Journal of Inequalities in Pure and
Applied Mathematics
AN IDENTITY FOR THE CHEBYCHEV FUNCTIONAL
AND SOME RAMIFICATIONS
volume 3, issue 1, article 4,
2002.
P. CERONE
School of Communications and Informatics
Victoria University of Technology
PO Box 14428
Melbourne City MC
8001 Victoria, Australia
EMail: pc@matilda.vu.edu.au
URL: http://rgmia.vu.edu.au/cerone
Received 15 April, 2001;
accepted 17 July, 2001.
Communicated by: R.P. Agarwal
Abstract
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c 2000 Victoria University
ISSN (electronic): 1443-5756
034-01
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Abstract
An identity for the Chebychev functional is presented in which a RiemannStieltjes integral is involved. This allows bounds for the functional to be obtained
for functions that are of bounded variation, Lipschitzian and monotone. Some
applications are presented to produce bounds for moments of functions about
a general point γ and for moment generating functions.
2000 Mathematics Subject Classification: Primary 26D15, 26D20; Secondary
65Xxx.
Key words: Chebychev functional, Bounds, Riemann-Stieltjes, Moments, Moment
Generating Function.
The author undertook this work while on sabbatical at the Division of Mathematics,
La Trobe University, Bendigo. Both Victoria University and the host University are
commended for giving the author the time and opportunity to think.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
An Identity for the Chebychev Functional . . . . . . . . . . . . . . . . 5
3
Results Involving Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4
Approximations for the Moment Generating Function . . . . . . 25
References
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
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1.
Introduction
For two measurable functions f, g : [a, b] → R, define the functional, which is
known in the literature as Chebychev’s functional, by
(1.1)
T (f, g) := M (f g) − M (f ) M (g) ,
where the integral mean is given by
(1.2)
1
M (f ) =
b−a
Z
b
f (x) dx.
a
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
The integrals in (1.1) are assumed to exist.
Further, the weighted Chebychev functional is defined by
(1.3)
T (f, g; p) := M (f, g; p) − M (f ; p) M (g; p) ,
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where the weighted integral mean is given by
Rb
p (x) f (x) dx
(1.4)
M (f ; p) = a R b
.
p (x) dx
a
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We note that,
T (f, g; 1) ≡ T (f, g)
and
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M (f ; 1) ≡ M (f ) .
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It is the aim of this article to obtain bounds on the functionals (1.1) and (1.3)
in terms of one of the functions, say f , being of bounded variation, Lipschitzian
or monotonic nondecreasing.
This is accomplished by developing identities involving a Riemann-Stieltjes
integral. These identities seem to be new. The main results are obtained in
Section 2, while in Section 3 bounds for moments about a general point γ are
obtained for functions of bounded variation, Lipschitzian and monotonic. In
a previous article, Cerone and Dragomir [2] obtained bounds in terms of the
kf 0 kp , p ≥ 1 where it necessitated the differentiability of the function f . There
is no need for such assumptions in the work covered by the current development.
A further application is given in Section 4 in which the moment generating
function is approximated.
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
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2.
An Identity for the Chebychev Functional
It is worthwhile noting that a number of identities relating to the Chebychev
functional already exist. The reader is referred to [7] Chapters IX and X. Korkine’s identity is well known, see [7, p. 296] and is given by
Z bZ b
1
(2.1)
T (f, g) =
(f (x) − f (y)) (g (x) − g (y)) dxdy.
2 (b − a)2 a a
It is identity (2.1) that is often used to prove an inequality of Grüss for functions
bounded above and below, [7].
The Grüss inequality is given by
(2.2)
|T (f, g)| ≤
1
(Φf − φf ) (Φg − φg )
4
where φf ≤ f (x) ≤ Φf for x ∈ [a, b].
If we let S (f ) be an operator defined by
(2.3)
S (f ) (x) := f (x) − M (f ) ,
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
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which shifts a function by its integral mean, then the following identity holds.
Namely,
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T (f, g) = T (S (f ) , g) = T (f, S (g)) = T (S (f ) , S (g)) ,
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(2.4)
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and so
(2.5)
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T (f, g) = M (S (f ) g) = M (f S (g)) = M (S (f ) S (g))
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since M (S (f )) = M (S (g)) = 0.
For the last term in (2.4) or (2.5) only one of the functions needs to be shifted
by its integral mean. If the other were to be shifted by any other quantity, the
identities would still hold. A weighted version of (2.5) related to T (f, g) =
M ((f (x) − κ) S (g)) for κ arbitrary was given by Sonin [8] (see [7, p. 246]).
The interested reader is also referred to Dragomir [5] and Fink [6] for extensive treatments of the Grüss and related inequalities.
The following lemma presents an identity for the Chebychev functional that
involves a Riemann-Stieltjes integral.
Lemma 2.1. Let f, g : [a, b] → R, where f is of bounded variation and g is
continuous on [a, b], then
Z b
1
(2.6)
T (f, g) =
ψ (t) df (t) ,
(b − a)2 a
ψ (t) = (t − a) A (t, b) − (b − t) A (a, t)
with
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Z
(2.8)
P. Cerone
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where
(2.7)
On an Identity for the
Chebychev Functional and
Some Ramifications
A (a, b) =
b
g (x) dx.
a
Proof. From (2.6) integrating the Riemann-Stieltjes integral by parts produces
Z b
1
ψ (t) df (t)
(b − a)2 a
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1
=
(b − a)2
(
1
=
(b − a)2
b
Z
−
ψ (t) f (t)
a
)
b
f (t) dψ (t)
a
Z
ψ (b) f (b) − ψ (a) f (a) −
b
f (t) ψ (t) dt
0
a
since ψ (t) is differentiable. Thus, from (2.7), ψ (a) = ψ (b) = 0 and so
Z b
Z b
1
1
ψ (t) df (t) =
[(b − a) g (t) − A (a, b)] f (t) dt
(b − a)2 a
(b − a)2 a
Z b
1
=
[g (t) − M (g)] f (t) dt
b−a a
= M (f S (g))
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
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from which the result (2.6) is obtained on noting identity (2.5).
The following well known lemmas will prove useful and are stated here for
lucidity.
Lemma 2.2. Let g, v : [a, b] → R be such that g is continuous and v is of
Rb
bounded variation on [a, b]. Then the Riemann-Stieltjes integral a g (t) dv (t)
exists and is such that
Z b
b
_
(2.9)
g (t) dv (t) ≤ sup |g (t)| (v) ,
a
where
Wb
a
t∈[a,b]
(v) is the total variation of v on [a, b].
a
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Lemma 2.3. Let g, v : [a, b] → R be such that g is Riemann-integrable on [a, b]
and v is L−Lipschitzian on [a, b]. Then
Z b
Z b
(2.10)
g (t) dv (t) ≤ L
|g (t)| dt
a
a
with v is L−Lipschitzian if it satisfies
|v (x) − v (y)| ≤ L |x − y|
for all x, y ∈ [a, b].
Lemma 2.4. Let g, v : [a, b] → R be such that g is continuous on [a, b] and v is
monotonic nondecreasing on [a, b]. Then
Z b
Z b
(2.11)
g (t) dv (t) ≤
|g (t)| dv (t) .
a
a
It should be noted that if v is nonincreasing then −v is nondecreasing.
Theorem 2.5. Let f, g : [a, b] → R, where f is of bounded variation and g is
continuous on [a, b]. Then
(b − a)2 |T (f, g)|

b
W



sup
|ψ
(t)|
(f ) ,


a
 t∈[a,b]
Rb
≤
L a |ψ (t)| dt,
for f L − Lipschitzian,





 Rb
|ψ (t)| df (t) ,
for f monotonic nondecreasing,
a
W
where ba (f ) is the total variation of f on [a, b].
(2.12)
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
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Proof. Follows directly from Lemmas 2.1 – 2.4. That is, from the identity (2.6)
and (2.9) – (2.11).
The following lemma gives an identity for the weighted Chebychev functional that involves a Riemann-Stieltjes integral.
Lemma 2.6. Let f, g, p : [a, b] → R, where f is of bounded variation and g, p
Rb
are continuous on [a, b]. Further, let P (b) = a p (x) dx > 0, then
1
T (f, g; p) = 2
P (b)
(2.13)
Z
b
Ψ (t) df (t) ,
a
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
where T (f, g; p) is as given in (1.3),
Ψ (t) = P (t) Ḡ (t) − P̄ (t) G (t)
(2.14)
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with
(2.15)
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


P (t) =
Rt
G (t) =
Rt
a
p (x) dx,
P̄ (t) = P (b) − P (t)
and
a
p (x) g (x) dx, Ḡ (t) = G (b) − G (t) .
Proof. The proof follows closely that of Lemma 2.1.
We first note that Ψ (t) may be represented in terms of only P (·) and G (·).
Namely,
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(2.16)
Ψ (t) = P (t) G (b) − P (b) G (t) .
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It may further be noticed that Ψ (a) = Ψ (b) = 0. Thus, integrating from (2.13)
and using either (2.14) or (2.16) gives
Z b
1
Ψ (t) df (t)
P 2 (b) a
Z b
−1
= 2
f (t) dΨ (t)
P (b) a
Z b
1
= 2
[P (b) G0 (t) − P 0 (t) G (b)] f (t) dt
P (b) a
Z b
1
G (b)
=
p (t) g (t) −
p (t) f (t) dt
P (b) a
P (b)
Z b
Z b
G (b)
1
1
p (t) g (t) f (t) dt −
·
p (t) f (t) dt
=
P (b) a
P (b) P (b) a
= M (f, g; p) − M (g; p) M (f ; p)
= T (f, g; p) ,
where we have used the fact that
G (b)
= M (g; p) .
P (b)
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
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Theorem 2.7. Let the conditions of Lemma 2.6 on f , g and p continue to hold.
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Then
(2.17) P 2 (b) |T (f, g; p)|

b
W



sup
|Ψ
(t)|
(f ) ,


a
t∈[a,b]



Rb
≤

L a |Ψ (t)| dt,
for f L − Lipschitzian,






 Rb
|Ψ (t)| df (t) ,
for f monotonic nondecreasing.
a
On an Identity for the
Chebychev Functional and
Some Ramifications
where T R(f, g; p) is as given by
R t(1.3) and Ψ (t) = P (t) G (b) − P (b) G (t), with
t
P (t) = a p (x) dx, G (t) = a p (x) g (x) dx.
Proof. The proof uses Lemmas 2.1 – 2.4 and follows closely that of Theorem
2.5.
Remark 2.1. If we take p (x) ≡ 1 in the above results involving the weighted
Chebychev functional, then the results obtained earlier for the unweighted Chebychev functional are recaptured.
Grüss type inequalities obtained from bounds on the Chebychev functional
have been applied in a variety of areas including in obtaining perturbed rules in
numerical integration, see for example [4]. In the following section the above
work will be applied to the approximation of moments. For other related results
see also [1] and [3].
P. Cerone
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Remark 2.2. If f is differentiable then the identity (2.6) would become
Z b
1
(2.18)
T (f, g) =
ψ (t) f 0 (t) dt
2
(b − a) a
and so

kψk1 kf 0 k∞ ,





2
kψkq kf 0 kp ,
(b − a) |T (f, g)| ≤





kψk∞ kf 0 k1 ,
f 0 ∈ L∞ [a, b] ;
f 0 ∈ Lp [a, b] ,
p > 1, p1 + 1q = 1;
f 0 ∈ L1 [a, b] ;
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
where the Lebesgue norms k·k are defined in the usual way as
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Z
kgkp :=
p1
b
p
|g (t)| dt ,
for g ∈ Lp [a, b] , p ≥ 1,
a
1 1
+ =1
p q
and
kgk∞ := ess sup |g (t)| ,
for g ∈ L∞ [a, b] .
t∈[a,b]
The identity for the weighted integral means (2.13) and the corresponding
bounds (2.17) will not be examined further here.
Theorem 2.8. Let g : [a, b] → R be absolutely continuous on [a, b] then for
(2.19)
D (g; a, t, b) := M (g; t, b) − M (g; a, t) ,
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(2.20)
|D (g; a, t, b)|
 b−a 0
kg k∞ ,
g 0 ∈ L∞ [a, b] ;

2






i1
h


(t−a)q +(b−t)q q

kg 0 kp , g 0 ∈ Lp [a, b] ,


q+1



p > 1, p1 + 1q = 1;
≤
kg 0 k1 ,
g 0 ∈ L1 [a, b] ;






Wb


g of bounded variation;

a (g) ,




 b−a 
L,
g is L − Lipschitzian.
2
Proof. Let the kernel r (t, u) be defined by

u−a


, u ∈ [a, t] ,

 t−a
(2.21)
r (t, u) :=


b−u


, u ∈ (t, b]
b−t
then a straight forward integration by parts argument of the Riemann-Stieltjes
integral over each of the intervals [a, t] and (t, b] gives the identity
Z b
(2.22)
r (t, u) dg (u) = D (g; a, t, b) .
a
Now for g absolutely continuous then
Z b
(2.23)
D (g; a, t, b) =
r (t, u) g 0 (u) du
a
On an Identity for the
Chebychev Functional and
Some Ramifications
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and so
Z
b
|g 0 (u)| du,
|D (g; a, t, b)| ≤ ess sup |r (t, u)|
u∈[a,b]
for g 0 ∈ L1 [a, b] ,
a
where from (2.21)
ess sup |r (t, u)| = 1
(2.24)
u∈[a,b]
and so the third inequality in (2.20) results. Further, using the Hölder inequality
gives
Z
(2.25)
|D (g; a, t, b)| ≤
b
|r (t, u)|q du
1q Z
b
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1 1
for p > 1, + = 1,
p q
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where explicitly from (2.21)
Z
b
q
|r (t, u)| du
(2.26)
a
1q
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p1
p
0
|g (t)| dt
a
a
On an Identity for the
Chebychev Functional and
Some Ramifications
q 1q
b−u
=
du +
du
b−t
a
t
Z 1
1q
q
q 1q
q
= [(t − a) + (b − t) ]
u du
Z t u−a
t−a
q
Z b
0
q
q 1q
(t − a) + (b − t)
=
q+1
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Also
(2.27)
0
Z
|D (g; a, t, b)| ≤ ess sup |g (u)|
u∈[a,b]
b
|r (t, u)| du,
a
and so from (2.26) with q = 1 gives the first inequality in (2.20).
Now, for g (u) of bounded variation on [a, b] then from Lemma 2.2, equation
(2.9) and identity (2.22) gives
|D (g; a, t, b)| ≤ ess sup |r (t, u)|
u∈[a,b]
b
_
(g)
a
producing the fourth inequality in (2.20) on using (2.24). From (2.10) and (2.22)
we have, by associating g with v and r (t, ·) with g (·),
Z b
|D (g; a, t, b)| ≤ L
|r (t, u)| du
a
and so from (2.26) with q = 1 gives the final inequality in (2.20).
Remark 2.3. The results of Theorem 2.8 may be used to obtain bounds on ψ (t)
since from (2.7) and (2.19)
ψ (t) = (t − a) (b − t) D (g; a, t, b) .
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
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Hence, upper bounds on the Chebychev functional may be obtained from (2.12)
and (2.18) for general functions g. The following two sections investigate the
exact evaluation (2.12) for specific functions for g (·).
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3.
Results Involving Moments
In this section bounds on the nth moment about a point γ are investigated. Define
for n a nonnegative integer,
Z b
(3.1)
Mn (γ) :=
(x − γ)n h (x) dx, γ ∈ R.
a
If γ = 0 then Mn (0) are the moments about the origin while taking γ = M1 (0)
gives the central moments. Further the expectation of a continuous random
variable is given by
Z b
(3.2)
E (X) =
h (x) dx,
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
a
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where h (x) is the probability density function of the random variable X and so
E (X) = M1 (0). Also, the variance of the random variable X, σ 2 (X) is given
by
Z b
2
2
(3.3)
σ (X) = E (X − E (X)) =
(x − E (X))2 h (x) dx,
a
which may be seen to be the second moment about the mean, namely
σ 2 (X) = M2 (M1 (0)) .
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The following corollary is valid.
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Corollary 3.1. Let f : [a, b] → R be integrable on [a, b], then
B n+1 − An+1
(3.4) Mn (γ) −
M (f )
n+1

b
W

1

(f ) , for f of bounded variation on [a, b] ,
sup
|φ
(t)|
·

n+1


a
t∈[a,b]





L Rb
≤
|φ (t)| dt,
for f L − Lipschitzian,


n+1 a






Rb

 1
|φ (t)| df (t) ,
for f monotonic nondecreasing.
n+1 a
where Mn (γ) is as given by (3.1), M (f ) is the integral mean of f as defined
in (1.2),
B = b − γ, A = a − γ
and
(3.5)
φ (t) = (t − γ)n −
t−a
b−a
(b − γ)n+1 +
b−t
b−a
(a − γ)n+1 .
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
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Proof. From (2.12) taking g (t) = (t − γ) then using (1.1) and (1.2) gives
(b − a) |T (f, (t − γ)n )| = Mn (γ) −
B
n+1
−A
n+1
n+1
M (f ) .
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n
The right hand side is obtained on noting that for g (t) = (t − γ) , φ (t) =
− ψ(t)
.
b−a
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Remark 3.1. It should be noted here that Cerone and Dragomir [2] obtained
bounds on the left hand expression for f 0 ∈ Lp [a, b], p ≥ 1. They obtained
the following Lemmas which will prove useful in procuring expressions for the
bounds in (3.4) in a more explicit form.
Lemma 3.2. Let φ (t) be as defined by (3.5), then


n odd, any γ and t ∈ (a, b)









<0


γ < a,
t ∈ (a, b)



 n even
a < γ < b, t ∈ [c, b)
(3.6)
φ (t)






γ > b,
t ∈ (a, b)


 > 0, n even
a < γ < b, t ∈ (a, c)
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
Title Page
where φ (c) = 0, a < c < b and

> γ, γ <





c = γ, γ =





< γ, γ >
Contents
a+b
2
a+b
2
a+b
.
2
Lemma 3.3. For φ (t) as given by (3.5) then
Z
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b
|φ (t)| dt
(3.7)
a
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=





















B−A
2
[B
n+1
−A
n+1
]−
B n+2 −An+2
,
n+2
n odd and any γ
;
n even and γ < a
2C n+2 −B n+2 −An+2
n+2
1
+ 2(b−a)
(b − a)2 − 2 (c − a)2 B n+1
+ 2 (b − c)2 − (b − a)2 An+1 , n even and a < γ < b;
B n+2 −An+2
n+2
−
B−A
2
[B n+1 − An+1 ] ,
n even and γ > b,
where
(3.8)











B = b − γ, A = a − γ, C = c − γ,
C1 =
Rc
a
with C (t) =
C (t) dt, C2 =
t−a
b−a
B n+1 +
Rb
c
b−t
b−a
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
C (t) dt,
An+1
and φ (c) = 0 with a < c < b.
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Contents
Lemma 3.4. For φ (t) as defined by (3.5), then

B n+1 −An+1

C (t∗ ) − (n+1)(B−A)
, n odd, n even and γ < a;





B n+1 −An+1
(3.9)
sup φ̃ (t) =
− C (t∗ ) n even and γ > b;
(n+1)(B−A)

t∈[a,b]




 m1 +m2
2
+ m1 −m
n even and a < γ < b,
2
2
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where
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(3.10)
(t∗ − γ)n =
B n+1 − An+1
,
(n + 1) (B − A)
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C (t) is as defined in (3.8), m1 = φ̃ (t∗1 ), m2 = −φ̃ (t∗2 ) and t∗ , t∗1 , t∗2 satisfy
(3.10) with t∗1 < t∗2 .
The following lemma is required to determine the bound in (3.4) when f is
monotonic nondecreasing. This was not covered in Cerone and Dragomir [2]
since they obtained bounds assuming that f were differentiable.
Lemma 3.5. The following result holds for φ (t) as defined by (3.5),
Z b
1
(3.11)
|φ (t)| df
n+1 a

χn (a, b) ,
n odd or n even and γ < a,




−χn (a, b) ,
n even and γ > b,
=




χn (c, b) − χn (a, c) , n even and a < γ < b
and for f : [a, b] → R, monotonic nondecreasing
Z b
1
(3.12)
|φ (t)| df
n+1 a
 B(B n −1)−A(An −1)
f (b) ,

n+1





n
n −1)


 A(A −1)−B(B
f (b) ,
n+1
≤
i
h


 B n+1 − C n+1 − (B n −An ) (b − c) f (b)


b−a
n+1

h
i


 + (B n −An ) (c − a) − (C n+1 − An+1 ) f (a) ,
b−a
n+1
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
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n odd or n even
and γ < a;
n even and γ > b;
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a < γ < b,
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where
(3.13)
Z b
χn (a, b) =
(t − γ)n −
(B n − An )
f (t) dt,
(n + 1) (b − a)
a
A = a − γ, B = b − γ, C = c − γ.
Proof. Let α, β ∈ [a, b] and
χn (α, β)
Z β
1
=
|φ (t)| df
n+1 α
Z β
φ (α) f (α) − φ (β) f (β)
(B n − An )
n
−
(t − γ) −
f (t) dt
=
n+1
(n + 1) (b − a)
α
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
Title Page
and χn (a, b) is as given by (3.13) since φ (a) = φ (b) = 0.
Further, using the results of Lemma 3.2 as represented in (3.6), and, the fact
that

Z β
φ (t) < 0, t ∈ [α, β]
 χ (α, β) ,
1
|φ (t)| df =

n+1 α
−χ (α, β) , φ (t) > 0, t ∈ [α, β]
gives the results as stated.
We now use the fact that f is monotonic nondecreasing so that from (3.13)
Z b
B n − An
n
dt.
χn (a, b) ≤ f (b)
(t − γ) −
(n + 1) (b − a)
a
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Further,
Z b
B n − An
n
χn (c, b) ≤ f (b)
dt
(t − γ) −
(n + 1) (b − a)
c
n+1
B
− C n+1 (B n − An ) (b − c)
= f (b)
−
n+1
(n + 1) (b − a)
and
Z c
B n − An
n
χn (a, c) ≥ f (a)
(t − γ) −
dt
(n + 1) (b − a)
a
n+1
C
− An+1 (B n − An ) (c − a)
−
f (a)
=
n+1
(n + 1) (b − a)
so that the proof of the lemma is now complete.
The following corollary gives bounds for the expectation.
Corollary 3.6. Let f : [a, b] → R+ be a probability density function associated
with a random variable X. Then the expectation E (X) satisfies the inequalities
(3.14)
a+b
2

b
(b−a)3 W


(f ) ,
f of bounded variation,


 6 a
≤
b−a 2 L
· 2,
f L − Lipschitzian,

2



 b−a
[a + b − 1] f (b) , f monotonic nondecreasing.
2
E (X) −
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P. Cerone
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Proof. Taking n = 1 in Corollary 3.1 and using Lemmas 3.2 – 3.5 gives the
results after some straightforward algebra. In particular,
2
φ (t) = t − (a + b) t + ab =
a+b
t−
2
and t∗ the one solution of φ0 (t) = 0 is t∗ =
2
+
b−a
2
2
a+b
.
2
The following corollary gives bounds for the variance.
We shall assume that a < γ = E [X] < b.
Corollary 3.7. Let f : [a, b] → R+ be a p.d.f. associated with a random
variable X. The variance σ 2 (X) is such that
(3.15) σ 2 (X) − S

Wb
a (f )


[m
+
m
+
|m
−
m
|]
,
f of bounded variation,
1
2
2
1

6


 n


3
2 3

C2
1
3

 4 − b−a (c − a) B − (b − c) A
o
(AB)2
≤
b−a 2
2
2
+
(B
+
A
)
−
· L3 ,
f is L − Lipschitzian,

2
2







[B 3 − C 3 − (a + b) (b − c)] f (b)

3

 + [(a + b) (c − a) − (C 3 − A3 )] f (a) , f monotonic nondecreasing.
3
where
(b − E (X))3 + (E (X) − a)3
,
S=
3 (b − a)
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
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1
1
m1 = φ E (X) − S 2 , m2 = φ E (X) + S 2 ,
b−t
t−a
3
3
φ (t) = (t − γ) +
(γ − a) −
(b − γ)3 ,
b−a
b−a
A = a − γ, B = b − γ, C = c − γ, φ (c) = 0, a < c < b
and γ = E (X).
Proof. Taking n = 2 in Corollary 3.1 gives from (3.5)
t−a
b−t
3
3
φ (t) = (t − γ) +
A −
B3
b−a
b−a
where a < γ = E (X) < b.
From Lemma 3.4 and the third inequality in (3.9) with n = 2 gives
1
1
t∗1 = E [X] − S 2 , t∗2 = E [X] + S 2 ,
and hence the first inequality is shown from the first inequality in (3.4).
Now, if f is Lipschitzian, then from the second inequality in (3.4) and since
n = 2 and a < γ = E (X) < b, the second identity in (3.7) produces the
reported result given in (3.15) after some simplification.
The last inequality is obtained from (3.12) of Lemma 3.5 with n = 2 and
hence the corollary is proved.
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
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4.
Approximations for the Moment Generating Function
Let X be a random variable on [a, b] with probability density function h (x) then
the moment generating function MX (p) is given by
(4.1)
MX (p) = E epX =
Z
b
epx h (x) dx.
a
The following lemma will prove useful, in the proof of the subsequent corollary, as it examines the behaviour of the function θ (t)
On an Identity for the
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Some Ramifications
P. Cerone
(4.2)
(b − a) θ (t) = tAp (a, b) − [aAp (t, b) + bAp (a, t)] ,
Title Page
where
(4.3)
ebp − eap
.
Ap (a, b) =
p
Lemma 4.1. Let θ (t) be as defined by (4.2) and (4.3) then for any a, b ∈ R,
θ (t) has the following characteristics:
(i) θ (a) = θ (b) = 0,
(ii) θ (t) is convex for p < 0 and concave for p > 0,
Ap (a,b)
1
∗
(iii) there is one turning point at t = p ln b−a and a ≤ t∗ ≤ b.
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Proof. The result (i) is trivial from (4.2) using standard properties of the definite
integral to give θ (a) = θ (b) = 0.
Now,
(4.4)
θ0 (t) =
Ap (a, b)
− ept ,
b−a
θ00 (t) = −pept
giving θ00 (t) > 0 for p < 0 and θ00 (t) < 0 for p > 0 and (ii) holds.
Further, from (4.4) θ0 (t∗ ) = 0 where
Ap (a, b)
1
∗
t = ln
.
p
b−a
On an Identity for the
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Some Ramifications
To show that a ≤ t∗ ≤ b it suffices to show that
P. Cerone
θ0 (a) θ0 (b) < 0
since the exponential is continuous. Here θ0 (a) is the right derivative at a and
θ0 (b) is the left derivative at b.
Now,
Ap (a, b)
Ap (a, b)
0
0
ap
bp
θ (a) θ (b) =
−e
−e
b−a
b−a
but
Z b
Ap (a, b)
1
=
ept dt,
b−a
b−a a
the integral mean over [a, b] so that θ0 (a) > 0, and θ0 (b) < 0 for p > 0 and
θ0 (a) < 0 and θ0 (b) > 0 for p < 0, giving that there is a point t∗ ∈ [a, b] where
θ (t∗ ) = 0.
Thus the lemma is now completely proved.
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Corollary 4.2. Let f : [a, b] → R be of bounded variation on [a, b] then
Z b
(4.5)
ept f (t) dt − Ap (a, b) M (f )
a Wb
beap −aebp
a (f )

,
m
(ln
(m)
−
1)
+

b−a
|p|



L
≤
(b − a) m b−a
p
−
1
for f L − Lipschitzian on [a, b] ,
2
|p|




 p (b − a) m [f (b) − f (a)] , f monotonic nondecreasing,
|p|
where
(4.6)
m=
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
Ap (a, b)
ebp − eap
=
.
b−a
p (b − a)
Proof. From (2.12) taking g (t) = ept and using (1.1) and (1.2) gives
(4.7) (b − a) T f, ept
Z b
=
ept f (t) dt − Ap (a, b) M (f )
a

Wb
sup
|θ
(t)|

a (f ) , for f of bounded variation on [a, b] ,


 t∈[a,b]
Rb
≤
L a |θ (t)| dt,
for f L − Lipschitzian on [a, b] ,


 Rb

|θ (t)| df (t) ,
f monotonic nondecreasing on [a, b] ,
a
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pt
where the bounds are obtained from (2.12) on noting that for g (t) = e , θ (t) =
ψ(t)
is as given by (4.2) – (4.3).
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Now, using the properties of θ (t) as expounded in Lemma 4.1 will aid in
obtaining explicit bounds from (4.7).
Firstly, from (4.2), (4.3) and (4.6)
sup |θ (t)| = |θ (t∗ )|
t∈[a,b]
=
=
=
Ap (a, t∗ )
Ap (t∗ , b)
+b
t m− a
b−a
b−a
bp
m
a e −m
b m − eap
ln (m) −
−
p
p
b−a
p
b−a
ap
bp
m
be − ae
(ln (m) − 1) +
.
p
p (b − a)
∗
In the above we have used the fact that m ≥ 0 and that pt∗ = ln (m). Using from
Lemma 4.1 the result that θ (t) is positive or negative for t ∈ [a, b] depending
on whether p > 0 or p < 0 respectively, the first inequality in (4.5) results.
For the second inequality we have that from (4.2), (4.3) and Lemma 4.1,
#
Z b"
Z b
a ebp − etp + b (etp − eap )
1
pmt −
dt
|θ (t)| dt =
|p| a
b−a
a
2
Z b
1
b − a2
bp
ap
pt
=
pm
− ae − be −
e dt
|p|
2
a
2
1
b − a2
bp
ap
=
pm
− ae − be − (b − a) m
|p|
2
On an Identity for the
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Some Ramifications
P. Cerone
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1
a+b
bp
ap
=
(b − a) m
p − 1 − ae − be
|p|
2
1 ebp − eap a + b
bp
ap
=
p − 1 − ae − be
|p|
p
2
b−a 1
1 bp
ap
e −e
−
.
=
|p|
2
p
Using (4.6) gives the second result in (4.5) as stated.
Rb
For the final inequality in (4.5) we need to determine a |θ (t)| df (t) for f
monotonic nondecreasing. Now, from (4.2) and (4.3)
Z b
Z b
beap − aebp ept
|θ (t)| df (t) =
mt −
−
df (t)
p (b − a)
p
a
a
Z b
1
beap − aebp
pt
pmt +
− e df (t) ,
=
|p| a
b−a
where we have used the fact that sgn (θ (t)) = sgn (p).
Integration by parts of the Riemann-Stieltjes integral gives
(4.8)
Z b
|θ (t)| df (t)
a
(
b
Z b
1
beap − aebp
pt
pt
=
pmt +
−e
f (t) − p
m − e f (t) dt
|p|
b−a
a
a
Z b
p
=
ept − m f (t) dt.
|p| a
On an Identity for the
Chebychev Functional and
Some Ramifications
P. Cerone
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Now,
Z
b
Z
tp
e f (t) dt ≤ f (b)
a
a
and
Z
−m
b
etp dt =
ebp − eap
f (b) = (b − a) mf (b)
p
b
f (t) dt ≤ −m (b − a) f (a)
a
so that combining with (4.8) gives the inequalities for f monotonic nondecreasing.
Remark 4.1. If f is a probability density function then M (f ) =
non-negative.
1
b−a
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Some Ramifications
and f is
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References
[1] N.S. BARNETT AND S.S. DRAGOMIR, Some elementary inequalities for
the expectation and variance of a random variable whose pdf is defined
on a finite interval, RGMIA Res. Rep. Coll., 2(7), Article 12. [ONLINE]
http://rgmia.vu.edu.au/v2n7.html,(1999).
[2] P. CERONE AND S.S. DRAGOMIR, On some inequalities arising from
Montgomery’s identity, J. Comput. Anal. Applics., (accepted).
[3] P. CERONE AND S.S. DRAGOMIR, On some inequalities for the expectation and variance, Korean J. Comp. & Appl. Math., 8(2) (2000), 357–380.
[4] P. CERONE AND S.S. DRAGOMIR, Three point quadrature rules, involving, at most, a first derivative, RGMIA Res. Rep. Coll., 2(4), Article 8. [ONLINE] (1999). http://rgmia.vu.edu.au/v2n4.html
[5] S.S. DRAGOMIR, Some integral inequalities of Grüss type, Indian J. of
Pure and Appl. Math., (accepted).
[6] A.M. FINK, A treatise on Grüss’ inequality, T.M. Rassias (Ed.), Kluwer
Academic Publishers, (1999).
[7] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and New
Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
On an Identity for the
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Some Ramifications
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[8] N.Ja. SONIN, O nekotoryh neravenstvah otnosjašcihsjak opredelennym integralam, Zap. Imp. Akad. Nauk po Fiziko-matem, Otd.t., 6 (1898), 1–54.
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