Journal of Inequalities in Pure and
Applied Mathematics
ON CEBYSEV-GRÜSS TYPE INEQUALITIES VIA PEČARIĆ’S
EXTENSION OF THE MONTGOMERY IDENTITY
volume 7, issue 1, article 11,
2006.
B.G. PACHPATTE
57 Shri Niketan Colony
Near Abhinay Talkies
Aurangabad 431 001
(Maharashtra) India.
Received 15 August, 2005;
accepted 19 January, 2006.
Communicated by: J. Sándor
EMail: bgpachpatte@gmail.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
022-06
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Abstract
In the present note we establish new Čebyšev-Grüss type inequalities by using
Pečarič’s extension of the Montgomery identity.
2000 Mathematics Subject Classification: 26D15, 26D20.
Key words: Čebyšev-Grüss type inequalities, Pečarić’s extension, Montgomery identity.
On Cebysev-Grüss Type
Inequalities via Pecarić’s
Extension of the Montgomery
Identity
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Proofs of Theorems 2.1 and 2.2 . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
4
7
B.G. Pachpatte
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J. Ineq. Pure and Appl. Math. 7(1) Art. 11, 2006
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1.
Introduction
For two absolutely continuous functions f, g : [a, b] → R consider the functional
Z b
1
T (f, g) =
f (x) g (x) dx
b−a a
Z b
Z b
1
1
−
f (x) dx
g (x) dx ,
b−a a
b−a a
where the involved integrals exist. In 1882, Čebyšev [1] proved that if f 0 , g 0 ∈
L∞ [a, b], then
(1.1)
1
|T (f, g)| ≤
(b − a)2 kf 0 k∞ kg 0 k∞ .
12
|T (f, g)| ≤
B.G. Pachpatte
Title Page
In 1935, Grüss [2] showed that
(1.2)
On Cebysev-Grüss Type
Inequalities via Pecarić’s
Extension of the Montgomery
Identity
1
(M − m) (N − n) ,
4
provided m, M, n, N are real numbers satisfying the condition −∞ < m ≤
M < ∞, −∞ < n ≤ N < ∞ for x ∈ [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 new inequalities similar
to those of Čebyšev and Grüss inequalities by using Pečarič’s extension of the
Montgomery identity given in [6].
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2.
Statement of Results
Let f : [a, b] → R be differentiable on [a, b] and f 0 : [a, b] → R is integrable on
[a, b]. Then the Montgomery identity holds [3]:
(2.1)
1
f (x) =
b−a
Z
b
Z
f (t) dt +
a
b
P (x, t) f 0 (t) dt,
a
where P (x, t) is the Peano kernel defined by
( t−a
, a ≤ t ≤ x,
b−a
(2.2)
P (x, t) =
t−b
, x < t ≤ b.
b−a
Let w : [a, b] → [0, ∞) be some probability density function, that is, an
Rb
Rt
integrable function satisfying a w (t) dt = 1, and W (t) = a w (x) dx for
t ∈ [a, b], W (t) = 0 for t < a, and W (t) = 1 for t > b. In [6] Pečarić has given
the following weighted extension of the Montgomery identity:
Z b
Z b
(2.3)
f (x) =
w (t) f (t) dt +
Pw (x, t) f 0 (t) dt,
a
a
where Pw (x, t) is the weighted Peano kernel defined by
(
W (t) ,
a ≤ t ≤ x,
(2.4)
Pw (x, t) =
W (t) − 1, x < t ≤ b.
On Cebysev-Grüss Type
Inequalities via Pecarić’s
Extension of the Montgomery
Identity
B.G. Pachpatte
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We use the following notation to simplify the details of presentation. For some
suitable functons w, f, g : [a, b] → R,we set
Z
T (w, f, g) =
a
b
w (x) f (x) g (x) dx
Z b
Z b
−
w (x) f (x) dx
w (x) g (x) dx ,
a
a
and define k·k∞ as the usual Lebesgue norm on L∞ [a, b] that is, khk∞ :=
ess sup |h (t)| for h ∈ L∞ [a, b] .
t∈[a,b]
Our main results are given in the following theorems.
B.G. Pachpatte
0
0
Theorem 2.1. Let f, g : [a, b] → R be differentiable on [a, b] and f , g :
[a, b] → R are integrable on [a, b]. Let w : [a, b] → [0, ∞) be an integrable
Rb
function satisfying a w (t) dt = 1. Then
(2.5)
|T (w, f, g)| ≤ kf 0 k∞ kg 0 k∞
Z
b
w (x) H 2 (x) dx,
a
where
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Z
(2.6)
On Cebysev-Grüss Type
Inequalities via Pecarić’s
Extension of the Montgomery
Identity
b
|Pw (x, t)| dt
H (x) =
a
for x ∈ [a, b] and Pw (x, t) is the weighted Peano kernel given by (2.4).
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Theorem 2.2. Let f, g, f 0 , g 0 , w be as in Theorem 2.1. Then
Z
1 b
(2.7) |T (w, f, g)| ≤
w (x) [|g (x)| kf 0 k∞ + |f (x)| kg 0 k∞ ]H (x) dx,
2 a
where H(x) is defined by (2.6).
On Cebysev-Grüss Type
Inequalities via Pecarić’s
Extension of the Montgomery
Identity
B.G. Pachpatte
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3.
Proofs of Theorems 2.1 and 2.2
Proof of Theorem 2.1. From the hypotheses the following identities hold [6]:
Z b
Z b
f (x) =
(3.1)
w (t) f (t) dt +
Pw (x, t) f 0 (t) dt,
a
a
Z b
Z b
(3.2)
g (x) =
w (t) g (t) dt +
Pw (x, t) g 0 (t) dt,
a
a
From (3.1) and (3.2) we observe that
Z b
Z b
f (x) −
w (t) f (t) dt g (x) −
w (t) g (t) dt
a
a
Z b
Z b
0
0
=
Pw (x, t) f (t) dt
Pw (x, t) g (t) dt ,
a
a
i.e.,
On Cebysev-Grüss Type
Inequalities via Pecarić’s
Extension of the Montgomery
Identity
B.G. Pachpatte
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Contents
Z
b
Z
b
(3.3) f (x) g (x) − f (x)
w (t) g (t) dt − g (x)
w (t) f (t) dt
a
a
Z b
Z b
+
w (t) f (t) dt
w (t) g (t) dt
a
a
Z b
Z b
0
0
=
Pw (x, t) f (t) dt
Pw (x, t) g (t) dt .
a
a
Multiplying both sides of (3.3) by w(x) and then integrating both sides of
the resulting identity with respect to x from a to b and using the fact that
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Rb
a
w (t) dt = 1 , we have
Z
(3.4)
T (w, f, g) =
b
Z
w (x)
a
a
b
Pw (x, t) f (t) dt
Z b
0
×
Pw (x, t) g (t) dt dx.
0
a
From (3.4) and using the properties of modulus we observe that
Z b
Z b
Z b
0
0
|T (w, f, g)| ≤
w (x)
|Pw (x, t)| |f (t)| dt
|Pw (x, t)| |g (t)| dt dx
a
a
a
Z b
0
0
≤ kf k∞ kg k∞
w (x) H 2 (x) dx.
On Cebysev-Grüss Type
Inequalities via Pecarić’s
Extension of the Montgomery
Identity
B.G. Pachpatte
a
This completes the proof of Theorem 2.1.
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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)
Z b
Z b
1
=
w (x) g (x)
w (t) f (t) dt + w (x) f (x)
w (t) g (t) dt
2
a
a
Z b
1
+
w (x) g (x)
Pw (x, t) f 0 (t) dt
2
a
Z b
0
+w (x) f (x)
Pw (x, t) g (t) dt .
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a
J. Ineq. Pure and Appl. Math. 7(1) Art. 11, 2006
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Integrating both sides of (3.5) with respect to x from a to b and rewriting we
have
Z Z b
1 b
(3.6)
T (w, f, g) =
w (x) g (x)
Pw (x, t) f 0 (t) dt
2 a
a
Z b
0
+ w (x) f (x)
Pw (x, t) g (t) dt dx.
a
From (3.6) and using the properties of modulus we observe that
|T (w, f, g)|
Z
Z b
1 b
w (x) |g (x)|
|Pw (x, t)| |f 0 (t)| dt
≤
2 a
a
Z b
0
+ |f (x)|
|Pw (x, t)| |g (t)| dt dx
a
Z
1 b
w (x) [|g (x)| kf 0 (t)k∞ + |f (x)| kg 0 (t)k∞ ] H (x) dx.
≤
2 a
The proof of Theorem 2.2 is complete.
On Cebysev-Grüss Type
Inequalities via Pecarić’s
Extension of the Montgomery
Identity
B.G. Pachpatte
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References
[1] P.L. ČEBYŠEV, 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. RGRÜSS, Über das maximum
absoluten Betrages von
Rb
Rdes
b
b
1
1
f (x) g (x) dx− (b−a)2 a f (x) dx a g (x) dx, Math. Z., 39 (1935),
b−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 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. PEČARIĆ, On the Čebyšev inequality, Bul. Şti. Tehn. Inst. Politehn.
"Train Vuia" Timişora, 25(39)(1) (1980), 5–9.
On Cebysev-Grüss Type
Inequalities via Pecarić’s
Extension of the Montgomery
Identity
B.G. Pachpatte
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