Volume 8 (2007), Issue 2, Article 55, 4 pp.
ON GENERALIZATION OF ˇCEBYŠEV TYPE INEQUALITIES
K. BOUKERRIOUA AND A. GUEZANE-LAKOUD DEPARTMENT OFMATHEMATICS
UNIVERSITY OFGUELMA
GUELMA, ALGERIA
Received 19 April, 2006; accepted 24 January, 2007 Communicated by N.S. Barnett
ABSTRACT. A generalization of Peˇcari´c’s extension of Montgomery’s identity is established and used to derive new ˇCebyšev type inequalities.
Key words and phrases: Montgomery and ˇCebyšev-Grüss type inequalities, Peˇcari´c’s extension, Montgomery identity.
2000 Mathematics Subject Classification. 26D15, 26D20.
1. INTRODUCTION
In the present work we establish a generalization of Peˇcari´c’s extension of ‘Montgomery’s’
identity and use it to derive new ˇCebyšev type inequalities.
We recall the ˇCebyšev inequality [1], given by the following:
(1.1) |T (f, g)| ≤ 1
12(b−a)2kf0k∞kg0k∞,
wheref, g : [a, b] → R are absolutely continuous functions, whose first derivativesf0 andg0 are bounded,
(1.2) T (f, g) = 1 b−a
Z b a
f(x)g(x)dx− 1
b−a Z b
a
f(x)dx 1 b−a
Z b a
g(x)dx
andk·k∞denotes the norm inL∞[a, b]defined askpk∞ =esssup
t∈[a,b]
|p(t)|.
Pachpatte in [6] established new inequalities of the ˇCebyšev type by using Peˇcari˙c’s extension of the Montgomery identity [7].
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2 K. BOUKERRIOUA ANDA. GUEZANE-LAKOUD
2. STATEMENT OFRESULTS
From [3], iff : [a, b] →Ris differentiable on[a, b]with the first derivativef0(t)integrable on[a, b],then the Montgomery identity holds:
(2.1) f(x) = 1
b−a Z b
a
f(t)dt+ Z b
a
P(x, t)f0(t)dt, whereP (x, t)is the Peano kernel defined by
P (x, t) =
t−a
b−a, a≤t≤x t−b
b−a, x < t≤b.
We assume thatw: [a, b]→ [0,+∞[is some probability density function, i.e. Rb
aw(t)dt = 1, and set W(t) = Rt
aw(x)dx for a ≤ t ≤ b, W(t) = 0 for t < a and for t > b. We then have the following identity given by Peˇcari´c in [7], that is the weighted generalization of the Montgomery identity:
(2.2) f(x) =
Z b a
w(t)f(t)dt+ Z b
a
Pw(x, t)f0(t)dt, where the weighted Peano kernelPwis:
(2.3) Pw(x, t) =
( 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 functionsw, f, g: [a, b]→R,we set
(2.4) T (w, f, g, ϕ0) = Z b
a
w(x)ϕ0 Z x
a
w(t)dt
f(x)g(x)dx
− 1 ϕ(1)
Z b a
w(x)ϕ0 Z x
a
w(t)dt
f(x)dx Z b
a
w(x)ϕ0 Z x
a
w(t)dt
g(x)dx
. Theorem 2.1. Letf : [a, b]→Rbe differentiable andf0(t)integrable on[a, b],then,
(2.5) f(x) = 1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt+ 1 ϕ(1)
Z b a
Pw,ϕ(x, t)f0(t)dt, wherePw,ϕis 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/
GENERALIZATION OFCˇEBYŠEVTYPEINEQUALITIES 3
Proof. Using the hypothesis onϕ, Z b
a
Pw,ϕ(x, t)f0(t)dt (2.7)
= Z x
a
ϕ(W(t))f0(t)dt+ Z b
x
(ϕ(W(t))−ϕ(1))f0(t)dt
= Z b
a
ϕ(W(t))f0(t)dt−ϕ(1) Z b
x
f0(t)dt
= [ϕ(W(t))f(t)]ba− Z b
a
w(t)ϕ0(W(t))f(t)dt−ϕ(1) [f(b)−f(x)]
=ϕ(1)f(x)− Z b
a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt.
Multiplying both sides by1/ϕ(1), we obtain, f(x) = 1
ϕ(1) Z b
a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt+ 1 ϕ(1)
Z b a
Pw,ϕ(x, t)f0(t)dt
and this completes the proof.
Theorem 2.2. Letf, g [a, b] → R be differentiable on[a, b]and f0, g0 be integrable on[a, b]
and letw, ϕbe as in Theorem 2.1, then,
|T(w, f, g, ϕ0)| ≤ 1
ϕ2(1)kf0k∞kg0k∞kϕ0k∞ Z b
a
w(x)H2(x)dx, whereH(x) = Rb
a |Pw,ϕ(x, t)|dtandkϕ0k∞=esssup
t∈[0,1]
|ϕ0(t)|.
Since the functionsf andg satisfy the hypothesis of Theorem 2.1, the following identities hold:
(2.8) f(x) = 1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt+ 1 ϕ(1)
Z b a
Pw,ϕ(x, t)f0(t)dt and
(2.9) g(x) = 1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
g(t)dt+ 1 ϕ(1)
Z b a
Pw,ϕ(x, t)g0(t)dt. Using (2.8) and (2.9) we obtain,
f(x)− 1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt
×
g(x)− 1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
g(t)dt
= 1
ϕ2(1) Z b
a
Pw,ϕ(x, t)f0(t)dt Z b
a
Pw,ϕ(x, t)g0(t)dt
.
J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 55, 4 pp. http://jipam.vu.edu.au/
4 K. BOUKERRIOUA ANDA. GUEZANE-LAKOUD
Consequently,
(2.10) f(x)g(x)− 1
ϕ(1)f(x) Z b
a
w(t)ϕ0 Z t
a
w(s)ds
g(t)dt
− 1
ϕ(1)g(x) Z b
a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt+ 1 ϕ2(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt
× Z b
a
w(t)ϕ0 Z t
a
w(s)ds
g(t)dt
= 1
ϕ2(1) Z b
a
Pw,ϕ(x, t)f0(t)dt Z b
a
Pw,ϕ(x, t)g0(t)dt
. Multiplying both sides of (2.10) by w(x)ϕ0 Rx
a w(s)ds
and then integrating the resultant identity with respect toxfromatob, we get,
(2.11) T (w, f, g, ϕ0) = 1 ϕ2(1)
Z b a
w(x)ϕ0 Z x
a
w(t)dt
× Z b
a
Pw,ϕ(x, t)f0(t)dt Z b
a
Pw,ϕ(x, t)g0(t)dt
dx.
Finally,
|T(w, f, g, ϕ0)| ≤ 1
ϕ2(1)kf0k∞kg0k∞kϕ0k∞ Z b
a
w(x)H2(x)dx.
REFERENCES
[1] P.L. ˇCEBYŠEV, 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. GRÜSS. Über das maximum des absoluten Betrages von b−a1 Rb
af(x)g(x)dx −
1 (b−a)2
Rb
af(x)dxRb
ag(x)dx,Math. Z., 39 (1935), 215–226.
[3] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´C AND A.M. FINK, Inequalities Involving Functions and their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991.
[4] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´C ANDA.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 ˇCebysev-Grüss type inequalities via Peˇcari´c’s extention of the Montgomery identity, J. Inequal. Pure and Appl. Math., 7(1) (2006), Art 108. [ONLINE:http://jipam.vu.
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[7] J.E. PE ˇCARI ´C, On the ˇCebysev inequality, Bul. Sti. Tehn. Inst. Politehn. "Tralan Vuia" Timi¸sora (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/
