ON GENERALIZATION OF ČEBYŠEV TYPE
INEQUALITIES
K. BOUKERRIOUA AND A. GUEZANE-LAKOUD
Department of Mathematics
University of Guelma
Guelma, Algeria
EMail: khaledV2004@yahoo.fr
vol. 8, iss. 2, art. 55, 2007
and
a_guezane@yahoo.fr
Received:
19 April, 2006
Accepted:
24 January, 2007
Communicated by:
N.S. Barnett
2000 AMS Sub. Class.:
Primary 30C45.
Key words:
Montgomery and Čebyšev-Grüss type inequalities, Pečarić’s extension, Montgomery
identity.
Abstract:
Generalization of Čebyšev
Type Inequalities
K. Boukerrioua and
A. Guezane-Lakoud
A generalization of Pečarić’s extension of Montgomery’s identity is established and
used to derive new Čebyšev type inequalities.
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Contents
1
Introduction
3
2
Statement of Results
4
Generalization of Čebyšev
Type Inequalities
K. Boukerrioua and
A. Guezane-Lakoud
vol. 8, iss. 2, art. 55, 2007
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1.
Introduction
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,
(1.2)
1
T (f, g) =
b−a
Z
a
Generalization of Čebyšev
Type Inequalities
K. Boukerrioua and
A. Guezane-Lakoud
vol. 8, iss. 2, art. 55, 2007
b
f (x) g (x) dx
Z b
Z b
1
1
−
f (x) dx
g (x) dx
b−a a
b−a a
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and k·k∞ denotes the norm in L∞ [a, b] defined as kpk∞ = ess sup |p (t)|.
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Pachpatte in [6] established new inequalities of the Čebyšev type by using Pečariċ’s
extension of the Montgomery identity [7].
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t∈[a,b]
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2.
Statement of Results
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:
Z b
Z b
1
(2.1)
f (x) =
f (t) dt +
P (x, t) f 0 (t) dt,
b−a a
a
where P (x, t) is the Peano kernel defined by

t−a



 b − a, a ≤ t ≤ x
P (x, t) =


t−b


, x < t ≤ b.
b−a
We assume that w : [a, b] → [0, +∞[ is some probability density function, i.e.
Rb
Rt
w (t) dt = 1, and set W (t) = a w (x) dx for a ≤ t ≤ b, W (t) = 0 for t < a
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
Z b
(2.2)
f (x) =
w (t) f (t) dt +
Pw (x, t) f 0 (t) dt,
a
a
where the weighted Peano kernel Pw is:
(
W (t) ,
a≤t≤x
(2.3)
Pw (x, t) =
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,
Generalization of Čebyšev
Type Inequalities
K. Boukerrioua and
A. Guezane-Lakoud
vol. 8, iss. 2, art. 55, 2007
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g : [a, b] → R, we set
b
Z
0
Z
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
1
0
−
w (x) ϕ
w (t) dt f (x) dx
ϕ (1) a
a
Z b
Z x
0
×
w (x) ϕ
w (t) dt g (x) dx .
0
a
a
Theorem 2.1. Let f : [a, b] → R be differentiable and f 0 (t) integrable on [a, b] ,
then,
Z t
Z b
1
0
(2.5)
f (x) =
w (t) ϕ
w (s) ds f (t) dt
ϕ (1) a
a
Z b
1
Pw,ϕ (x, t) f 0 (t)dt,
+
ϕ (1) a
where Pw,ϕ is a generalization of the weighted Peano kernel defined by:
(
ϕ (W (t)) ,
a ≤ t ≤ x;
(2.6)
Pw,ϕ (x, t) =
ϕ (W (t)) − ϕ (1) , x < t ≤ b.
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
Generalization of Čebyšev
Type Inequalities
K. Boukerrioua and
A. Guezane-Lakoud
vol. 8, iss. 2, art. 55, 2007
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Z
b
0
Z
b
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
w (s) ds f (t) dt +
Pw,ϕ (x, t) f 0 (t)dt
ϕ (1) a
ϕ
(1)
a
a
Generalization of Čebyšev
Type Inequalities
K. Boukerrioua and
A. Guezane-Lakoud
vol. 8, iss. 2, art. 55, 2007
and this completes the proof.
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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
Rb
where H (x) = a |Pw,ϕ (x, t)| dt and kϕ0 k∞ = ess sup |ϕ0 (t)| .
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t∈[0,1]
Since the functions f and g satisfy the hypothesis of Theorem 2.1, the following
identities hold:
Z t
Z b
1
0
(2.8)
f (x) =
w (t) ϕ
w (s) ds f (t) dt
ϕ (1) a
a
Z b
1
+
Pw,ϕ (x, t) f 0 (t)dt
ϕ (1) a
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and
(2.9)
1
g (x) =
ϕ (1)
Z
b
w (t) ϕ
0
a
Z
a
t
w (s) ds g (t) dt
Z b
1
Pw,ϕ (x, t) g 0 (t) dt .
+
ϕ (1) a
Using (2.8) and (2.9) we obtain,
f (x) −
b
Z
0
Z
t
w (t) ϕ
w (s) ds f (t) dt
a
a
Z t
Z b
1
0
w (t) ϕ
w (s) ds g (t) dt
× g (x) −
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
1
ϕ (1)
Consequently,
Generalization of Čebyšev
Type Inequalities
K. Boukerrioua and
A. Guezane-Lakoud
vol. 8, iss. 2, art. 55, 2007
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Z t
Z b
1
0
(2.10) f (x) g (x) −
f (x)
w (t) ϕ
w (s) ds g (t) dt
ϕ (1)
a
a
Z t
Z b
1
0
−
g (x)
w (t) ϕ
w (s) ds f (t) dt
ϕ (1)
a
a
Z b
Z t
1
0
+ 2
w (t) ϕ
w (s) ds f (t) dt
ϕ (1)
a
a
Z b
Z t
0
×
w (t) ϕ
w (s) ds g (t) dt
a
a
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1
= 2
ϕ (1)
Z
b
Z b
0
Pw,ϕ (x, t) f (t)dt
Pw,ϕ (x, t) g (t) dt .
0
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
(2.11) T (w, f, g, ϕ ) = 2
w (x) ϕ
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.
0
a
Generalization of Čebyšev
Type Inequalities
K. Boukerrioua and
A. Guezane-Lakoud
vol. 8, iss. 2, art. 55, 2007
a
Finally,
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|T (w, f, g, ϕ0 )| ≤
1
kf 0 k∞ kg 0 k∞ kϕ0 k∞
ϕ2 (1)
Z
a
b
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w (x) H 2 (x) dx.
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References
[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.
[2] G. R GRÜSS. Über das Rmaximum R des absoluten Betrages von
b
b
b
1
1
f (x) g (x) dx − (b−a)
f (x) dx a g (x) dx, Math. Z., 39 (1935),
2
b−a a
a
215–226.
[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.
Generalization of Čebyšev
Type Inequalities
K. Boukerrioua and
A. Guezane-Lakoud
vol. 8, iss. 2, art. 55, 2007
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