#
2
1
xi + xi+1
(3δ − 1) h2i
+
(1 − δ) ξi −
−
(f 0 (xi+1 ) − f 0 (xi ))
2
2
24
Z xi+1
−
f (t) dt
xi
2
1
1
x
+
x
i
i+1
,
≤ (Φ − ϕ) hi
(1 − δ) hi + ξi −
8
2
2
1
≤ (Φ − ϕ) (1 − δ)2 h3i
8
as
x
+
x
i
i+1
ξi −
≤ (1 − δ) hi for all i ∈ {0, 1, ..., n − 1}
2
2
A Generalized Ostrowski-Grüss
Type Inequality for Twice
Differentiable Mappings and
Applications
A. Rafiq, N.A. Mir and Fiza Zafar
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for any choice ξi of the intermediate points.
Summing the above inequalities over i from 0 to n − 1, and using the generalized triangle inequality, we get the desired estimation (3.3) .
Corollary 3.2. The following perturbed midpoint rule holds:
Z b
(3.4)
f (x) dx = M (f, f 0 , In ) + RM (f, f 0 , In ) ,
a
where
0
(3.5) M (f, f , In ) =
n−1
X
i=0
hi f
xi + xi+1
2
n−1
+
1X 2 0
h (f (xi+1 ) − f 0 (xi ))
24 i=0 i
A Generalized Ostrowski-Grüss
Type Inequality for Twice
Differentiable Mappings and
Applications
A. Rafiq, N.A. Mir and Fiza Zafar
and the remainder term RM (f, f 0 , In ) satisfies the estimation:
n−1
X
1
(Φ − ϕ)
h3i .
|RM (f, f 0 , In )| ≤
32
i=0
(3.6)
Corollary 3.3. The following perturbed trapezoid rule holds
Z b
(3.7)
f (x) dx = T (f, f 0 , In ) + RT (f, f 0 , In ) ,
a
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where
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0
(3.8) T (f, f , In ) =
n−1
X
i=0
n−1
f (xi ) + f (xi+1 )
1X 2 0
hi
−
hi (f (xi+1 ) − f 0 (xi ))
2
12 i=0
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and the remainder term RT (f, f 0 , In ) satisfies the estimation:
n−1
(3.9)
|RT (f, f 0 , In )| ≤
X
1
(Φ − ϕ)
h3i .
32
i=0
Remark 4. Note that the above mentioned perturbed midpoint formula (3.5)
and perturbed trapezoid formula (3.8) can offer better approximations of the
Rb
integral a f (x) dx for general classes of mappings as discussed in Remarks 1
and 2.
A Generalized Ostrowski-Grüss
Type Inequality for Twice
Differentiable Mappings and
Applications
A. Rafiq, N.A. Mir and Fiza Zafar
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References
[1] N.S. BARNETT, P. CERONE, S.S. DRAGOMIR, J. ROUMELIOTIS AND
A. SOFO, A survey on Ostrowski type inequalities for twice differentiable
mappings and applications, Inequality Theory and Applications, 1 (2001),
33–86.
[2] S.S. DRAGOMIR AND S. WANG, An inequality of Ostrwski-Grüss type
and its applications to the estimation of error bounds for some special
means and for some numerical quadrature rules, Computers Math. Appl.,
33 (1997), 15–20.
[3] S.S. DRAGOMIR AND N.S. BARNETT, An Ostrowski type inequality for
mappings whose second derivatives are bounded and applications, RGMIA
Research Report of Collection, 1(2) (1998), 61–63.
[4] S.S. DRAGOMIR AND Th.M. RASSIAS (Eds.), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publishers 2002.
A Generalized Ostrowski-Grüss
Type Inequality for Twice
Differentiable Mappings and
Applications
A. Rafiq, N.A. Mir and Fiza Zafar
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