Journal of Inequalities in Pure and Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 1, Article 11, 2006
ON ˇCEBYŠEV-GRÜSS TYPE INEQUALITIES VIA PE ˇCARI ´C’S EXTENSION OF THE MONTGOMERY IDENTITY
B.G. PACHPATTE 57 SHRINIKETANCOLONY
NEARABHINAYTALKIES
AURANGABAD431 001 (MAHARASHTRA) INDIA
Received 15 August, 2005; accepted 19 January, 2006 Communicated by J. Sándor
ABSTRACT. In the present note we establish new ˇCebyšev-Grüss type inequalities by using Peˇcariˇc’s extension of the Montgomery identity.
Key words and phrases: ˇCebyšev-Grüss type inequalities, Peˇcari´c’s extension, Montgomery identity.
2000 Mathematics Subject Classification. 26D15, 26D20.
1. INTRODUCTION
For two absolutely continuous functionsf, g: [a, b]→Rconsider the functional 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
,
where the involved integrals exist. In 1882, ˇCebyšev [1] proved that iff0, g0 ∈L∞[a, b], then
(1.1) |T (f, g)| ≤ 1
12(b−a)2kf0k∞kg0k∞. In 1935, Grüss [2] showed that
(1.2) |T (f, g)| ≤ 1
4(M −m) (N −n),
provided m, M, n, N are real numbers satisfying the condition−∞ < m ≤ M < ∞,−∞ <
n≤N <∞forx∈[a, b].
Many researchers have given considerable attention to the inequalities (1.1), (1.2) and various generalizations, extensions and variants of these inequalities have appeared in the literature, to mention a few, see [4, 5] and the references cited therein. The aim of this note is to establish two
ISSN (electronic): 1443-5756
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022-06
2 B.G. PACHPATTE
new inequalities similar to those of ˇCebyšev and Grüss inequalities by using Peˇcariˇc’s extension of the Montgomery identity given in [6].
2. STATEMENT OFRESULTS
Letf : [a, b]→ Rbe differentiable on[a, b]andf0 : [a, b] →Ris integrable on[a, b]. Then the Montgomery identity holds [3]:
(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
(2.2) P(x, t) =
( t−a
b−a, a≤t ≤x,
t−b
b−a, x < t≤b.
Let w : [a, b] → [0,∞) be some probability density function, that is, an integrable function satisfying Rb
a w(t)dt = 1, and W(t) = Rt
aw(x)dx for t ∈ [a, b], W(t) = 0 for t < a, and W(t) = 1 for t > b. In [6] Peˇcari´c has given the following weighted extension of the Montgomery identity:
(2.3) f(x) =
Z b
a
w(t)f(t)dt+ Z b
a
Pw(x, t)f0(t)dt, wherePw(x, t)is the weighted Peano kernel defined by
(2.4) Pw(x, t) =
( W(t), a≤t≤x, W(t)−1, x < t≤b.
We use the following notation to simplify the details of presentation. For some suitable functons w, f, g: [a, b]→R,we set
T (w, f, g) = Z b
a
w(x)f(x)g(x)dx− Z b
a
w(x)f(x)dx
Z b
a
w(x)g(x)dx
,
and definek·k∞ as the usual Lebesgue norm onL∞[a, b]that is,khk∞ :=ess sup
t∈[a,b]
|h(t)|for h∈L∞[a, b].
Our main results are given in the following theorems.
Theorem 2.1. Let f, g : [a, b] → R be differentiable on [a, b] and f0, g0 : [a, b] → R are integrable on[a, b]. Letw: [a, b]→[0,∞)be an integrable function satisfyingRb
a w(t)dt= 1.
Then
(2.5) |T (w, f, g)| ≤ kf0k∞kg0k∞ Z b
a
w(x)H2(x)dx, where
(2.6) H(x) =
Z b
a
|Pw(x, t)|dt
forx∈[a, b]andPw(x, t)is the weighted Peano kernel given by (2.4).
Theorem 2.2. Letf, g, f0, g0, wbe as in Theorem 2.1. Then
(2.7) |T (w, f, g)| ≤ 1 2
Z b
a
w(x) [|g(x)| kf0k∞+|f(x)| kg0k∞]H(x)dx, whereH(x)is defined by (2.6).
J. Inequal. Pure and Appl. Math., 7(1) Art. 11, 2006 http://jipam.vu.edu.au/
CˇEBYŠEV-GRÜSSTYPEINEQUALITIES 3
3. PROOFS OFTHEOREMS2.1AND 2.2
Proof of Theorem 2.1. From the hypotheses the following identities hold [6]:
f(x) = Z b
a
w(t)f(t)dt+ Z b
a
Pw(x, t)f0(t)dt, (3.1)
g(x) = Z b
a
w(t)g(t)dt+ Z b
a
Pw(x, t)g0(t)dt, (3.2)
From (3.1) and (3.2) we observe that
f(x)− Z b
a
w(t)f(t)dt g(x)− Z b
a
w(t)g(t)dt
= Z b
a
Pw(x, t)f0(t)dt Z b
a
Pw(x, t)g0(t)dt
,
i.e.,
(3.3) f(x)g(x)−f(x) Z b
a
w(t)g(t)dt−g(x) Z b
a
w(t)f(t)dt
+ Z b
a
w(t)f(t)dt
Z b
a
w(t)g(t)dt
= Z b
a
Pw(x, t)f0(t)dt Z b
a
Pw(x, t)g0(t)dt
.
Multiplying both sides of (3.3) byw(x)and then integrating both sides of the resulting identity with respect toxfromatoband using the fact thatRb
a w(t)dt= 1, we have (3.4) T (w, f, g) =
Z b
a
w(x) Z b
a
Pw(x, t)f0(t)dt Z b
a
Pw(x, t)g0(t)dt
dx.
From (3.4) and using the properties of modulus we observe that
|T(w, f, g)| ≤ Z b
a
w(x) Z b
a
|Pw(x, t)| |f0(t)|dt Z b
a
|Pw(x, t)| |g0(t)|dt
dx
≤ kf0k∞kg0k∞ Z b
a
w(x)H2(x)dx.
This completes the proof of Theorem 2.1.
Proof of Theorem 2.2. Multiplying both sides of (3.1) and (3.2) by w(x)g(x) and w(x)f(x), adding the resulting identities and rewriting we have
(3.5) w(x)f(x)g(x)
= 1 2
w(x)g(x) Z b
a
w(t)f(t)dt+w(x)f(x) Z b
a
w(t)g(t)dt
+ 1 2
w(x)g(x) Z b
a
Pw(x, t)f0(t)dt+w(x)f(x) Z b
a
Pw(x, t)g0(t)dt
.
J. Inequal. Pure and Appl. Math., 7(1) Art. 11, 2006 http://jipam.vu.edu.au/
4 B.G. PACHPATTE
Integrating both sides of (3.5) with respect toxfromatoband rewriting we have (3.6) T (w, f, g) = 1
2 Z b
a
w(x)g(x) Z b
a
Pw(x, t)f0(t)dt
+w(x)f(x) Z b
a
Pw(x, t)g0(t)dt
dx.
From (3.6) and using the properties of modulus we observe that
|T (w, f, g)|
≤ 1 2
Z b
a
w(x)
|g(x)|
Z b
a
|Pw(x, t)| |f0(t)|dt+|f(x)|
Z b
a
|Pw(x, t)| |g0(t)|dt
dx
≤ 1 2
Z b
a
w(x) [|g(x)| kf0(t)k∞+|f(x)| kg0(t)k∞]H(x)dx.
The proof of Theorem 2.2 is complete.
REFERENCES
[1] P.L. ˇCEBYŠEV, Sue les expressions approxmatives 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 ´CAND 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 multivariate 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] J.E. PE ˇCARI ´C, On the ˇCebyšev inequality, Bul. ¸Sti. Tehn. Inst. Politehn. "Train Vuia" Timi¸sora, 25(39)(1) (1980), 5–9.
J. Inequal. Pure and Appl. Math., 7(1) Art. 11, 2006 http://jipam.vu.edu.au/
