Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 3, Issue 1, Article 4, 2002
ON AN IDENTITY FOR THE CHEBYCHEV FUNCTIONAL AND SOME
RAMIFICATIONS
P. CERONE
S CHOOL OF C OMMUNICATIONS AND I NFORMATICS
V ICTORIA U NIVERSITY OF T ECHNOLOGY
PO B OX 14428
M ELBOURNE C ITY MC
V ICTORIA 8001, AUSTRALIA .
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
A BSTRACT. An identity for the Chebychev functional is presented in which a Riemann-Stieltjes
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.
Key words and phrases: Chebychev functional, Bounds, Riemann-Stieltjes, Moments, Moment Generating Function.
2000 Mathematics Subject Classification. Primary 26D15, 26D20; Secondary 65Xxx.
1. I NTRODUCTION
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
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) ,
ISSN (electronic): 1443-5756
c 2002 Victoria University. All rights reserved.
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.
034-01
2
P. C ERONE
where the weighted integral mean is given by
Rb
(1.4)
M (f ; p) =
a
p (x) f (x) dx
.
Rb
p
(x)
dx
a
We note that,
T (f, g; 1) ≡ T (f, g)
and
M (f ; 1) ≡ M (f ) .
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.
2. A N I DENTITY FOR THE C HEBYCHEV F UNCTIONAL
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
1
(2.2)
|T (f, g)| ≤ (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 ) ,
which shifts a function by its integral mean, then the following identity holds. Namely,
(2.4)
T (f, g) = T (S (f ) , g) = T (f, S (g)) = T (S (f ) , S (g)) ,
and so
(2.5)
T (f, g) = M (S (f ) g) = M (f S (g)) = M (S (f ) S (g))
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.
J. Inequal. Pure and Appl. Math., 3(1) Art. 4, 2002
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A N I DENTITY FOR
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C HEBYCHEV F UNCTIONAL
3
The following lemma presents an identity for the Chebychev functional that involves a RiemannStieltjes 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
where
ψ (t) = (t − a) A (t, b) − (b − t) A (a, t)
(2.7)
with
Z
A (a, b) =
(2.8)
b
g (x) dx.
a
Proof. From (2.6) integrating the Riemann-Stieltjes integral by parts produces
(
)
b Z b
Z b
1
1
ψ (t) df (t) =
ψ (t) f (t) −
f (t) dψ (t)
(b − a)2 a
(b − a)2
a
a
Z b
1
0
=
ψ (b) f (b) − ψ (a) f (a) −
f (t) ψ (t) dt
(b − a)2
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))
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 bounded variation on
Rb
[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) ,
t∈[a,b]
a
a
Wb
where a (v) is the total variation of v on [a, b].
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].
J. Inequal. Pure and Appl. Math., 3(1) Art. 4, 2002
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4
P. C ERONE
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
b
Z
g (t) dv (t) ≤
(2.11)
a
|g (t)| dv (t) .
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
W



(f ) ,
sup
|ψ
(t)|



a
t∈[a,b]


2
Rb
(b − a) |T (f, g)| ≤

L a |ψ (t)| dt,
for f L − Lipschitzian,






 R b |ψ (t)| df (t) ,
for f monotonic nondecreasing,
a
(2.12)
where
b
W
(f ) is the total variation of f on [a, b].
a
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 are continuous
Rb
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
where T (f, g; p) is as given in (1.3),
Ψ (t) = P (t) Ḡ (t) − P̄ (t) G (t)
(2.14)
with
(2.15)



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,
(2.16)
Ψ (t) = P (t) G (b) − P (b) G (t) .
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A N I DENTITY FOR
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C HEBYCHEV F UNCTIONAL
5
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
Z b
1
−1
Ψ (t) df (t) =
f (t) dΨ (t)
P 2 (b) a
P 2 (b) a
Z b
1
=
[P (b) G0 (t) − P 0 (t) G (b)] f (t) dt
P 2 (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)
Theorem 2.7. Let the conditions of Lemma 2.6 on f , g and p continue to hold. Then

b
W


 sup |Ψ (t)| (f ) ,



a
t∈[a,b]


Rb
(2.17)
P 2 (b) |T (f, g; p)| ≤

L a |Ψ (t)| dt,
for f L − Lipschitzian,






 R b |Ψ (t)| df (t) ,
for f monotonic nondecreasing.
a
where
T (f, g; p) is Ras given by (1.3) and Ψ (t) = P (t) G (b) − P (b) G (t), with P (t) =
Rt
t
p
(x)
dx, G (t) = a p (x) g (x) dx.
a
Proof. The proof uses Lemmas 2.1 – 2.4 and follows closely that of Theorem 2.5.
Remark 2.8. 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].
Remark 2.9. 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
(b − a)2 a
and so

kψk1 kf 0 k∞ ,





2
kψkq kf 0 kp ,
(b − a) |T (f, g)| ≤





kψk∞ kf 0 k1 ,
J. Inequal. Pure and Appl. Math., 3(1) Art. 4, 2002
f 0 ∈ L∞ [a, b] ;
f 0 ∈ Lp [a, b] ,
p > 1, p1 + 1q = 1;
f 0 ∈ L1 [a, b] ;
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6
P. C ERONE
where the Lebesgue norms k·k are defined in the usual way as
Z b
p1
1 1
p
kgkp :=
|g (t)| dt ,
for g ∈ Lp [a, b] , p ≥ 1, + = 1
p q
a
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 2.10. Let g : [a, b] → R be absolutely continuous on [a, b] then for
D (g; a, t, b) := M (g; t, b) − M (g; a, t) ,
(2.19)
(2.20)
 b−a


kg 0 k∞ ,
g 0 ∈ L∞ [a, b] ;


2







q
q 1q

(t
−
a)
+
(b
−
t)



kg 0 kp , g 0 ∈ Lp [a, b] ,


q
+
1


p > 1, p1 + 1q = 1;
|D (g; a, t, b)| ≤

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
b
Z
r (t, u) g 0 (u) du
D (g; a, t, b) =
(2.23)
a
and so
Z
|D (g; a, t, b)| ≤ ess sup |r (t, u)|
u∈[a,b]
b
|g 0 (u)| du,
for g 0 ∈ L1 [a, b] ,
a
where from (2.21)
(2.24)
ess sup |r (t, u)| = 1
u∈[a,b]
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A N I DENTITY FOR
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C HEBYCHEV F UNCTIONAL
7
and so the third inequality in (2.20) results. Further, using the Hölder inequality gives
Z b
1q Z b
p1
p
q
|D (g; a, t, b)| ≤
|r (t, u)| du
(2.25)
|g 0 (t)| dt
a
a
1 1
for p > 1, + = 1,
p q
where explicitly from (2.21)
Z b
1q
Z t q
q 1q
Z b
u−a
b−u
q
du +
du
(2.26)
|r (t, u)| du
=
t−a
b−t
a
a
t
Z 1
1q
q
q 1q
q
= [(t − a) + (b − t) ]
u du
0
=
q 1q
q
(t − a) + (b − t)
q+1
.
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
b
_
|D (g; a, t, b)| ≤ ess sup |r (t, u)| (g)
u∈[a,b]
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 2.11. The results of Theorem 2.10 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) .
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 (·).
3. R ESULTS I NVOLVING M OMENTS
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,
a
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P. C ERONE
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)) .
The following corollary is valid.
Corollary 3.1. Let f : [a, b] → R be integrable on [a, b], then
(3.4)
B n+1 − An+1
Mn (γ) −
M (f )
n+1

b
W

1

sup
|φ
(t)|
·
(f ) , for f of bounded variation on [a, b] ,

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)
n
φ (t) = (t − γ) −
t−a
b−a
n+1
(b − γ)
+
b−t
b−a
n+1
(a − γ)
.
Proof. From (2.12) taking g (t) = (t − γ)n then using (1.1) and (1.2) gives
(b − a) |T (f, (t − γ)n )| = Mn (γ) −
B n+1 − An+1
M (f ) .
n+1
The right hand side is obtained on noting that for g (t) = (t − γ)n , φ (t) = − ψ(t)
.
b−a
Remark 3.2. 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 3.3. 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)
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A N I DENTITY FOR
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C HEBYCHEV F UNCTIONAL
9
where φ (c) = 0, a < c < b and

> γ, γ <





c = γ, γ =





< γ, γ >
a+b
2
a+b
2
a+b
.
2
Lemma 3.4. For φ (t) as given by (3.5) then
Z b
(3.7)
|φ (t)| dt
a


n odd and any γ
B−A
B n+2 −An+2
n+1
n+1


;
[B
−A ]−
,

2
n+2

n even and γ < a




 n+2 n+2 n+2
2
2 n+1
2C
−B
−A
1
+
(b
−
a)
−
2
(c
−
a)
B
=
n+2
2(b−a)

2
2
n+1


+ 2 (b − c) − (b − a)
A , n even and a < γ < b;






 B n+2 −An+2 − B−A [B n+1 − An+1 ] ,
n even and γ > b,
n+2
2
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
C (t) dt,
An+1
and φ (c) = 0 with a < c < b.
Lemma 3.5. 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 + m1 −m2
n even and a < γ < b,
2
2
where
(3.10)
(t∗ − γ)n =
B n+1 − An+1
,
(n + 1) (B − A)
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 3.6. The following result holds for φ (t) as defined by (3.5),


χn (a, b) ,
n odd or n even and γ < a,



Z b

1
−χn (a, b) ,
n even and γ > b,
(3.11)
|φ (t)| df =

n+1 a



 χ (c, b) − χ (a, c) , n even and a < γ < b
n
J. Inequal. Pure and Appl. Math., 3(1) Art. 4, 2002
n
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10
P. C ERONE
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 odd or n even


n+1



and γ < a;



 A (An − 1) − B (B n − 1)


f (b) ,
n even and γ > b;

n+1
≤



f (b)
(B n − An )

n+1
n+1


B
−C
−
(b − c)
n even and


b−a
n+
1



(B n − An )
f (a)


 +
(c − a) − (C n+1 − An+1 )
, a < γ < b,
b−a
n+1
where
Z b
(B n − An )
n
(3.13)
χn (a, b) =
(t − γ) −
f (t) dt,
(n + 1) (b − a)
a
A = a − γ, B = b − γ, C = c − γ.
Proof. Let α, β ∈ [a, b] and
Z β
1
χn (α, β) =
|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
and χn (a, b) is as given by (3.13) since φ (a) = φ (b) = 0.
Further, using the results of Lemma 3.3 as represented in (3.6), and, the fact that

Z β
φ (t) < 0, t ∈ [α, β]
 χ (α, β) ,
1
|φ (t)| df =
 −χ (α, β) , φ (t) > 0, t ∈ [α, β]
n+1 α
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
χn (a, b) ≤ f (b)
(t − γ) −
dt.
(n + 1) (b − a)
a
Further,
Z b
B n − An
n
χn (c, b) ≤ f (b)
(t − γ) −
dt
(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.
J. Inequal. Pure and Appl. Math., 3(1) Art. 4, 2002
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A N I DENTITY FOR
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11
The following corollary gives bounds for the expectation.
Corollary 3.7. Let f : [a, b] → R+ be a probability density function associated with a random
variable X. Then the expectation E (X) satisfies the inequalities

b

(b − a)3 W


(f ) ,
f of bounded variation,


6

a




 2
a+b
b−a
L
(3.14)
E (X) −
≤
· ,
f L − Lipschitzian,

2

2
2







b−a


[a + b − 1] f (b) , f monotonic nondecreasing.
2
Proof. Taking n = 1 in Corollary 3.1 and using Lemmas 3.3 – 3.6 gives the results after some
straightforward algebra. In particular,
2 2
a+b
b−a
2
φ (t) = t − (a + b) t + ab = t −
+
2
2
and t∗ the one solution of φ0 (t) = 0 is t∗ =
a+b
.
2
The following corollary gives bounds for the variance.
We shall assume that a < γ = E [X] < b.
Corollary 3.8. 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 2


3
2 3

C
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
,
3 (b − a)
1
1
2
2
m1 = φ E (X) − S , m2 = φ E (X) + S ,
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
S =
and γ = E (X).
Proof. Taking n = 2 in Corollary 3.1 gives from (3.5)
b−t
t−a
3
3
φ (t) = (t − γ) +
A −
B3
b−a
b−a
where a < γ = E (X) < b.
J. Inequal. Pure and Appl. Math., 3(1) Art. 4, 2002
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12
P. C ERONE
From Lemma 3.5 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 3.6 with n = 2 and hence the corollary
is proved.
4. A PPROXIMATIONS FOR THE M OMENT G ENERATING F UNCTION
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
Z b
pX (4.1)
MX (p) = E e
=
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)
(4.2)
(b − a) θ (t) = tAp (a, b) − [aAp (t, b) + bAp (a, t)] ,
where
Ap (a, b) =
(4.3)
ebp − eap
.
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
0,
p >
Ap (a,b)
1
∗
(iii) there is one turning point at t = p ln b−a and a ≤ t∗ ≤ b.
Proof. The result (i) is trivial from (4.2) using standard properties of the definite integral to give
θ (a) = θ (b) = 0.
Now,
Ap (a, b)
− ept , θ00 (t) = −pept
b−a
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
1
Ap (a, b)
∗
t = ln
.
p
b−a
(4.4)
θ0 (t) =
To show that a ≤ t∗ ≤ b it suffices to show that
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)
ap
bp
0
0
θ (a) θ (b) =
−e
−e
b−a
b−a
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A N I DENTITY FOR
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C HEBYCHEV F UNCTIONAL
13
but
Ap (a, b)
1
=
b−a
b−a
Z
b
ept dt,
a
0
the integral mean over [a, b] so that θ (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.
Corollary 4.2. Let f : [a, b] → R be of bounded variation on [a, b] then
Z
(4.5)
a
b
ept f (t) dt − Ap (a, b) M (f )
 Wb

beap − aebp

a (f )

m
(ln
(m)
−
1)
+
,


b−a
|p|



b−a
L
≤
(b − a) m
p−1
for f L − Lipschitzian on [a, b] ,


2
|p|



p


(b − a) m [f (b) − f (a)] ,
f monotonic nondecreasing,

|p|
where
m=
(4.6)
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

W
sup |θ (t)| ba (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
is as given
where the bounds are obtained from (2.12) on noting that for g (t) = ept , θ (t) = ψ(t)
b−a
by (4.2) – (4.3).
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 (t∗ , b)
Ap (a, t∗ )
t m− a
+b
b−a
b−a
bp
m
a e −m
b m − eap
ln (m) −
−
p
p
b−a
p
b−a
m
beap − aebp
(ln (m) − 1) +
.
p
p (b − a)
J. Inequal. Pure and Appl. Math., 3(1) Art. 4, 2002
∗
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14
P. C ERONE
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
|θ (t)| dt =
pmt −
dt
|p| a
b−a
a
2
Z b
1
b − a2
bp
ap
pt
=
pm
− ae − be −
e dt
|p|
2
a
2
b − a2
1
bp
ap
=
pm
− ae − be − (b − a) m
|p|
2
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. R
b
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
−
df (t)
|θ (t)| df (t) =
mt −
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
Z b
(4.8)
|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
Now,
Z
b
Z
tp
e f (t) dt ≤ f (b)
a
a
b
etp dt =
ebp − eap
f (b) = (b − a) mf (b)
p
and
Z
−m
b
f (t) dt ≤ −m (b − a) f (a)
a
so that combining with (4.8) gives the inequalities for f monotonic nondecreasing.
Remark 4.3. If f is a probability density function then M (f ) =
J. Inequal. Pure and Appl. Math., 3(1) Art. 4, 2002
1
b−a
and f is non-negative.
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A N I DENTITY FOR
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C HEBYCHEV F UNCTIONAL
15
R EFERENCES
[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.
[8] N.Ja. SONIN, O nekotoryh neravenstvah otnosjašcihsjak opredelennym integralam, Zap. Imp. Akad.
Nauk po Fiziko-matem, Otd.t., 6 (1898), 1–54.
J. Inequal. Pure and Appl. Math., 3(1) Art. 4, 2002
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