Volume 8 (2007), Issue 2, Article 55, 4 pp.
ON GENERALIZATION OF ČEBYŠEV TYPE INEQUALITIES
K. BOUKERRIOUA AND A. GUEZANE-LAKOUD
D EPARTMENT OF M ATHEMATICS
U NIVERSITY OF G UELMA
G UELMA , A LGERIA
khaledV2004@yahoo.fr
a_guezane@yahoo.fr
Received 19 April, 2006; accepted 24 January, 2007
Communicated by N.S. Barnett
A BSTRACT. A generalization of Pečarić’s extension of Montgomery’s identity is established
and used to derive new Čebyšev type inequalities.
Key words and phrases: Montgomery and Čebyšev-Grüss type inequalities, Pečarić’s extension, Montgomery identity.
2000 Mathematics Subject Classification. 26D15, 26D20.
1. I NTRODUCTION
In the present work we establish a generalization of Pečarić’s extension of ‘Montgomery’s’
identity and use it to derive new Čebyšev type inequalities.
We recall the Čebyšev inequality [1], given by the following:
(1.1)
|T (f, g)| ≤
1
(b − a)2 kf 0 k∞ kg 0 k∞ ,
12
where f, g : [a, b] → R are absolutely continuous functions, whose first derivatives f 0 and g 0
are bounded,
Z b
Z b
Z b
1
1
1
(1.2)
T (f, g) =
f (x) g (x) dx −
f (x) dx
g (x) dx
b−a a
b−a a
b−a a
and k·k∞ denotes the norm in L∞ [a, b] defined as kpk∞ = ess sup |p (t)|.
t∈[a,b]
Pachpatte in [6] established new inequalities of the Čebyšev type by using Pečariċ’s extension
of the Montgomery identity [7].
114-06
2
K. B OUKERRIOUA AND A. G UEZANE -L AKOUD
2. S TATEMENT OF R ESULTS
From [3], if f : [a, b] → R is differentiable on [a, b] with the first derivative f 0 (t) integrable
on [a, b] , then the Montgomery identity holds:
1
f (x) =
b−a
(2.1)
b
Z
b
Z
P (x, t) f 0 (t) dt,
f (t) dt +
a
a
where P (x, t) is the Peano kernel defined by
P (x, t) =

t−a



 b − a, a ≤ t ≤ x


t−b


, x < t ≤ b.
b−a
Rb
We assume that w : [a, b] → [0, +∞[ is some probability density function, i.e. a w (t) dt = 1,
Rt
and set W (t) = a w (x) dx for a ≤ t ≤ b, W (t) = 0 for t < a and for t > b. We then
have the following identity given by Pečarić in [7], that is the weighted generalization of the
Montgomery identity:
Z
b
f (x) =
(2.2)
Z
w (t) f (t) dt +
a
b
Pw (x, t) f 0 (t) dt,
a
where the weighted Peano kernel Pw is:
(
Pw (x, t) =
(2.3)
W (t) ,
a≤t≤x
W (t) − 1, x < t ≤ b
Let ϕ : [0, 1] → R be a differentiable function on [0, 1], with ϕ (0) = 0, ϕ (1) 6= 0 and ϕ0
integrable on [0, 1]. To simplify the notation, for some given functions w, f, g : [a, b] → R, we
set
b
x
(2.4) T (w, f, g, ϕ ) =
w (x) ϕ
w (t) dt f (x) g (x) dx
a
a
Z b
Z x
Z b
Z x
1
0
0
−
w (x) ϕ
w (t) dt f (x) dx
w (x) ϕ
w (t) dt g (x) dx .
ϕ (1) a
a
a
a
Z
0
0
Z
Theorem 2.1. Let f : [a, b] → R be differentiable and f 0 (t) integrable on [a, b] , then,
(2.5)
1
f (x) =
ϕ (1)
Z
b
w (t) ϕ
0
t
Z
a
w (s) ds f (t) dt +
a
1
ϕ (1)
Z
b
Pw,ϕ (x, t) f 0 (t)dt,
a
where Pw,ϕ is a generalization of the weighted Peano kernel defined by:
(
(2.6)
Pw,ϕ (x, t) =
ϕ (W (t)) ,
a ≤ t ≤ x;
ϕ (W (t)) − ϕ (1) , x < t ≤ b.
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 55, 4 pp.
http://jipam.vu.edu.au/
G ENERALIZATION OF Č EBYŠEV T YPE I NEQUALITIES
3
Proof. Using the hypothesis on ϕ,
Z b
(2.7)
Pw,ϕ (x, t) f 0 (t)dt
a
Z x
Z b
0
=
ϕ (W (t)) f (t)dt +
(ϕ (W (t)) − ϕ (1)) f 0 (t)dt
a
x
Z b
Z b
0
=
ϕ (W (t)) f (t)dt − ϕ (1)
f 0 (t)dt
a
x
Z b
= [ϕ (W (t)) f (t)]ba −
w (t) ϕ0 (W (t)) f (t) dt − ϕ (1) [f (b) − f (x)]
a
Z t
Z b
0
= ϕ (1) f (x) −
w (t) ϕ
w (s) ds f (t) dt.
a
a
Multiplying both sides by 1/ϕ (1), we obtain,
Z t
Z b
Z b
1
1
0
f (x) =
w (t) ϕ
Pw,ϕ (x, t) f 0 (t)dt
w (s) ds f (t) dt +
ϕ (1) a
ϕ
(1)
a
a
and this completes the proof.
Theorem 2.2. Let f, g [a, b] → R be differentiable on [a, b] and f 0 , g 0 be integrable on [a, b]
and let w, ϕ be as in Theorem 2.1, then,
Z b
1
0
0
0
0
|T (w, f, g, ϕ )| ≤ 2
kf k∞ kg k∞ kϕ k∞
w (x) H 2 (x) dx,
ϕ (1)
a
where H (x) =
Rb
a
|Pw,ϕ (x, t)| dt and kϕ0 k∞ = ess sup |ϕ0 (t)| .
t∈[0,1]
Since the functions f and g satisfy the hypothesis of Theorem 2.1, the following identities
hold:
Z t
Z b
Z b
1
1
0
(2.8)
f (x) =
w (t) ϕ
w (s) ds f (t) dt +
Pw,ϕ (x, t) f 0 (t)dt
ϕ (1) a
ϕ
(1)
a
a
and
(2.9)
1
g (x) =
ϕ (1)
Z
b
w (t) ϕ
0
a
Z
t
a
1
w (s) ds g (t) dt +
ϕ (1)
Z
b
Pw,ϕ (x, t) g 0 (t) dt .
a
Using (2.8) and (2.9) we obtain,
f (x) −
1
ϕ (1)
b
0
Z
t
w (t) ϕ
w (s) ds f (t) dt
a
a
Z t
Z b
1
0
× g (x) −
w (t) ϕ
w (s) ds g (t) dt
ϕ (1) a
a
Z b
Z b
1
0
0
= 2
Pw,ϕ (x, t) f (t)dt
Pw,ϕ (x, t) g (t) dt .
ϕ (1) a
a
Z
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 55, 4 pp.
http://jipam.vu.edu.au/
4
K. B OUKERRIOUA AND A. G UEZANE -L AKOUD
Consequently,
Z t
Z b
1
0
f (x)
w (t) ϕ
w (s) ds g (t) dt
(2.10) f (x) g (x) −
ϕ (1)
a
a
Z t
Z b
Z t
Z b
1
1
0
0
−
g (x)
w (t) ϕ
w (s) ds f (t) dt+ 2
w (t) ϕ
w (s) ds f (t) dt
ϕ (1)
ϕ (1)
a
a
a
a
Z b
Z t
0
×
w (t) ϕ
w (s) ds g (t) dt
a
a
Z b
Z b
1
0
0
= 2
Pw,ϕ (x, t) f (t)dt
Pw,ϕ (x, t) g (t) dt .
ϕ (1) a
a
Rx
Multiplying both sides of (2.10) by w (x) ϕ0 a w (s) ds and then integrating the resultant
identity with respect to x from a to b, we get,
Z x
Z b
1
0
0
w (x) ϕ
(2.11) T (w, f, g, ϕ ) = 2
w (t) dt
ϕ (1) a
a
Z b
Z b
0
0
×
Pw,ϕ (x, t) f (t)dt
Pw,ϕ (x, t) g (t) dt dx.
a
a
Finally,
1
kf 0 k∞ kg 0 k∞ kϕ0 k∞
|T (w, f, g, ϕ )| ≤ 2
ϕ (1)
0
Z
b
w (x) H 2 (x) dx.
a
R EFERENCES
[1] P.L. ČEBYŠEV, Sur les expressions approximatives des intégrales définies par les autres prises entre
les mêmes limites, Proc. Math. Soc. Charkov, 2 (1882), 93–98.
Rb
1
[2] G. GRÜSS. Über das maximum des absoluten Betrages von b−a
a f (x) g (x) dx −
Rb
Rb
1
f (x) dx a g (x) dx, Math. Z., 39 (1935), 215–226.
(b−a)2 a
[3] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequalities Involving Functions and their
Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991.
[4] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and New Inequalities in Analysis,
Kluwer Academic Publishers, Dordrecht, 1993.
[5] B.G. PACHPATTE, New weighted multivariable Grüss type inequalities, J. Inequal. Pure and Appl.
Math., 4(5) (2003), Art 108. [ONLINE: http://jipam.vu.edu.au/article.php?sid=
349].
[6] B.G. PACHPATTE, On Čebysev-Grüss type inequalities via Pečarić’s extention of the Montgomery
identity, J. Inequal. Pure and Appl. Math., 7(1) (2006), Art 108. [ONLINE: http://jipam.vu.
edu.au/article.php?sid=624].
[7] J.E. PEČARIĆ, On the Čebysev inequality, Bul. Sti. Tehn. Inst. Politehn. "Tralan Vuia" Timişora
(Romania), 25(39) (1) (1980), 5–9.
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 55, 4 pp.
http://jipam.vu.edu.au/