Tamsui Oxford Journal of Mathematical Sciences 26(1) (2010) 61-76
Aletheia University
A Generalzation of Ostrowski-grüss Type
Inequality for First Differentiable Mappings
Fiza Zafar†
CASPAM, Bahauddin Zakariya University,
Multan 60800, Pakistan.
and
Nazir Ahmad Mir‡
Dept. of Mathematics, COMSATS Institute of
Information Technology, Islamabad 44000, Pakistan.
Received September 15, 2008, Accepted October 20, 2008.
Abstract
In this paper, we improve and further generalize some OstrowskiGrüss type inequalities involving first differentiable functions and apply
them to probability density functions, generalized beta random variable
and special means.
Keywords and Phrases: Ostrowski-Grüss inequality, Euclidean norm.
∗
2000 Mathematics Subject Classification. Primary 26D15; Secondary 41A55.
Corresponding author. E-mail: [email protected]
‡
E-mail: [email protected]
†
∗
62
Fiza Zafar and Nazir Ahmad Mir
1. Introduction
In 1938, A. Ostrowski [8] proved the following integral inequality.
Theorem 1. Let f : I ⊆ R → R be a differentiable mapping in I 0 (interior
0
of I), and let a, b ∈ I 0 with a < b. If f : (a, b) → R is bounded on (a, b) with
0
sup f (t) ≤ M, then we have
t∈[a,b]
#
"
Z b
a+b 2
x
−
1
2
f (x) − 1
(b − a) M,
+
f (t) dt ≤
b−a a
4
(b − a)2
(1.1)
f or all x ∈ [a, b] .
The constant 41 is sharp in the sense that it cannot be replaced by a smaller
one.
This inequality provides an upper bound for the approximation of integral mean of a function f by the functional value f (x) at x ∈ [a, b] .In 1997,
Dragomir and Wang [3], by the use of the Grüss inequality proved the following Ostrowski-Grüss type integral inequality.
Theorem 2. Let f : I → R, where I ⊆ R is an interval, be a mapping
differentiable in the interior I 0 of I, and let a, b ∈ I 0 with a < b. If γ ≤
0
f (x) ≤ Γ, x ∈ [a, b] for some constants γ, Γ ∈ R, then
Zb
f (b) − f (a)
a + b f (x) − 1
f (t) dt −
x−
b−a
b−a
2
a
1
≤
(b − a) (Γ − γ) ,
4
(1.2)
for all x ∈ [a, b] .
This inequality provides a connection between Ostrowski inequality [8] and
the Grüss inequality [5]. In 2000, M. Matić, J. Pecarić and N. Ujević [7], by
the use of pre-Grüss inequality improved the factor of the right membership
of (1.2) with 4√1 3 as follows.
Generalzation of Ostrowski-grüss Type Inequality
63
Theorem 3. Under the assumption of Theorem 2, we have
Zb
f (b) − f (a)
a + b f (x) − 1
f
(t)
dt
−
x
−
b−a
b−a
2
a
≤
1
√ (Γ − γ) (b − a) ,
4 3
(1.3)
for all x ∈ [a, b] .
In 2000, Barnett et al.[1], by the use of Chebyshev’s functional, improved
the Matić-Pecarić-Ujević result by providing first membership of the right side
of (1.3) in terms of Euclidean norm as follows.
Theorem 4. Let f : [a, b] → R be an absolutely continuous function whose
0
derivative f ∈ L2 [a, b] . Then we have the inequality
Zb
a + b f (b) − f (a)
f (x) − 1
x−
f (t) dt −
b−a
b−a
2
a
#1
2 f (b) − f (a) 2 2
(b − a)
1 0
√
≤
,
f −
b
−
a
b−a
2
2 3
(b − a) (Γ − γ)
0
√
if γ ≤ f (t) ≤ Γ for a.e t on [a, b].
(1.4)
≤
4 3
"
for all x ∈ [a, b] .
We define for two mappings f, g : [a, b] → R, the Chebyshev functional as
Z b
Z b
Z b
1
1
1
T (f, g) =
f (t) g (t) dt −
f (t) dt
g (t) dt ,
b−a a
b−a a
b−a a
provided that f, g and f g are integrable on [a, b] .
Also in [1] we can find the pre-Grüss inequality as
T 2 (f, g) ≤ T (f, f ) T (g, g) ,
where f, g ∈ L2 [a, b] and T (f, g) is the Chebyshev functional as defined above.
In this paper, we give a generalization of (1.4) and then apply it to probability density functions, generalized beta random variable and special means.
64
Fiza Zafar and Nazir Ahmad Mir
2. Main Results
Theorem 5. Let f : [a, b] → R be an absolutely continuous function whose
0
first derivative f ∈ L2 [a, b] . Then, we have the inequality
Zb
f (a) + f (b)
1
(1 − h) f (x) − f (b) − f (a) x − a + b
+
h
−
f
(t)
dt
b
−
a
2
2
b
−
a
a
"
2 # 12
(b − a)2
a
+
b
≤
3h2 − 3h + 1 + h (1 − h) x −
12
2
1
"
2 # 2
1 f (b) − f (a)
0 2
×
,
f −
b−a
b−a
2
"
2 # 21
2
a+b
(b − a)
1
(Γ − γ)
3h2 − 3h + 1 + h (1 − h) x −
≤
,
2
12
2
0
if γ ≤ f (t) ≤ Γ almost everywhere t on (a, b) .
(2.1)
for all x ∈ a + h b−a
, b − h b−a
and h ∈ [0, 1] .
2
2
Proof. We consider the kernel p : [a, b]2 → R as defined in [2]:
t − a + h b−a
, if t ∈ [a, x]
2
p (x, t) =
b−a
t − b − h 2 , if t ∈ (x, b].
Using Korkine’s identity
1
T (f, g) :=
2 (b − a)2
Zb Zb
(f (t) − f (s)) (g (t) − g (s)) dtds,
a
a
we obtain
1
b−a
Zb
1
p (x, t) f (t) dt −
b−a
0
a
=
1
2 (b − a)2
Zb
a
Zb
a
Zb
a
1
p (x, t) dt
b−a
Zb
0
f (t) dt
a
0
0
(p (x, t) − p (x, s)) f (t) − f (s) dtds.
(2.2)
65
Generalzation of Ostrowski-grüss Type Inequality
Since,
1
b−a
Zb
f (a) + f (b)
1
p (x, t) f (t) dt = (1 − h) f (x) + h
−
2
b−a
0
a
Zb
f (t) dt,
a
1
b−a
Zb
a+b
p (x, t) dt = (1 − h) x −
,
2
a
and
Zb
1
b−a
0
f (t) dt =
f (b) − f (a)
,
b−a
a
then by (2.2) we get the following identity
f (b) − f (a)
a+b
(1 − h) f (x) −
x−
b−a
2
Zb
f (a) + f (b)
1
+h
−
f (t) dt
2
b−a
a
=
1
2 (b − a)2
Zb Zb
a
0
0
(p (x, t) − p (x, s)) f (t) − f (s) dtds,
(2.3)
a
, b − h b−a
for all x ∈ a + h b−a
and h ∈ [0, 1] .
2
2
Using the Cauchy-Bunaikowski-Schwartz inequality for double integrals,we
may write
b b
Z Z
1
0
0
(p
(x,
t)
−
p
(x,
s))
f
(t)
−
f
(s)
dtds
2 (b − a)2 a

≤ 
1
2 (b − a)2
a
Zb Zb
a

×
1
2 (b − a)2
 21
(p (x, t) − p (x, s))2 dtds
a
Zb Zb a
a
 12
2
0
0
f (t) − f (s) dtds .
(2.4)
66
Fiza Zafar and Nazir Ahmad Mir
However,
1
2 (b − a)2
a
(p (x, t) − p (x, s))2 dtds
a

Zb
2
1
p (x, t) dt
b−a
a
"a
#
3
3
3
b−a
3
+ b − h b−a
−
x
x− a+h 2
1
h
(b
−
a)
2
=
+
b−a
3
12
2
a+b
2
− (1 − h) x −
.
(2.5)
2
=
1
b−a
Zb
Zb Zb
p2 (x, t) dt − 
In addition simple calculations shows that
3
3 b−a
b−a
−x
+ b−h
x− a+h
2
2
#
" 2
(1 − h)2 (b − a)2
a+b
+
,
= (b − a) (1 − h) 3 x −
2
4
(2.6)
and
1
2 (b − a)2
Zb Zb a
2
0
0
f (t) − f (s) dtds
a
2
f (b) − f (a)
1 0 2
.
=
f −
b−a
b−a
2
(2.7)
Using (2.3)−(2.7),we deduce the first inequality.
0
Moreover, if γ ≤ f (t) ≤ Γ almost everywhere t on (a, b) , then, by using
Grüss inequality, we have

2
b
Zb Z
2
1
1
1
0
0
0≤
f (t) dt − 
f (t) dt ≤ (Γ − γ)2 ,
b−a
b−a
4
a
which proves the last inequality of (2.1).
a
Generalzation of Ostrowski-grüss Type Inequality
67
Remark 1. Since
3h2 − 3h + 1 ≤ 1, ∀ h ∈ [0, 1] .
and is minimum for h = 21 .
Thus, (2.1) shows an overall improvement in the inequality obtained by
Barnett et al. [5].
The following corollary contains some special cases of (2.1).
Corollary 1. 1. For h = 1, i.e., x =
a+b
,
2
(2.1) gives
Zb
f
(a)
+
f
(b)
(b − a)
− f (t) dt
2
a
"
#1
2 f (b) − f (a) 2 2
(b − a)2
1 0
√
≤
,
f −
b
−
a
b−a
2
2 3
1
≤ √ (Γ − γ) (b − a)2 .
4 3
0
if γ ≤ f (t) ≤ Γ almost everywhere t on (a, b) .
(2.8)
which is trapezoid inequality.
2. For h = 0 and x = a+b
, (2.1) gives
2
Zb
a
+
b
(b − a) f
−
f
(t)
dt
2
a
"
2 # 12
2
2
1 0
f (b) − f (a)
(b − a)
√
≤
,
f −
b−a
b−a
2
2 3
≤
1
√ (Γ − γ) (b − a)2 .
4 3
0
if γ ≤ f (t) ≤ Γ almost everywhere t on (a, b) .
which is mid-point inequality.
(2.9)
68
Fiza Zafar and Nazir Ahmad Mir
3. For h =
1
2
and x =
a+b
,
2
(2.1) gives
Zb
f (a) + 2f a+b
+
f
(b)
1
2
−
f
(t)
dt
4
b
−
a
a
1
"
#
2 2
f (b) − f (a)
(b − a)2
1 0 2
√
≤
,
f −
b−a
b−a
2
4 3
≤
1
√ (Γ − γ) (b − a)2 .
8 3
0
if γ ≤ f (t) ≤ Γ almost everywhere t on (a, b) .
(2.10)
which is an averaged mid-point and trapezoid inequality.
, (2.1) gives
4. For h = 31 and x = a+b
2
Zb
a+b
(b − a) f (a) + 4f 2 + f (b) − f (t) dt
6
a
"
2 # 12
(b − a)2
1 f (b) − f (a)
0 2
≤
,
f −
6
b−a
b−a
2
≤
1
(Γ − γ) (b − a)2 .
12
0
if γ ≤ f (t) ≤ Γ almost everywhere t on (a, b) .
(2.11)
0
which is a variant of Simpson’s inequality for first differentiable function f, f
0
is integrable and there exist constants γ, Γ ∈ R such that γ ≤ f (t) ≤ Γ, t ∈
(a, b).
Generalzation of Ostrowski-grüss Type Inequality
69
3. Applications
3.1
Application for P.D.F’s
Let X be a random variable having the p.d.f f : [a, b] → R+ and the cumulative
distribution function F : [a, b] → [0, 1] , i.e.,
Zx
F (x) =
b−a
b−a
,b − h
⊂ [a, b] ,
f (t) dt, x ∈ a + h
2
2
a
and
Zb
tf (t) dt,
E (X) =
a
is the expectation of the random variable X on the interval [a, b] . Then, we
may have the following.
Theorem 6. Under the above assumptions and if the p.d.f belongs to L2 [a, b] ,
then, we have the inequality:
a+b
h b − E (X) (1 − h) F (x) − 1
x−
+ −
b−a
2
2
b−a "
2 # 12
1
a
+
b
1
≤
3h2 − 3h + 1 + h (1 − h) x −
b − a 12
2
1
× (b − a) kf k22 − 1 2 ,
"
2 # 21
(M − m) 1
a+b
≤
3h2 − 3h + 1 + h (1 − h) x −
,
2 (b − a) 12
2
(if m ≤ f ≤ M almost everywhere on [a, b] .)
(3.1)
for all x ∈ a + h b−a
.
, b − h b−a
2
2
Proof. Put in (2.1) F instead of f to get (3.1) and the details are omitted.
70
Fiza Zafar and Nazir Ahmad Mir
Corollary 2. Under the above assumptions, we have
h
b
−
E
(X)
a
+
b
(1 − h) Pr X ≤
+ −
2
2
b−a 1 1
1
≤ √ 3h2 − 3h + 1 2 (b − a) kf k22 − 1 2 ,
2 3
1
1
≤ √ 3h2 − 3h + 1 2 (M − m) , m ≤ f ≤ M as above.
4 3
3.2
(3.2)
Applications for generalized beta random variable
If X is a beta random variable with parameters β3 > −1, β4 > −1 and for
β2 > 0 and any β1 , the generalized beta random variable
Y = β1 + β2 X,
is said to have a generalized beta distribution [4] and the probability density
function of the generalized beta distribution of beta random variable is,
(
(x−β1 )β3 (β1 +β2 −x)β4
(β +β +1) , for β1 < x < β1 + β2
β(β3 +1,β4 +1)β2 3 4
f (x) =
,
0,
otherwise.
where β (l, m) is the beta function with l, m > 0 and is defined as
Z1
β (l, m) =
xl−1 (1 − x)m−1 dx.
0
For p, q > 0 and h ∈ [0, 1), we choose,
h
,
2
= (1 − h) ,
= p − 1,
= q − 1.
β1 =
β2
β3
β4
Then, the probability density function associated with generalized beta random variable
h
Y = + (1 − h) X,
2
71
Generalzation of Ostrowski-grüss Type Inequality
takes the form
(
p−1
(x− h2 )
f (x) =
(1− h2 −x)
β(p,q)(1−h)
q−1
p+q−1
,
0,
h
2
<x<1−
otherwise.
h
2
Now,
1− h
2
Z
E (Y ) =
xf (x) dx
h
2
= (1 − h)
h
p
+ .
p+q 2
(3.3)
and
1
β (2p − 1, 2q − 1) .
(1 − h) β 2 (p, q)
Then, by Theorem 6, we may state the following.
kf (.; p, q)k22 =
Proposition 1. Let X be a beta random variable with parameters (p, q) . Then
for generalized beta random variable
h
Y = + (1 − h) X,
2
we have the inequality
Pr (Y ≤ x) − x + 1 − q 2
p + q
"
2 # 12
1
1
≤
3h2 − 3h + 1 + h (1 − h) x −
12
2
1
×
[β (2p − 1, 2q − 1) − (1 − h) β 2 (p, q)] 2
3
.
(3.4)
(1 − h) 2 β (p, q)
for all x ∈ h2 , 1 − h2 .
In particular, for x = 21 in (3.4), we have:
1
q
Pr Y ≤
−
2
p + q
1
1 [β (2p − 1, 2q − 1) − (1 − h) β 2 (p, q)] 2
1
≤ √ 3h2 − 3h + 1 2
.
3
2 3
(1 − h) 2 β (p, q)
72
3.3
Fiza Zafar and Nazir Ahmad Mir
Applications for Special Means
Example 1. Consider the mapping f (x) = xp , p ∈ R \ {−1, 0} . Then
Z b
1
f (t) dt = Lpp (a, b) ,
b−a a
f (b) − f (a)
= pLp−1
p−1 ,
b−a
f (a) + f (b)
ap + b p
=
= A (ap , bp ) ,
2
2
and
Z b
1 1
0 2
0 2
2(p−1)
f =
f (t) dt = p2 L2(p−1) .
b−a
b−a a
2
Therefore, (2.1) takes the form
p−1
(1 − h) xp − pLp−1
(x − A (a, b)) + hA (ap , bp ) − Lpp "
# 21
(b − a)2
3h2 − 3h + 1 + h (1 − h) (x − A (a, b))2
≤ |p|
12
i1
h
2(p−1)
2(p−1) 2
× L2(p−1) − Lp−1
(3.5)
Choose x = A (a, b) in (3.5), to get
(1 − h) Ap (a, b) + hA (ap , bp ) − Lpp h
i1
1
(b − a)
2(p−1)
2(p−1) 2
√
.
≤
3h2 − 3h + 1 2 |p| L2(p−1) − Lp−1
2 3
which is minimum for h = 12 . Moreover for h = 1,
A (ap , bp ) − Lpp i1
(b − a) h 2(p−1)
2(p−1) 2
√
≤
|p| L2(p−1) − Lp−1
.
2 3
Example 2. Consider the mapping f (x) = x1 , x ∈ a + h b−a
, b − h b−a
⊂ (0, ∞) .
2
2
Generalzation of Ostrowski-grüss Type Inequality
73
Then,
1
b−a
Zb
f (t) dt =
1
,
L
a
f (b) − f (a)
1
= − 2,
b−a
G
A (a, b)
f (a) + f (b)
=
,
2
G2
Zb a2 + ab + b2
1
0 2
dt
=
f
(t)
,
b−a
3a3 b3
a
and
1
b−a
2
Zb f (b) − f (a)
(b − a)2
0 2
=
.
f (t) dt −
b−a
3a3 b3
a
Therefore, (2.1) becomes,
1
1
A
(a,
b)
1
(1 − h)
+
(x
−
A
(a,
b))
+
h
−
x G2
G2
L
"
# 21
2
(b − a)
≤
3h2 − 3h + 1 + h (1 − h) (x − A (a, b))2
12
(b − a)
×√
.
3G3
Choosing x = A (a, b) in (3.6),
(1 − h) 1 + h A (a, b) − 1 A
G2
L
21
(b − a)2
2
3h
−
3h
+
1
≤
.
6G3
(3.6)
74
Fiza Zafar and Nazir Ahmad Mir
If we choose x = L in (3.6),we get
L
A
(a,
b)
1
(1 − h)
+
(2h
−
1)
−
h
G2
G2
L
# 21
"
2
(b − a)
3h2 − 3h + 1 + h (1 − h) (L − A (a, b))2
≤
12
(b − a)
.
×√
3G3
Example 3. Finally,
the
let us consider
mapping
b−a
f (x) = ln x, x ∈ a + h b−a
⊂ (0, ∞) . Then
,
b
−
h
2
2
1
b−a
Zb
f (t) dt = ln I,
a
1
f (b) − f (a)
=
,
b−a
L
f (a) + f (b)
= ln G,
2
Zb 1
1
0 2
,
f (t) dt =
b−a
G2
a
and
1
b−a
2
Zb f (b) − f (a)
L2 − G2
0 2
.
=
f (t) dt −
b−a
L2 G2
a
Thus, (2.1) takes the form,
x(1−h) Gh
x
−
A
(a,
b)
ln
− (1 − h)
I
L
"
# 21
(b − a)2
≤
3h2 − 3h + 1 + h (1 − h) (x − A (a, b))2
12
1
(L2 − G2 ) 2
×
.
LG
(3.7)
Generalzation of Ostrowski-grüss Type Inequality
75
For x = A (a, b) in (3.7), we get:
(A (a, b))(1−h) Gh ln
I
1
1
(b − a) (3h2 − 3h + 1) 2 (L2 − G2 ) 2
√
.
≤
LG
2 3
which for h = 1, takes the form
G
ln I
1
(b − a) (L2 − G2 ) 2
√
≤
.
2 3LG
Also, choosing x = I, we get
Gh
I
−
A
(a,
b)
ln
I h − (1 − h)
L
# 12
"
(b − a)2
3h2 − 3h + 1 + h (1 − h) (I − A (a, b))2
≤
12
1
(L2 − G2 ) 2
.
×
LG
References
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no.3 (2001), 533–542 Preprint RGMIA Res. Rep. Coll., 3 no. 1 (2000),
Article 11.
[2] S. S. Dragomir, P. Cerone, and J. Roumeliotis, A new generalization of Ostrowski’s integral inequality for mappings whose derivatives are bounded
and applications in numerical integration, Appl. Math. Lett.,13 (2000),
19-25.
76
Fiza Zafar and Nazir Ahmad Mir
[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, Comput. Math. Appl., 33 no.
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