Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 4, Article 124, 2006
A GENERALIZED OSTROWSKI-GRÜSS TYPE INEQUALITY FOR TWICE
DIFFERENTIABLE MAPPINGS AND APPLICATIONS
A. RAFIQ, N. A. MIR, AND FIZA ZAFAR
COMSATS I NSTITUTE OF I NFORMATION T ECHNOLOGY
I SLAMABAD , PAKISTAN
arafiq@comsats.edu.pk
namir@comsats.edu.pk
CASPAM, B. Z. U NIVERSITY
M ULTAN , PAKISTAN
fizazafar@gmail.com
Received 15 September, 2005; accepted 10 May, 2006
Communicated by N.S. Barnett
A BSTRACT. 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.
Key words and phrases: Ostrowski inequality, Grüss inequality.
1991 Mathematics Subject Classification. 26D15.
1. I NTRODUCTION
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
b−a a
b−a a
a
1
≤ (ϕ − Ψ) (Γ − γ) ,
4
1
where the constant 4 is sharp.
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
The authors express thanks to Prof. N.S. Barnett for giving valuable suggestions during the preparation of this manuscript.
275-05
2
A. R AFIQ , N. A. M IR ,
AND
F IZA Z AFAR
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)
Z b
f (b) − f (a)
a + b 1
f (x) − 1
f (t)dt −
x−
≤ 4 (b − a) (Γ − γ)
b−a a
b−a
2
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
(b − a)2 1
a+b
f (b) − f 0 (a)
+
x−
f (x) +
24
2
2
b−a
Z b
a+b
1
0
− x−
f (x) −
f (t)dt
2
b−a a
2
1
a + b 1
(b − a) + x −
≤ (Φ − ϕ)
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.
2. M AIN R ESULTS
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,
a+b
f (a) + f (b)
0
(2.1) (1 − h) f (x) − x −
f (x) + h
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
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 − h b−a
] and h ∈ [0, 1] .
2
2
J. Inequal. Pure and Appl. Math., 7(4) Art. 124, 2006
http://jipam.vu.edu.au/
O STROWSKI -G RÜSS TYPE INEQUALITY
3
Proof. The proof uses the following identity,
Z
(2.2)
a
b
f (t)dt = (b − a) (1 − h) f (x)
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
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
0 ≤ K(x, t) ≤
(2.4)
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
Applying the Grüss inequality for the mappings f 00 (·) and K (x, ·) we get,
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) ×
a+b
 1
4
b−a 2
b−a
x
−
a
+
h
,
x
∈
,
b
−
h
.
2
2
2
2
(2.5)
Observe that,
Z
a
(2.6)
b
2
Z b
b−a 2
t − a + h b−a
t
−
b
−
h
2
2
K(x, t)dt =
dt +
dt
2
2
a
x
"
#
3 3
1
b−a
b−a
h3 (b − a)3
=
x− a+h
+
b−h
−x +
6
2
2
4
"
#
2
(b − a)2 (1 − h)2 1
a+b
= (b − a) (1 − h)
+
x−
24
2
2
Z
x
h3 (b − a)3
+
.
24
J. Inequal. Pure and Appl. Math., 7(4) Art. 124, 2006
http://jipam.vu.edu.au/
4
A. R AFIQ , N. A. M IR ,
AND
F IZA Z AFAR
Using (2.6) in (2.5) , we get
"
2
1 Z b
(b − a)2 (1 − h)3 1
a+b
00
K (x, t) f (t) dt −
+ (1 − h) x −
b − a a
24
2
2
#
f 0 (b) − f 0 (a) h3 (b − a)2
+
24
b−a
 1
2
b − h b−a
− x , x ∈ a + h b−a
, a+b

2
2
2
2
1
≤ (Φ − ϕ) ×
a+b
 1
4
b−a 2
b−a
x
−
a
+
h
,
x
∈
.
,
b
−
h
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
 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
Since,
(
max
2 )
b−a 2
b − h b−a
−
x
x
−
a
+
h
2
2
,
2
2
 1
2
− x , x ∈ a + h b−a
, a+b
 2 b − h b−a
2
2
2
=
a+b
 1
b−a 2
b−a
x
−
a
+
h
,
x
∈
,
b
−
h
,
2
2
2
2
but on the other hand,
(
max
2 )
b−a 2
b − h b−a
−
x
x
−
a
+
h
2
2
,
2
2
2
1 1
a + b =
(b − a) (1 − h) + x −
,
2 2
2 inequality (2.1) is proved.
Remark 2.2. For h = 0 in (2.1), we obtain (1.3).
J. Inequal. Pure and Appl. Math., 7(4) Art. 124, 2006
http://jipam.vu.edu.au/
O STROWSKI -G RÜSS TYPE INEQUALITY
5
Corollary 2.3. If f is as in Theorem 2.1, then we have the following perturbed midpoint
inequality:
f (a) + f (b)
a+b
+h
(2.7) (1 − h) f
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
f a + b + (b − a) (f 0 (b) − f 0 (a)) − 1
≤ 1 (Φ − ϕ) (b − a)2 ,
f
(t)dt
32
2
24
b−a a
for h = 0.
Remark 2.4. The classical midpoint inequality states that
Z b
a
+
b
1
1
f
(2.9)
−
f (t) dt ≤
(b − a)2 kf 00 k∞ .
2
b−a a
24
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 ≤ 34 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∞ .
Corollary 2.5. Let f be as in Theorem 2.1, then,
Z b
f (a) + f (b) (b − a) 0
1
0
−
(f (b) − f (a)) −
f (t)dt
(2.10)
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 2.6. Let f be as in Theorem 2.1, then we have the following perturbed Trapezoid
inequality:
Z b
f (a) + f (b) (b − a) 0
1
0
(2.11)
−
(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 2.7. The classical Trapezoid inequality states that
Z b
f (a) + f (b)
1
≤ 1 (b − a)2 kf 00 k .
(2.12)
−
f
(t)
dt
∞
12
2
b−a a
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).
J. Inequal. Pure and Appl. Math., 7(4) Art. 124, 2006
http://jipam.vu.edu.au/
6
A. R AFIQ , N. A. M IR ,
AND
F IZA Z AFAR
3. A PPLICATIONS IN N UMERICAL I NTEGRATION
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),
then,
Z
b
(3.1)
f (t)dt = A (f, f 0 , In , ξ, δ) + R (f, f 0 , In , ξ, δ) ,
a
where
(3.2) A (f, f 0 , In , ξ, δ) = (1 − δ)
n−1
X
hi f (ξi )
i=0
− (1 − δ)
n−1
X
i=0
n−1
X
f (xi ) + f (xi+1 )
xi + xi−1
0
f (ξi ) + δ
hi
hi ξi −
2
2
i=0
"
2
n−1
X
1
xi + xi+1
(1 − δ) ξi −
+
2
2
i=0
(3δ − 1) h2i
(f 0 (xi+1 ) − f 0 (xi ))
−
24
and the remainder R (f, f 0 , In , ξ, δ) satisfies the estimation:
2
n−1
X
1
(1 − δ)
x
+
x
i
i+1
0
|R (f, f , In , ξ, δ)| ≤ (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:
x
+
x
f
(x
)
+
f
(x
)
i
i+1
i
i+1
0
(1 − δ) hi f (ξi ) − hi ξi −
f (ξi ) + δhi
2
2
"
#
2
Z xi+1
2
1
xi + xi+1
(3δ − 1) hi
0
0
+
(1 − δ) ξi −
−
(f (xi+1 ) − f (xi )) −
f (t) dt
2
2
24
xi
2
1
1
xi + xi+1 ≤ (Φ − ϕ) hi
(1 − δ) hi + ξi −
,
8
2
2
1
≤ (Φ − ϕ) (1 − δ)2 h3i
8
as
ξi − xi + xi+1 ≤ (1 − δ) hi for all i ∈ {0, 1, ..., n − 1}
2
2
for any choice ξi of the intermediate points.
J. Inequal. Pure and Appl. Math., 7(4) Art. 124, 2006
http://jipam.vu.edu.au/
O STROWSKI -G RÜSS TYPE INEQUALITY
7
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
(3.5)
0
M (f, f , In ) =
n−1
X
hi f
i=0
xi + xi+1
2
n−1
1X 2 0
hi (f (xi+1 ) − f 0 (xi ))
+
24 i=0
and the remainder term RM (f, f 0 , In ) satisfies the estimation:
n−1
X
1
|RM (f, f , In )| ≤
(Φ − ϕ)
h3i .
32
i=0
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
where
(3.8)
0
T (f, f , In ) =
n−1
X
i=0
n−1
1X 2 0
f (xi ) + f (xi+1 )
−
h (f (xi+1 ) − f 0 (xi ))
hi
2
12 i=0 i
and the remainder term RT (f, f 0 , In ) satisfies the estimation:
n−1
(3.9)
X
1
|RT (f, f , In )| ≤
(Φ − ϕ)
h3i .
32
i=0
0
Remark 3.4. Note that the above mentioned perturbed midpoint formula (3.5) and perturbed
Rb
trapezoid formula (3.8) can offer better approximations of the integral a f (x) dx for general
classes of mappings as discussed in Remarks 2.2 and 2.4.
R EFERENCES
[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.
J. Inequal. Pure and Appl. Math., 7(4) Art. 124, 2006
http://jipam.vu.edu.au/