Journal of Inequalities in Pure and
Applied Mathematics
ON GRÜSS TYPE INEQUALITIES OF DRAGOMIR AND FEDOTOV
J.E. PEČARIĆ AND B. TEPEŠ
Faculty of Textile Technology
University of Zagreb,
Pirottijeva 6, 1000 Zagreb,
Croatia.
EMail: pecaric@mahazu.hazu.hr
URL: http://mahazu.hazu.hr/DepMPCS/indexJP.html
Faculty of Philosophy,
I. Lučića 3, 1000 Zagreb, Croatia.
volume 4, issue 5, article 91,
2003.
Received 20 May, 2003;
accepted 18 September, 2003.
Communicated by: S.S. Dragomir
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Victoria University
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Abstract
Weighted versions of Grüss type inequalities of Dragomir and Fedotov are
given. Some related results are also obtained.
2000 Mathematics Subject Classification: 26D15.
Key words: Grüss type inequalities.
On Grüss Type Inequalities of
Dragomir and Fedotov
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
6
J.E. Pečarić and B. Tepeš
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J. Ineq. Pure and Appl. Math. 4(5) Art. 91, 2003
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1.
Introduction
In 1935, G. Grüss proved the following inequality:
(1.1)
Z b
Z b
Z b
1
1
1
f (x) g (x) dx −
f (x) dx ·
g (x) dx
b − a
b−a a
b−a a
a
1
6 (Φ − ϕ) (Γ − γ) ,
4
provided that f and g are two integrable functions on [a, b] satisfying the condition
(1.2)
ϕ 6 f (x) 6 Φ and γ 6 g(x) 6 Γ for all x ∈ [a, b] .
The constant
1
4
is best possible and is achieved for
a+b
.
f (x) = g (x) = sgn x −
2
The following result of Grüss type was proved by S.S. Dragomir and I. Fedotov
[1]:
Theorem 1.1. Let f, u : [a, b] → R be such that u is L−Lipschitzian on [a, b],
i.e.,
(1.3)
|u (x) − u (y)| 6 L |x − y| for all x ∈ [a, b] ,
On Grüss Type Inequalities of
Dragomir and Fedotov
J.E. Pečarić and B. Tepeš
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f is Riemann integrable on [a, b] and there exist the real numbers m, M so that
J. Ineq. Pure and Appl. Math. 4(5) Art. 91, 2003
(1.4)
m 6 f (x) 6 M for all x ∈ [a, b] .
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Then we have the inequality
Z b
Z
1
u (b) − u (a) b
(1.5) f (x) du (x) −
f (t) dt 6 L (M − m) (b − a) ,
b−a
2
a
a
and the constant
one.
1
2
is sharp, in the sense that it cannot be replaced by a smaller
The following result of Grüss’ type was proved by S.S. Dragomir and I.
Fedotov [2]:
Theorem 1.2. Let f, u : [a, b] → R be such that u is L−lipschitzian
Wb on [a, b],
and f is a function of bounded variation on [a, b]. Denote by a f the total
variation of f on [a, b]. Then the following inequality holds:
(1.6)
Z b
Z b
b
_
1
f (b) − f (a)
u (x) df (x) −
·
u (x) dx 6 L (b − a) f.
b−a
2
a
a
a
The constant 21 is sharp, in the sense that it cannot be replaced by a smaller one.
Remark 1.1. For other related results see [3].
Let us also state that the weighted version of (1.1) is well known, that is we
have with condition (1.2) the following generalization of (1.1):
On Grüss Type Inequalities of
Dragomir and Fedotov
J.E. Pečarić and B. Tepeš
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(1.7)
1
|D (f, g; w)| 6 (Φ − ϕ) (Γ − γ) ,
4
where
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J. Ineq. Pure and Appl. Math. 4(5) Art. 91, 2003
D (f, g; w) = A (f, g; w) − A (f ; w) A (g; w) ,
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and
Rb
A (f ; w) =
a
w (x) f (x) dx
.
Rb
w (x) dx
a
So, in this paper we shall show that corresponding weighted versions of (1.5)
and (1.6) are also valid. Some related results will be also given.
On Grüss Type Inequalities of
Dragomir and Fedotov
J.E. Pečarić and B. Tepeš
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2.
Results
Theorem 2.1. Let f, u : [a, b] → R be such that f is Riemann integrable on
[a, b] and u is L−Lipschitzian on [a, b], i.e. (1.3) holds true. If w : [a, b] → R is
a positive weight function, then
Z b
(2.1)
|T (f, u; w)| 6 L
w (x) |f (x) − A (f ; w)| dx,
a
where
On Grüss Type Inequalities of
Dragomir and Fedotov
Z
(2.2) T (f, u; w) =
a
b
w (x) f (x) du (x)
Z b
Z b
1
w (x) du (x)
w (x) f (x) dx.
− Rb
w
(x)
dx
a
a
a
Moreover, if there exist the real numbers m, M such that (1.4) is valid, then
Z b
L
(2.3)
|T (f, u; w)| 6 (M − m)
w (x) dx.
2
a
J.E. Pečarić and B. Tepeš
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Proof. As in [1], we have
Z b
|T (f, u; w)| = w (x) [f (x) − A (f ; w)] du (x)
a
Z b
6L
w (x) |f (x) − A (f ; w)| dx.
a
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J. Ineq. Pure and Appl. Math. 4(5) Art. 91, 2003
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That is, (2.1) is valid. Furthermore, from an application of Cauchy’s inequality
we have:
Z
(2.4)
|T (f, u; w)| 6 L
b
Z
2
w (x) (f (x) − A (f ; w)) dx
w (x) dx
a
b
12
,
a
from where we obtain
(2.5)
1
2
Z
b
|T (f, u; w)| 6 L · (D (f, f ; w)) ·
w (x) dx.
a
From (1.7) for g ≡ f we get:
(2.6)
J.E. Pečarić and B. Tepeš
1
(D (f, f ; w)) 2 6
1
(Φ − ϕ) .
2
Now, (2.4) and (2.5) give (2.3).
Now, we shall prove the following result.
Theorem 2.2. Let f : [a, b] → R be M −Lipschitzian on [a, b] and u : [a, b] →
R be L−Lipschitzian on [a, b]. If w : [a, b] → R is a positive weight function,
then
RbRb
w (x) w (x) |x − y| dxdy
(2.7)
|T (f, u; w)| 6 L · M · a a
.
Rb
w
(y)
dy
a
Proof. It follows from (2.1)
On Grüss Type Inequalities of
Dragomir and Fedotov
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R b
w (y) (f (x) − f (y)) dy a
|T (f, u; w)| 6 L ·
w (x) dx
Rb
w
(y)
dy
a
a
R
Z b
b
w (y) |f (x) − f (y)| dy
a
6L·
w (x)
dx
Rb
w (y) dy
a
a
RbRb
w (x) w (x) |x − y| dxdy
6L·M · a a
.
Rb
w (y) dy
a
b
Z
On Grüss Type Inequalities of
Dragomir and Fedotov
If in the previous result we set w (x) ≡ 1, then we can obtain the following
corollary:
Corollary 2.3. Let f and u be as in Theorem 2.2, then,
Z b
Z b
L · M · (b − a)2
u
(b)
−
u
(a)
6
f
(x)
du
(x)
−
f
(t)
dt
.
b−a
3
a
a
Proof. The proof follows by the fact that
Z bZ b
Z b Z b
|x − y| dxdy =
|x − y| dx dy
a
a
a
a
Z b Z y
Z b
=
(y − x) dx +
(x − y) dx dy
a
1
=
2
a
Z
a
b
y
1
(y − a)2 + (b − y)2 dy = (b − a)3 .
3
J.E. Pečarić and B. Tepeš
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Theorem 2.4. Let f, u : [a, b] → R be such that u is L−Lipschitzian on [a, b],
and f is a function of bounded variation on [a, b]. If w : [a, b] → R is a positive
weight function, then the following inequality holds:
|T (u, f ; w)| 6 M L
b
_
g 6 WML
a
b
_
f,
a
T (u, f ; w) is defined by (2.2), g : [a, b] → R is the function g (x) =
Rwhere
x
w
(t)
df (t),
a
)
(R b
Rb
w
(t)
(t
−
a)
dt
w
(t)
(b
−
t)
dt
a
, a Rb
W = supx∈[a,b] w (x) , M = max
,
Rb
w
(t)
dt
w (t) dt
a
a
and
Wb
a g and
J.E. Pečarić and B. Tepeš
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Wb
a f denote the total variation of g and f on [a, b], respectively.
Proof. We have
T (u, f ; w)
Z b
=
w (x) u (x) df (x) − R b
a
Rb
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Z
b
Z
w (x) df (x)
w (x) dx a
!
b
w
(t)
u
(t)
dt
=
w (x) u (x) − a R b
df (x)
w
(t)
dt
a
a
!
Z b Rb
w
(t)
(u
(x)
−
u
(t))
dt
a
=
w (x) df (x).
Rb
w
(t)
dt
a
a
a
Z
On Grüss Type Inequalities of
Dragomir and Fedotov
b
w (x) u (x) dx
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Using the fact that u is L−Lipschitzian on [a, b], we can state that:
Z
!
b R b w (t) (u (x) − u (t)) dt
a
w
(x)
df
(x)
|T (u, f ; w)| = Rb
a
w
(t)
dt
a
Z
!
Z x
b R b w (t) (u (x) − u (t)) dt
a
d
w (t) df (t) =
Rb
a
w (t) dt
a
a
!
Rb
b Z x
w (t) |x − t| dt _
a
6 Lsupx∈[a,b]
w (t) df (t)
Rb
w
(t)
dt
a
a
a
= ML
b
_
On Grüss Type Inequalities of
Dragomir and Fedotov
J.E. Pečarić and B. Tepeš
g.
a
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The constant M has the value
Contents
Rb
M = supx∈[a,b]
a
w (t) |x − t| dt
Rb
w (t) dt
a
!
.
If we denote a new function y(x) as:
Z
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w (t) |x − t| dt
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a
Z
x
Z
w (t) (x − t) dt +
=
a
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y (x) =
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w (t) (t − x) dt,
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then the first derivative of this function is:
Z x
Z x
Z b
Z b
dy
d
=
x
w (t) tdt −
tw (t) dt +
w (t) tdt − x
w (t) dt
dx
dx
a
a
x
x
Z
x
Z
b
w (t) dt + w (x) x − w (x) x − w (x) x −
=
a
Z
x
Z
w (t) dt −
=
w (t) dt + w (x) x
x
a
b
On Grüss Type Inequalities of
Dragomir and Fedotov
w (t) dt;
x
J.E. Pečarić and B. Tepeš
and the second derivative is:
d2 y
= w (x) + w (x) = 2w (x) > 0.
dx2
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Obviously f is a convex function, so we have:
Rb
M = supx∈[a,b]
= supx∈[a,b]
a
y(x)
Rb
a
(R b
= max
a
w (t) |x − t| dt
Rb
w (t) dt
a
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w (t) dt
w (t) (b − t) dt
,
Rb
w
(t)
dt
a
II
I
Rb
a
w (t) (t − a) dt
Rb
w (t) dt
a
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)
.
J. Ineq. Pure and Appl. Math. 4(5) Art. 91, 2003
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That is:
Z
!
b R b w (t) (u (x) − u (t)) dt
a
|T (u, f ; w)| = w (x) df (x)
Rb
a
w (t) dt
a
Z
!
b R b w (t) (u (x) − u (t)) dt
a
=
w (x) df (x)
Rb
a
w (t) dt
a
Z b Rb
w (t) |u (x) − u (t)| dt
a
6
w (x) |df (x)|
Rb
w
(t)
dt
a
a
! b
Rb
_
w
(t)
|x
−
t|
dt
a
6 supx∈[a,b] w (x) Lsupx∈[a,b]
f
Rb
w (t) dt
a
a
= WML
b
_
On Grüss Type Inequalities of
Dragomir and Fedotov
J.E. Pečarić and B. Tepeš
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References
[1] S.S. DRAGOMIR AND I. FEDOTOV, An inequality of Grüss type for
Riemann-Stieltjes integral and applications for special means, Tamkang J.
of Math., 29(4) (1998), 287–292.
[2] S.S. DRAGOMIR AND I. FEDOTOV, A Grüss type inequality for mapping of bounded variation and applications to numerical analysis integral
and applications for special means, RGMIA Res. Rep. Coll., 2(4) (1999).
[ONLINE: http://rgmia.vu.edu.au/v2n4.html].
[3] S.S. DRAGOMIR, New inequalities of Grüss type for the Stieltjes integral, RGMIA Res. Rep. Coll., 5(4) (2002), Article 3. [ONLINE: http:
//rgmia.vu.edu.au/v5n4.html].
On Grüss Type Inequalities of
Dragomir and Fedotov
J.E. Pečarić and B. Tepeš
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