Journal of Inequalities in Pure and
Applied Mathematics
NEW UPPER AND LOWER BOUNDS FOR THE
ČEBYŠEV FUNCTIONAL
volume 3, issue 5, article 77,
2002.
P. CERONE AND S.S. DRAGOMIR
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
EMail: sever@matilda.vu.edu.au
URL: http://rgmia.vu.edu.au/SSDragomirWeb.html
Received 8 May, 2002;
accepted 5 November, 2002.
Communicated by: F. Qi
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
048-02
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Abstract
New bounds are developed for the Čebyšev functional utilising an identity involving a Riemann-Stieltjes integral. A refinement of the classical Čebyšev inequality is produced for f monotonic non-decreasing, g continuous and M (g; t, b)−
M (g; a, t) ≥ 0, for t ∈ [a, b] where M (g; c, d) is the integral mean over [c, d] .
2000 Mathematics Subject Classification: Primary 26D15; Secondary 26D10.
Key words: Čebyšev functional, Bounds, Refinement.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Integral Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
More on Čebyšev’s Functional . . . . . . . . . . . . . . . . . . . . . . . . . 13
References
New Upper and Lower Bounds
for the Čebyšev Functional
P. Cerone and S.S. Dragomir
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1.
Introduction
For two given integrable functions on [a, b] , define the Čebyšev functional ([2,
3, 4])
1
(1.1) T (f, g) :=
b−a
Z
b
f (x) g (x) dx
a
1
−
b−a
Z
a
b
1
f (x) dx ·
b−a
Z
b
g (x) dx.
a
In [1], P. Cerone has obtained the following identity that involves a Stieltjes
integral (Lemma 2.1, p. 3):
Lemma 1.1. Let f, g : [a, b] → R, where f is of bounded variation and g is
continuous on [a, b] , then the T (f, g) from (1.1) satisfies the identity,
(1.2)
T (f, g) =
1
(b − a)2
Z
b
Ψ (t) df (t) ,
a
where
(1.3)
New Upper and Lower Bounds
for the Čebyšev Functional
P. Cerone and S.S. Dragomir
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Ψ (t) := (t − a) A (t, b) − (b − t) A (a, t) ,
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with
Page 3 of 26
Z
(1.4)
A (c, d) :=
d
g (x) dx.
c
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Using this representation and the properties of Stieltjes integrals he obtained
the following result in bounding the functional T (·, ·) (Theorem 2.5, p. 4):
Theorem 1.2. With the assumptions in Lemma 1.1, we have:
(1.5)
|T (f, g)|

W

sup |Ψ (t)| ba (f ) ,


t∈[a,b]




1
Rb
≤
2 × L
|Ψ (t)| dt,
for L − Lipschitzian;
a
(b − a)





 Rb
|Ψ (t)| df (t) ,
for f monotonic nondecreasing,
a
where
Wb
a
(f ) denotes the total variation of f on [a, b] .
Cerone [1] also proved the following theorem, which will be useful for the
development of subsequent results, and is thus stated here for clarity. The notation M (g; c, d) is used to signify the integral mean of g over [c, d] . Namely,
(1.6)
A (c, d)
1
M (g; c, d) :=
=
d−c
d−c
Z
New Upper and Lower Bounds
for the Čebyšev Functional
P. Cerone and S.S. Dragomir
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d
f (t) dt.
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Theorem 1.3. Let g : [a, b] → R be absolutely continuous on [a, b] , then for
(1.7)
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D (g; a, t, b) := M (g; t, b) − M (g; a, t) ,
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 b−a 0
g 0 ∈ L∞ [a, b] ;
kg k∞ ,

2






h
i1


(t−a)q +(b−t)q q

kg 0 kp , g 0 ∈ Lp [a, b] ,


q+1



p > 1, p1 + 1q = 1;
(1.8) |D (g; a, t, b)| ≤
kg 0 k1 ,
g 0 ∈ L1 [a, b] ;




 W


b

g of bounded variation;

a (g) ,





 b−a L,
g is L − Lipschitzian.
2
Although the possibility of utilising Theorem 1.3 to obtain bounds on ψ (t) ,
as given by (1.3), was mentioned in [1], it was not capitalised upon. This aspect
will be investigated here since even though this will provide coarser bounds,
they may be more useful in practice.
A lower bound for the Čebyšev functional improving the classical result due
to Čebyšev is also developed and thus providing a refinement.
New Upper and Lower Bounds
for the Čebyšev Functional
P. Cerone and S.S. Dragomir
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2.
Integral Inequalities
Now, if we use the function ϕ : (a, b) → R,
Rb
(2.1)
ϕ (t) := D (g; a, t, b) =
t
g (x) dx
−
b−t
Rt
a
g (x) dx
,
t−a
then by (1.2) we may obtain the identity:
Z b
1
(t − a) (b − t) ϕ (t) df (t) .
(2.2)
T (f, g) =
(b − a)2 a
New Upper and Lower Bounds
for the Čebyšev Functional
P. Cerone and S.S. Dragomir
We may prove the following lemma.
Lemma 2.1. If g : [a, b] → R is monotonic nondecreasing on [a, b] , then ϕ as
defined by (2.1) is nonnegative on (a, b) .
Rb
Proof. Since g is nondecreasing, we have t g (x) dx ≥ (b − t) g (t) and thus
from (2.1)
Rt
Rt
g
(x)
dx
(t
−
a)
g
(t)
−
g (x) dx
a
(2.3)
ϕ (t) ≥ g (t) − a
=
≥ 0,
t−a
t−a
by the monotonicity of g.
The following result providing a refinement of the classical Čebyšev inequality holds.
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Theorem 2.2. Let f : [a, b] → R be a monotonic nondecreasing function on
[a, b] and g : [a, b] → R a continuous function on [a, b] so that ϕ (t) ≥ 0 for
each t ∈ (a, b) . Then one has the inequality:
Z b Z b
1
(2.4) T (f, g) ≥
(t
−
a)
g
(x)
dx
(b − a)2 a
t
Z t
− (b − t) g (x) dx df (t) ≥ 0.
a
Proof. Since ϕ (t) ≥ 0 and f is monotonic nondecreasing, one has successively
#
"R b
Rt
Z b
g
(x)
dx
g
(x)
dx
1
T (f, g) =
− a
df (t)
(t − a) (b − t) t
b−t
t−a
(b − a)2 a
R b
Z b
g (x) dx R t g (x) dx 1
t
=
− a
(t − a) (b − t) df (t)
2
b−t
t−a (b − a) a
R b
g (x) dx R t g (x) dx Z b
t
a
1
df (t)
≥
(t
−
a)
(b
−
t)
−
b
−
t
t
−
a
(b − a)2 a

R
 R b
t
Z b
g
(x)
dx
g
(x)
dx
t
a
1
 df (t)
≥
(t − a) (b − t) 
−
2 b−t
t−a
(b − a) a
Z b Z b
Z t
1
=
(t − a) g (x) dx − (b − t) g (x) dx df (t)
2 (b − a)
a
t
a
≥0
New Upper and Lower Bounds
for the Čebyšev Functional
P. Cerone and S.S. Dragomir
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and the inequality (1.5) is proved.
Remark 2.1. By Lemma 2.1, we may observe that for any two monotonic nondecreasing functions f, g : [a, b] → R, one has the refinement of Čebyšev inequality provided by (2.4).
We are able now to prove the following inequality in terms of f and the
function ϕ defined above in (2.1).
Theorem 2.3. Let f : [a, b] → R be a function of bounded variation and g :
[a, b] → R an absolutely continuous function so that ϕ is bounded on (a, b) .
Then one has the inequality:
New Upper and Lower Bounds
for the Čebyšev Functional
P. Cerone and S.S. Dragomir
b
|T (f, g)| ≤
(2.5)
_
1
kϕk∞ (f ) ,
4
a
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where ϕ is as given by (2.1) and
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kϕk∞ := sup |ϕ (t)| .
t∈(a,b)
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Proof. Using the first inequality in Theorem 1.2, we have
Close
|T (f, g)| ≤
=
1
sup |Ψ (t)|
(b − a)2 t∈[a,b]
b
_
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(f )
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a
1
sup |(t − a) (b − t) ϕ (t)|
(b − a)2 t∈[a,b]
b
_
a
(f )
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b
_
1
≤
sup
[(t
−
a)
(b
−
t)]
sup
|ϕ
(t)|
(f )
(b − a)2 t∈[a,b]
t∈(a,b)
a
b
_
1
≤ kϕk∞ (f ) ,
4
a
since, obviously, supt∈[a,b] [(t − a) (b − t)] =
(b−a)2
.
4
The case of Lipschitzian functions f : [a, b] → R is embodied in the following theorem as well.
Theorem 2.4. Let f : [a, b] → R be an L−Lipschitzian function on [a, b] and
g : [a, b] → R an absolutely continuous function on [a, b] . Then
(2.6)
|T (f, g)|

(b−a)3

L
kϕk∞

6




1
1
q [B (q + 1, q + 1)] q kϕk ,
≤
L
(b
−
a)
p





 L
kϕk1 ,
4
New Upper and Lower Bounds
for the Čebyšev Functional
P. Cerone and S.S. Dragomir
Title Page
if ϕ ∈ L∞ [a, b] ;
1
p
1
q
p > 1, + = 1
if ϕ ∈ Lp [a, b] ;
if ϕ ∈ L1 [a, b] ,
where k·kp are the usual Lebesgue p−norms on [a, b] and B (·, ·) is Euler’s Beta
function.
Proof. Using the second inequality in Theorem 1.2, we have
Z b
Z b
L
L
|T (f, g)| ≤
|Ψ (t)| dt =
(b − t) (t − a) |ϕ (t)| dt.
(b − a)2 a
(b − a)2 a
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Obviously
Z b
Z
(b − t) (t − a) |ϕ (t)| dt ≤ sup |ϕ (t)|
t∈[a,b]
a
b
(t − a) (b − t) dt
a
(b − a)3
=
kϕk∞ .
6
giving the first result in (2.6).
By Hölder’s integral inequality we have
Z
b
b
Z
1q
p1 Z b
q
[(b − t) (t − a)] dt
|ϕ (t)| dt
p
(b − t) (t − a) |ϕ (t)| dt ≤
a
New Upper and Lower Bounds
for the Čebyšev Functional
a
a
2+ 1q
= kϕkp (b − a)
Finally,
Z
1
[B (q + 1, q + 1)] q .
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b
Z
(b − t) (t − a) |ϕ (t)| dt ≤ sup [(b − t) (t − a)]
a
P. Cerone and S.S. Dragomir
t∈[a,b]
b
|ϕ (t)| dt
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a
(b − a)2
=
kϕk1
4
and the inequality (2.6) is thus completely proved.
We will use the following inequality for the Stieltjes integral in the subse-
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quent work, namely
(2.7)
Z b
h (t) k (t) df (t)
a
≤

Rb
sup |h (t)| a |k (t)| df (t)



t∈[a,b]



1
R
1 R



 ab |h (t)|p df (t) p ab |k (t)|q df (t) q ,

where p > 1,





Rb



 sup |k (t)| a |h (t)| df (t) ,
1
p
+
1
q
=1
New Upper and Lower Bounds
for the Čebyšev Functional
P. Cerone and S.S. Dragomir
t∈[a,b]
provided f is monotonic nondecreasing and h, k are continuous on [a, b] .
We note that a simple proof of these inequalities may be achieved by using
the definition of the Stieltjes integral for monotonic functions. The following
weighted inequalities for real numbers also hold,
(2.8)
n
X
ai bi wi Contents
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i=1
≤
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
n
P


|a
|
|bi | wi
max
i


i=1
 i=1,n
1q
n
p1 n


P
P

q
p


wi |ai |
wi |bi |
, p > 1,
i=1
i=1
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1
p
+
1
q
= 1,
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where ai , bi ∈ R and wi ≥ 0, i ∈ {1, . . . , n} .
Using (2.7), we may state and prove the following theorem.
Theorem 2.5. Let f : [a, b] → R be a monotonic nondecreasing function on
[a, b] . If g is continuous, then one has the inequality:
(2.9)
|T (f, g)|
 Rb
1

|ϕ (t)| df (t)

4 a




1 1


 1 2 R b [(b − t) (t − a)]q df (t) q R b |ϕ (t)|p df (t) p ,
a
a
(b−a)
≤

p > 1, p1 + 1q = 1;




Rb


1

 (b−a)2 sup |ϕ (t)| a (t − a) (b − t) df (t) .
New Upper and Lower Bounds
for the Čebyšev Functional
P. Cerone and S.S. Dragomir
Title Page
t∈[a,b]
Contents
Proof. From the third inequality in (1.5), we have
Z b
1
(2.10)
|T (f, g)| ≤
|Ψ (t)| df (t)
(b − a)2 a
Z b
1
=
(b − t) (t − a) |ϕ (t)| df (t) .
(b − a)2 a
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Using (2.7), the inequality (2.9) is thus obtained.
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3.
More on Čebyšev’s Functional
Using the representation (1.2) and the integration by parts formula for the Stieltjes integral, we have (see also [4, p. 268], for a weighted version) the identity,
Z b
Z t
1
(3.1) T (f, g) =
(b − t)
(u − a) dg (u) df (t)
(b − a)2 a
a
Z b
Z b
+
(t − a)
(b − u) dg (u) df (t) .
a
New Upper and Lower Bounds
for the Čebyšev Functional
t
The following result holds.
Theorem 3.1. Assume that f : [a, b] → R is of bounded variation and g :
[a, b] → R is continuous and of bounded variation on [a, b] . Then one has the
inequality:
b
b
If g : [a, b] → R is Lipschitzian with the constant L > 0, then
b
_
4
|T (f, g)| ≤
(b − a) L (f ) .
27
a
(3.3)
If g : [a, b] → R is continuous and monotonic nondecreasing, then
(3.4)
|T (f, g)|
≤
1
(b − a)2
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_
1_
|T (f, g)| ≤
(g) (f ) .
2 a
a
(3.2)
P. Cerone and S.S. Dragomir
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(
Z t
sup (b − t) (t − a) g (t) −
g (u) du
t∈[a,b]
a
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Z b
) _
b
g (u) du − g (t) (b − t)
+ sup (t − a)
(f )
t∈[a,b]
≤
t
a
(

Z t

1


sup (t − a) g (t) −
g (u) du


b − a t∈[a,b]

a



Z b
) _
b




+
sup
g
(u)
du
−
g
(t)
(b
−
t)
×
(f ) ,



t∈[a,b]
t

a
(

Z t


1
1


sup g (t) −
g (u) du



4 t∈[a,b]
t−a a



) _
Z b
b


1



+ sup
g (u) du − g (t)
(f ) .

b−t
t∈[a,b]
t
a
Proof. Denote the two terms in (3.1) by
Z t
Z b
1
I1 :=
(b − t)
(u − a) dg (u) df (t)
(b − a)2 a
a
New Upper and Lower Bounds
for the Čebyšev Functional
P. Cerone and S.S. Dragomir
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and by
Z b
1
I2 :=
(t − a)
(b − a)2 a
t
Taking the modulus, we have
Z t
_
b
1
|I1 | ≤
sup (b − t) (u − a) dg (u)
(f )
(b − a)2 t∈[a,b]
a
a
Z
b
(b − u) dg (u) df (t) .
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and
Z b
_
b
1
|I2 | ≤
sup (t − a) (b − u) dg (u)
(f ) .
(b − a)2 t∈[a,b]
t
a
However,
"
#
Z t
t
_
sup (b − t) (u − a) dg (u) ≤ sup (b − t) (t − a) (g)
t∈[a,b]
t∈[a,b]
a
a
≤ sup [(b − t) (t − a)] sup
t∈[a,b]
t∈[a,b] a
2
=
t
_
(b − a)
4
b
_
(g)
New Upper and Lower Bounds
for the Čebyšev Functional
P. Cerone and S.S. Dragomir
(g)
Title Page
a
Contents
and, similarly,
Z b
b
(b − a)2 _
sup (t − a) (b − u) dg (u) ≤
(g) .
4
t∈[a,b]
t
a
Thus, from (3.1),
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b
b
_
1_
|T (f, g)| ≤ |I1 | + |I2 | ≤
(g) (f )
2 a
a
and the inequality (3.2) is proved.
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If g is L−Lipschitzian, then we have
Z t
Z t
L (t − a)2
(u − a) dg (u) ≤ L
(u
−
a)
du
=
2
a
a
and
Z b
Z b
L (b − t)2
(b − u) dg (u) ≤ L
(b − u) du =
2
t
t
and thus
b
_
1
2
|I1 | ≤
L
sup
(b
−
t)
(t
−
a)
(f ) ,
2 (b − a)2 t∈[a,b]
a
and
P. Cerone and S.S. Dragomir
b
_
1
2
|I2 | ≤
L
sup
(t
−
a)
(b
−
t)
(f ) .
2 (b − a)2 t∈[a,b]
a
Since
sup (b − t) (t − a)2 =
t∈[a,b]
New Upper and Lower Bounds
for the Čebyšev Functional
a + 2b
b−
3
a + 2b
−a
3
2
4
=
(b − a)3 ,
27
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then
b
2 (b − a) _
|I1 | ≤
L (f )
27
a
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and, similarly,
b
2 (b − a) _
L (f ) .
|I2 | ≤
27
a
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Consequently
b
4 (b − a) _
|T (f, g)| ≤ |I1 | + |I2 | ≤
L (f )
27
a
and the inequality (3.3) is also proved.
If g is monotonic nondecreasing, then
Z t
Z t
Z t
(u − a) dg (u) ≤
(u − a) dg (u) = (t − a) g (t) −
g (u) du
a
a
a
and
New Upper and Lower Bounds
for the Čebyšev Functional
P. Cerone and S.S. Dragomir
Z b
Z b
Z b
(b − u) dg (u) ≤
(b − u) dg (u) =
g (u) du − g (t) (b − t) .
t
t
t
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Consequently,
_
Z t
b
1
|I1 | ≤
sup
(b
−
t)
(t
−
a)
g
(t)
−
g
(u)
du
(f )
(b − a)2 t∈[a,b]
a
a
h
iW

Rt
b
1
 b−a supt∈[a,b] (t − a) g (t) − a g (u) du a (f ) ,
h
iW
≤
Rt
b
 1 sup
1
g
(t)
−
g
(u)
du
t∈[a,b]
a (f ) ,
4
t−a a
and
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|I2 | ≤
1
(b − a)2
Z b
_
b
sup (t − a)
g (u) du − g (t) (b − t)
(f )
t∈[a,b]
t
a
J. Ineq. Pure and Appl. Math. 3(5) Art. 77, 2002
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hR
iW
b
supt∈[a,b] t g (u) du − g (t) (b − t) ba (f ) ,
h R
iW
≤
b
b
 1 sup
1
t∈[a,b] b−t t g (u) du − g (t)
a (f ) ,
4


1
b−a
and the inequality (3.4) is also proved.
The following result concerning a differentiable function g : [a, b] → R also
holds.
Theorem 3.2. Assume that f : [a, b] → R is of bounded variation and g :
[a, b] → R is differentiable on (a, b) . Then,
(3.5)
b
_
1
|T (f, g)| ≤
(f )
(b − a)2 a
h
i

0

sup
(b
−
t)
(t
−
a)
kg
k

t∈[a,b]
[a,t],1

h
i



0

+
sup
(b
−
t)
(t
−
a)
kg
k
if g 0 ∈ L1 [a, b] ;

t∈[a,b]
[t,b],1



n
h
i


1+ 1q

1
0

sup
(b
−
t)
(t
−
a)
kg
k
1

t∈[a,b]
[a,t],p

 (q+1) q
h
io
1
×
+ supt∈[a,b] (t − a) (b − t)1+ q kg 0 k[t,b],p





if g 0 ∈ Lp [a, b] , p > 1, p1 + 1q = 1;




i
n
h


2
1
0


sup
(b
−
t)
(t
−
a)
kg
k
t∈[a,b]

[a,t],∞
2

h
io



+ supt∈[a,b] (t − a) (b − t)2 kg 0 k[t,b],∞
if g 0 ∈ L∞ [a, b]
New Upper and Lower Bounds
for the Čebyšev Functional
P. Cerone and S.S. Dragomir
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≤
b
_
a
(f ) ×











1
2
kg 0 k[a,b],1
if g 0 ∈ L1 [a, b] ;
1
2q(q+1)(b−a) q
1 +2
(2q+1) q
4(b−a)
27
kg 0 k[a,b],p if g 0 ∈ Lp [a, b] , p > 1,
kg 0 k[a,b],∞
1
p
+
1
q
= 1;
if g 0 ∈ L∞ [a, b] ,
where the Lebesgue norms over an interval [c, d] are defined by
Z
khk[c,d],p :=
d
p1
|h (t)| dt , 1 ≤ p < ∞
New Upper and Lower Bounds
for the Čebyšev Functional
p
c
P. Cerone and S.S. Dragomir
and
khk[c,d],∞ := ess sup |h (t)| .
Title Page
t∈[c,d]
Proof. Since g is differentiable on (a, b) , we have
Z t
(3.6) (u − a) dg (u)
a
Z t
0
= (u − a) g (u) du
a

(t − a) kg 0 k[a,t],1



 1q
Rt
q
≤
(u
−
a)
du
kg 0 k[a,t],p , p > 1,
a



 Rt
(u − a) du kg 0 k[a,t],∞
a
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1
p
+
1
q
= 1;
Page 19 of 26
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
(t − a) kg 0 k[a,t],1





1+ 1
(t−a) q
=
kg 0 k[a,t],p , p > 1,
1
q

(q+1)



 (t−a)2 kg 0 k
[a,t],∞
2
1
p
+
1
q
= 1;
and, similarly,
(3.7)

(b − t) kg 0 k[t,b],1



Z b

 (b−t)1+ 1q
0
(b − u) dg (u) ≤
1 kg k[t,b],p , p > 1,
q

(q+1)
t



 (b−t)2 0
kg k[t,b],∞ .
2
1
p
+
With the notation in Theorem 3.1, we have on using (3.6)

(b − t) (t − a) kg 0 k[a,t],1




b

_
1+ 1
1
(b−t)(t−a) q
|I1 | ≤
(f
)
·
sup
kg 0 k[a,t],p , p > 1,
1
(q+1) q
(b − a)2 a
t∈[a,b] 



 (b−t)(t−a)2 0
kg k[a,t],∞
2
1
q
= 1;
P. Cerone and S.S. Dragomir
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1
p
+
1
q
= 1;
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and from (3.7)

(t − a) (b − t) kg 0 k[t,b],1




b

_
1+ 1
1
(t−a)(b−t) q
|I2 | ≤
(f
)
·
sup
kg 0 k[t,b],p , p > 1,
1
q

(q+1)
(b − a)2 a
t∈[a,b] 

 (t−a)(b−t)2 0

kg k[t,b],∞ .
2
New Upper and Lower Bounds
for the Čebyšev Functional
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1
p
+
1
q
= 1;
J. Ineq. Pure and Appl. Math. 3(5) Art. 77, 2002
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Further, since
|T (f, g)| ≤ |I1 | + |I2 | ,
we deduce the first inequality in (3.5).
Now, observe that
h
i
0
sup (b − t) (t − a) kg k[a,t],1 ≤ sup [(b − t) (t − a)] sup kg 0 k[a,t],1
t∈[a,b]
t∈[a,b]
t∈[a,b]
2
=
"
sup
t∈[a,b]
1+ 1q
(b − t) (t − a)
(q + 1)
≤
1
q
(b − a)
kg 0 k[a,b],1 ;
4
P. Cerone and S.S. Dragomir
#
kg 0 k[a,t],p
Title Page
1
h
1
q
1+ 1q
sup (b − t) (t − a)
i
(q + 1)
= Mq kg 0 k[a,b],p
where
Mq :=
New Upper and Lower Bounds
for the Čebyšev Functional
t∈[a,b]
1
(q + 1)
h
1
q
sup kg 0 k[a,t],p
t∈[a,b]
1+ 1q
sup (b − t) (t − a)
i
.
t∈[a,b]
Consider the arbitrary function ρ (t) = (b − t) (t − a)r+1 , r > 0. Then ρ0 (t) =
(t − a)r [(r + 1) b + a − (r + 2) t] showing that
a + (r + 1) b
(b − a)r+2 (r + 1)r+1
sup ρ (t) = ρ
=
.
r+2
(r + 2)r+2
t∈[a,b]
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Consequently,
Mq =
1
1
q
1
·
(b − a)2+ q (q + 1)1+ q
1
(2q + 1)2+ q
(q + 1) q
1
=
q (q + 1) (b − a)2+ q
1
(2q + 1)2+ q
.
Also,
"
sup
t∈[a,b]
#
(b − t) (t − a)2 0
1
kg k[a,t],∞ ≤ sup (b − t) (t − a)2 sup kg 0 k[a,t],∞
2
2 t∈[a,b]
t∈[a,b]
=
2 (b − a)3 0
kg k[a,b],∞ .
27
P. Cerone and S.S. Dragomir
In a similar fashion we have
Title Page
2
h
i (b − a)
kg 0 k[a,b],1 ;
sup (t − a) (b − t) kg 0 k[t,b],1 ≤
4
t∈[a,b]
"
sup
1+ 1q
(t − a) (b − t)
(q + 1)
t∈[a,b]
and
"
sup
t∈[a,b]
1
q
New Upper and Lower Bounds
for the Čebyšev Functional
#
kg 0 k[t,b],p ≤
2
2+ 1q
q (q + 1) (b − a)
(2q + 1)
#
2+ 1q
kg 0 k[a,b],p ,
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2 (b − a)
(t − a) (b − t)
kg 0 k[t,b],∞ ≤
kg 0 k[a,b],∞
2
27
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and the last part of (3.5) is thus completely proved.
J. Ineq. Pure and Appl. Math. 3(5) Art. 77, 2002
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Lemma 3.3. Let g : [a, b] → R be absolutely continuous on [a, b] then for
ϕ (t) = M (g; t, b) − M (g; a, t) ,
(3.8)
with M (g; c, d) defined by (1.6),
 b−a 0
kg k∞ ,

2




b−a

0

1 kg kα ,


β
 (β+1)
kg 0 k1 ,
(3.9)
kϕk∞ ≤


Wb




a (g) ,



 b−a L,
2
g 0 ∈ L∞ [a, b] ;
g 0 ∈ Lα [a, b] , α > 1,
1
α
+
1
β
= 1;
g 0 ∈ L1 [a, b] ;
New Upper and Lower Bounds
for the Čebyšev Functional
g of bounded variation;
P. Cerone and S.S. Dragomir
g is L − Lipschitzian,
Title Page
and for p ≥ 1
(3.10)
kϕkp ≤
Contents

1 0
b−a 1+ p

kg k∞ , g 0 ∈ L∞ [a, b] ;


2




h
p1


R b (t−a)β +(b−t)β i βp


dt kg 0 kα ,

β+1
a





g 0 ∈ Lα [a, b] , α > 1, α1 +
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1
β
= 1;
1


p kg 0 k ,

(b
−
a)
g 0 ∈ L1 [a, b] ;

1




1 W


(b − a) p ba (g) , g of bounded variation;






 b−a 1+ p1
L,
g is L − Lipschitzian.
2
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Proof. Identifying ϕ (t) with D (g; a, t, b) of (1.7) produces bounds for |ϕ (t)|
from (1.8). Taking the supremum over t ∈ [a, b] readily gives (3.9), a bound for
kϕk∞ .
The bound for kϕkp is obtained from (1.8) using the definition of the Lebesque
p−norms over [a, b] .
Remark 3.1. Utilising (3.9) of Lemma 3.3 in (2.5) produces a coarser upper
bound for |T (f, g)| . Making use of the whole of Lemma 3.3 in (2.6) produces
coarser bounds for (2.6) which may prove more amenable in practical situations.
Corollary 3.4. Let the conditions of Theorem 2.3 hold, then
 b−a 0
kg k∞ , g 0 ∈ L∞ [a, b] ;

2






b−a
0

g 0 ∈ Lα [a, b] ,
1 kg kα ,


β
(β+1)



b

α > 1, α1 + β1 = 1;
_
1
(3.11) |T (f, g)| ≤
(f ) kg 0 k ,
g 0 ∈ L1 [a, b] ;
1

4 a




Wb



g of bounded variation;

a (g) ,




 b−a L,
g is L − Lipschitzian.
2
Proof. Using (3.9) in (2.5) produces (3.11).
Remark 3.2. We note from the last two inequalities of (3.11) that the bounds
produced are sharper than those of Theorem 3.1, giving constants of 14 and 18
New Upper and Lower Bounds
for the Čebyšev Functional
P. Cerone and S.S. Dragomir
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4
compared with 12 and 27
of equations (3.2) and (3.3). For g differentiable then
we notice that the first and third results of (3.11) are sharper than the first and
third results in the second cluster of (3.5). The first cluster in (3.5) are sharper
where the analysis is done over the two subintervals [a, x] and (x, b].
New Upper and Lower Bounds
for the Čebyšev Functional
P. Cerone and S.S. Dragomir
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References
[1] P. CERONE, On an identity for the Chebychev functional and some ramifications, J. Ineq. Pure. & Appl. Math., 3(1) (2002), Article 4. [ONLINE]
http://jipam.vu.edu.au/v3n1/034_01.html
[2] S.S. DRAGOMIR, Some integral inequalities of Grüss type, Indian J. of
Pure and Appl. Math., 31(4) (2000), 397–415.
[3] A.M. FINK, A treatise on Grüss’ inequality, Th.M. Rassias and H.M. Srivastava (Ed.), Kluwer Academic Publishers, (1999), 93–114.
[4] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and New
Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
New Upper and Lower Bounds
for the Čebyšev Functional
P. Cerone and S.S. Dragomir
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