Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 3, Issue 5, Article 77, 2002
NEW UPPER AND LOWER BOUNDS FOR THE ČEBYŠEV FUNCTIONAL
P. CERONE AND S.S. DRAGOMIR
S CHOOL OF C OMMUNICATIONS AND I NFORMATICS
V ICTORIA U NIVERSITY OF T ECHNOLOGY
PO B OX 14428, MCMC 8001,
V ICTORIA , AUSTRALIA .
pc@matilda.vu.edu.au
URL: http://rgmia.vu.edu.au/cerone
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
A BSTRACT. 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] .
Key words and phrases: Čebyšev functional, Bounds, Refinement.
2000 Mathematics Subject Classification. Primary 26D15; Secondary 26D10.
1. I NTRODUCTION
For two given integrable functions on [a, b] , define the Čebyšev functional ([2, 3, 4])
Z b
Z b
Z b
1
1
1
(1.1)
T (f, g) :=
f (x) g (x) dx −
f (x) dx ·
g (x) dx.
b−a a
b−a a
b−a 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,
Z b
1
(1.2)
T (f, g) =
Ψ (t) df (t) ,
(b − a)2 a
where
(1.3)
Ψ (t) := (t − a) A (t, b) − (b − t) A (a, t) ,
ISSN (electronic): 1443-5756
c 2002 Victoria University. All rights reserved.
048-02
2
P. C ERONE AND S.S. D RAGOMIR
with
Z
A (c, d) :=
(1.4)
d
g (x) dx.
c
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:

W

sup |Ψ (t)| ba (f ) ,



t∈[a,b]



1
Rb
(1.5) |T (f, g)| ≤
2 × L
|Ψ (t)| dt,
for L − Lipschitzian;
a
(b − a)





 Rb
|Ψ (t)| df (t) ,
for f monotonic nondecreasing,
a
W
where ba (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,
Z d
A (c, d)
1
(1.6)
M (g; c, d) :=
=
f (t) dt.
d−c
d−c c
Theorem 1.3. Let g : [a, b] → R be absolutely continuous on [a, b] , then for
(1.7)
D (g; a, t, b) := M (g; t, b) − M (g; a, t) ,
(1.8)
 b−a 0
kg k∞ ,
g 0 ∈ L∞ [a, b] ;

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;
|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
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.
2. I NTEGRAL I NEQUALITIES
Now, if we use the function ϕ : (a, b) → R,
Rb
(2.1)
ϕ (t) := D (g; a, t, b) =
J. Inequal. Pure and Appl. Math., 3(5) Art. 77, 2002
t
g (x) dx
−
b−t
Rt
a
g (x) dx
,
t−a
http://jipam.vu.edu.au/
B OUNDS FOR THE Č EBYŠEV F UNCTIONAL
3
then by (1.2) we may obtain the identity:
(2.2)
1
T (f, g) =
(b − a)2
Z
b
(t − a) (b − t) ϕ (t) df (t) .
a
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) − a g (x) dx
a
(2.3)
ϕ (t) ≥ g (t) −
=
≥ 0,
t−a
t−a
by the monotonicity of g.
The following result providing a refinement of the classical Čebyšev inequality holds.
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
Z t
1
(2.4)
T (f, g) ≥
(t
−
a)
g
(x)
dx
−
(b
−
t)
g
(x)
dx
df
(t)
(b − a)2 a
t
a
≥ 0.
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) =
(t − a) (b − t) t
− a
df (t)
2
b
−
t
t−a
(b − a) a
R b
Z b
g (x) dx R t g (x) dx 1
=
(t − a) (b − t) t
− a
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
− (b − t) g (x) dx df (t)
=
(t
−
a)
g
(x)
dx
(b − a)2 a
t
a
≥0
and the inequality (1.5) is proved.
Remark 2.3. 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).
J. Inequal. Pure and Appl. Math., 3(5) Art. 77, 2002
http://jipam.vu.edu.au/
4
P. C ERONE AND S.S. D RAGOMIR
Theorem 2.4. 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:
b
_
1
|T (f, g)| ≤ kϕk∞ (f ) ,
4
a
(2.5)
where ϕ is as given by (2.1) and
kϕk∞ := sup |ϕ (t)| .
t∈(a,b)
Proof. Using the first inequality in Theorem 1.2, we have
b
_
1
|T (f, g)| ≤
sup
|Ψ
(t)|
(f )
(b − a)2 t∈[a,b]
a
b
_
1
=
sup
|(t
−
a)
(b
−
t)
ϕ
(t)|
(f )
(b − a)2 t∈[a,b]
a
b
≤
_
1
(f )
2 sup [(t − a) (b − t)] sup |ϕ (t)|
(b − a) 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 2.5. 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

3

L (b−a)
kϕk∞
if ϕ ∈ L∞ [a, b] ;

6




1
1
(2.6)
|T (f, g)| ≤
L (b − a) q [B (q + 1, q + 1)] q kϕkp , p > 1, p1 + 1q = 1



if ϕ ∈ Lp [a, b] ;


 L
kϕk
,
if ϕ ∈ L1 [a, b] ,
1
4
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
Obviously
Z
b
Z
(b − t) (t − a) |ϕ (t)| dt ≤ sup |ϕ (t)|
a
t∈[a,b]
b
(t − a) (b − t) dt
a
(b − a)3
=
kϕk∞ .
6
giving the first result in (2.6).
J. Inequal. Pure and Appl. Math., 3(5) Art. 77, 2002
http://jipam.vu.edu.au/
B OUNDS FOR THE Č EBYŠEV F UNCTIONAL
5
By Hölder’s integral inequality we have
Z b
p1 Z b
1q
Z b
p
q
(b − t) (t − a) |ϕ (t)| dt ≤
|ϕ (t)| dt
[(b − t) (t − a)] dt
a
a
a
2+ 1q
= kϕkp (b − a)
1
[B (q + 1, q + 1)] q .
Finally,
Z
b
Z
(b − t) (t − a) |ϕ (t)| dt ≤ sup [(b − t) (t − a)]
t∈[a,b]
a
(b − a)2
kϕk1
=
4
and the inequality (2.6) is thus completely proved.
b
|ϕ (t)| dt
a
We will use the following inequality for the Stieltjes integral in the subsequent work, namely
Z b
(2.7) h (t) k (t) df (t)
a

Rb
sup |h (t)| a |k (t)| df (t)



t∈[a,b]





 p1 R
1q
Rb
b
p
q
≤
|h
(t)|
df
(t)
|k
(t)|
df
(t)
, where p > 1, p1 + 1q = 1
a
a






Rb


 sup |k (t)| a |h (t)| df (t) ,
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,

n
 max |ai | P |bi | wi
n


i=1
X

 i=1,n
(2.8)
a
b
w
≤
i i i
n
p1 n
1q


P
P
i=1

p
q


wi |bi |
, p > 1, p1 + 1q = 1,
wi |ai |
i=1
i=1
where ai , bi ∈ R and wi ≥ 0, i ∈ {1, . . . , n} .
Using (2.7), we may state and prove the following theorem.
Theorem 2.6. Let f : [a, b] → R be a monotonic nondecreasing function on [a, b] . If g is
continuous, then one has the inequality:
 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)
(2.9)
|T (f, g)| ≤

p > 1, p1 + 1q = 1;




Rb


1

 (b−a)2 sup |ϕ (t)| a (t − a) (b − t) df (t) .
t∈[a,b]
J. Inequal. Pure and Appl. Math., 3(5) Art. 77, 2002
http://jipam.vu.edu.au/
6
P. C ERONE AND S.S. D RAGOMIR
Proof. From the third inequality in (1.5), we have
Z b
1
|T (f, g)| ≤
(2.10)
|Ψ (t)| df (t)
(b − a)2 a
Z b
1
=
(b − t) (t − a) |ϕ (t)| df (t) .
(b − a)2 a
Using (2.7), the inequality (2.9) is thus obtained.
3. M ORE ON Č EBYŠEV ’ S F UNCTIONAL
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
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
_
1_
|T (f, g)| ≤
(g) (f ) .
2 a
a
(3.2)
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
(
Z t
1
(3.4)
sup (b − t) (t − a) g (t) −
g (u) du
|T (f, g)| ≤
(b − a)2 t∈[a,b]
a
Z b
) _
b
+ sup (t − a)
g (u) du − g (t) (b − t)
(f )
t∈[a,b]
≤



































(
1
b−a
t
sup (t − a) g (t) −
t∈[a,b]
+ sup
hR
t∈[a,b]
(
1
4
a
h
h
sup g (t) −
t∈[a,b]
J. Inequal. Pure and Appl. Math., 3(5) Art. 77, 2002
+ sup
t∈[a,b]
a
g (u) du
i
b
g
(u)
du
−
g
(t)
(b
−
t)
t
1
t−a
h
Rt
i
Rt
1
b−t
g (u) du
a
Rb
t
)
×
Wb
a
(f ) ,
i
)
i W
b
g (u) du − g (t)
a (f ) .
http://jipam.vu.edu.au/
B OUNDS FOR THE Č EBYŠEV F UNCTIONAL
7
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
and by
1
I2 :=
(b − a)2
Taking the modulus, we have
Z
b
b
Z
(t − a)
a
(b − u) dg (u) df (t) .
t
Z t
b
_
1
(u − a) dg (u)
|I1 | ≤
sup
(b
−
t)
(f )
(b − a)2 t∈[a,b]
a
a
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
_
(g)
t∈[a,b] a
b
(b − a)2 _
=
(g)
4
a
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),
b
|T (f, g)| ≤ |I1 | + |I2 | ≤
b
_
1_
(g) (f )
2 a
a
and the inequality (3.2) is proved.
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
b
_
1
2
|I2 | ≤
L
sup
(t
−
a)
(b
−
t)
(f ) .
2 (b − a)2 t∈[a,b]
a
J. Inequal. Pure and Appl. Math., 3(5) Art. 77, 2002
http://jipam.vu.edu.au/
8
P. C ERONE AND S.S. D RAGOMIR
Since
sup (b − t) (t − a)2 =
t∈[a,b]
a + 2b
b−
3
a + 2b
−a
3
2
=
4
(b − a)3 ,
27
then
b
2 (b − a) _
|I1 | ≤
L (f )
27
a
and, similarly,
b
2 (b − a) _
|I2 | ≤
L (f ) .
27
a
Consequently
b
|T (f, g)| ≤ |I1 | + |I2 | ≤
4 (b − a) _
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
Z b
Z b
Z b
(b − u) dg (u) ≤
(b − u) dg (u) =
g (u) du − g (t) (b − t) .
t
t
t
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

sup
(t
−
a)
g
(t)
−
g
(u)
du

a (f ) ,
b−a
a


t∈[a,b]
≤
h
iW

Rt

b
1
1

sup
g
(t)
−
g
(u)
du
 4
a (f ) ,
t−a a
t∈[a,b]
and
Z b
_
b
1
sup (t − a)
g (u) du − g (t) (b − t)
(f )
|I2 | ≤
(b − a)2 t∈[a,b]
t
a
hR
iW

b
b
1

sup
g
(u)
du
−
g
(t)
(b
−
t)
 b−a
a (f ) ,
t


t∈[a,b]
≤
h R
iW


 1 sup 1 b g (u) du − g (t) b (f ) ,

a
4
b−t t
t∈[a,b]
and the inequality (3.4) is also proved.
The following result concerning a differentiable function g : [a, b] → R also holds.
J. Inequal. Pure and Appl. Math., 3(5) Art. 77, 2002
http://jipam.vu.edu.au/
B OUNDS FOR THE Č EBYŠEV F UNCTIONAL
9
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)
|T (f, g)|
b
_
1
≤
(f )
(b − a)2 a

h
i
0


sup
(b
−
t)
(t
−
a)
kg
k

[a,t],1


t∈[a,b]

h
i


0

+
sup
(b
−
t)
(t
−
a)
kg
k
if g 0 ∈ L1 [a, b] ;

[t,b],1


t∈[a,b]





(


h
i


1+ 1q
1
0


sup
(b
−
t)
(t
−
a)
kg
k
1

[a,t],p

q

t∈[a,b]
 (q+1)
)
h
i
1
×

+ sup (t − a) (b − t)1+ q kg 0 k[t,b],p
if g 0 ∈ Lp [a, b] ,



t∈[a,b]




p > 1, p1 + 1q = 1;

(


h
i


2

1
0

sup
(b
−
t)
(t
−
a)
kg
k

[a,t],∞
2


t∈[a,b]

)


h
i




+ sup (t − a) (b − t)2 kg 0 k[t,b],∞
if g 0 ∈ L∞ [a, b]


t∈[a,b]

1 0


kg k[a,b],1
if g 0 ∈ L1 [a, b] ;


2






b
 2q (q + 1) (b − a) 1q
_
kg 0 k[a,b],p if g 0 ∈ Lp [a, b] ,
≤
(f ) ×
1
+2
q

(2q + 1)

a



p > 1, p1 + 1q = 1;





 4 (b − a) kg 0 k[a,b],∞
if g 0 ∈ L∞ [a, b] ,
27
where the Lebesgue norms over an interval [c, d] are defined by
Z
khk[c,d],p :=
d
p1
|h (t)| dt , 1 ≤ p < ∞
p
c
and
khk[c,d],∞ := ess sup |h (t)| .
t∈[c,d]
Proof. Since g is differentiable on (a, b) , we have
Z t
(u − a) dg (u)
(3.6)
a
Z t
0
= (u − a) g (u) du
a
J. Inequal. Pure and Appl. Math., 3(5) Art. 77, 2002
http://jipam.vu.edu.au/
10
P. C ERONE AND S.S. D RAGOMIR

(t − a) kg 0 k[a,t],1




 1q
Rt
q
≤
(u
−
a)
du
kg 0 k[a,t],p , p > 1, p1 +
a




 R t (u − a) du kg 0 k
[a,t],∞
a

0
(t − a) kg k[a,t],1





1+ 1
(t−a) q
=
kg 0 k[a,t],p , p > 1, p1 + 1q = 1;
1
q

(q+1)



 (t−a)2 kg 0 k
[a,t],∞
2
1
q
= 1;
and, similarly,

(b − t) kg 0 k[t,b],1




Z b


 (b−t)1+ 1q 0
1 kg k[t,b],p , p > 1,
(b − u) dg (u) ≤
(q+1) q

t





 (b−t)2 0
kg k[t,b],∞ .
2
(3.7)
1
p
+
1
q
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
kg 0 k[a,t],p , p > 1,
1
|I1 | ≤
(f ) · sup
2
q
(q+1)
(b − a) a
t∈[a,b] 





 (b−t)(t−a)2 0
kg k[a,t],∞
2
= 1;
1
p
+
1
q
= 1;
1
p
+
1
q
= 1;
and from (3.7)

(t − a) (b − t) kg 0 k[t,b],1






b

1+ 1
_
1
(t−a)(b−t) q
kg 0 k[t,b],p , p > 1,
1
|I2 | ≤
(f ) · sup
2
q
(q+1)
(b − a) a
t∈[a,b] 





 (t−a)(b−t)2 0
kg k[t,b],∞ .
2
Further, since
|T (f, g)| ≤ |I1 | + |I2 | ,
we deduce the first inequality in (3.5).
Now, observe that
h
i
sup (b − t) (t − a) kg 0 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
=
J. Inequal. Pure and Appl. Math., 3(5) Art. 77, 2002
(b − a)
kg 0 k[a,b],1 ;
4
http://jipam.vu.edu.au/
B OUNDS FOR THE Č EBYŠEV F UNCTIONAL
"
sup
1
(b − t) (t − a)1+ q
(q + 1)
t∈[a,b]
#
kg 0 k[a,t],p
1
q
1
≤
11
h
1+ 1q
sup (b − t) (t − a)
1
q
i
(q + 1) t∈[a,b]
= Mq kg 0 k[a,b],p
sup kg 0 k[a,t],p
t∈[a,b]
where
Mq :=
1
h
1
(q + 1) q
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
(b − a)r+2 (r + 1)r+1
a + (r + 1) b
sup ρ (t) = ρ
=
.
r+2
(r + 2)r+2
t∈[a,b]
Consequently,
Mq =
1
q
1
·
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
In a similar fashion we have
h
i (b − a)2
sup (t − a) (b − t) kg 0 k[t,b],1 ≤
kg 0 k[a,b],1 ;
4
t∈[a,b]
"
sup
1
(t − a) (b − t)1+ q
(q + 1)
t∈[a,b]
1
q
#
0
kg k[t,b],p ≤
1
q (q + 1) (b − a)2+ q
(2q + 1)
2+ 1q
kg 0 k[a,b],p ,
and
"
sup
t∈[a,b]
#
(t − a) (b − t)2 0
2 (b − a)3 0
kg k[t,b],∞ ≤
kg k[a,b],∞
2
27
and the last part of (3.5) is thus completely proved.
Lemma 3.3. Let g : [a, b] → R be absolutely continuous on [a, b] then for
(3.8)
ϕ (t) = M (g; t, b) − M (g; a, t) ,
J. Inequal. Pure and Appl. Math., 3(5) Art. 77, 2002
http://jipam.vu.edu.au/
12
P. C ERONE AND S.S. D RAGOMIR
with M (g; c, d) defined by (1.6),
 b−a 0
kg k∞ ,

2






b−a
0


1 kg kα ,

β

(β+1)



(3.9)
kϕk∞ ≤
kg 0 k1 ,





Wb



 a (g) ,




 b−a L,
2
g 0 ∈ L∞ [a, b] ;
g 0 ∈ Lα [a, b] ,
α > 1, α1 + β1 = 1;
g 0 ∈ L1 [a, b] ;
g of bounded variation;
g is L − Lipschitzian,
and for p ≥ 1
(3.10)

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 , g 0 ∈ Lα [a, b] ,

β+1
a




α > 1, α1 + β1 = 1;
kϕkp ≤
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
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 3.4. 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 3.5. Let the conditions of Theorem 2.4 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 3.6. We note from the last two inequalities of (3.11) that the bounds produced are
4
of
sharper than those of Theorem 3.1, giving constants of 41 and 18 compared with 12 and 27
equations (3.2) and (3.3). For g differentiable then we notice that the first and third results of
J. Inequal. Pure and Appl. Math., 3(5) Art. 77, 2002
http://jipam.vu.edu.au/
B OUNDS FOR THE Č EBYŠEV F UNCTIONAL
13
(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].
R EFERENCES
[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.
J. Inequal. Pure and Appl. Math., 3(5) Art. 77, 2002
http://jipam.vu.edu.au/