Journal of Inequalities in Pure and
Applied Mathematics
NEW WEIGHTED MULTIVARIATE GRÜSS TYPE INEQUALITIES
B.G. PACHPATTE
57 Shri Niketan Colony,
Near Abhinay Talkies,
Aurangabad 431 001
(Maharashtra) India.
EMail: bgpachpatte@hotmail.com
volume 4, issue 5, article 108,
2003.
Received 09 October, 2002;
accepted 15 December, 2003.
Communicated by: P. Cerone
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
104-02
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Abstract
In this paper we establish some new weighted multidimensional Grüss type
integral and discrete inequalities by using a fairly elementary analysis .
2000 Mathematics Subject Classification: 26D15 , 26D20.
Key words: Multivariate Grüss type inequalities, Discrete inequalities, New estimates, Differentiable function, Partial derivatives, Forward difference operators, Mean value theorem .
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4
Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
References
New Weighted Multivariate
Grüss Type Inequalities
B.G. Pachpatte
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1.
Introduction
In 1935, G. Grüss [3] proved the following classical integral inequality (see,
also [4, p. 296]):
Z b
Z b
Z b
1
1
1
f
(x)
g
(x)
−
f
(x)
dx
g
(x)
dx
b − a
b−a
b−a
a
a
a
1
≤ (P − p) (Q − q) ,
4
New Weighted Multivariate
Grüss Type Inequalities
provided that f and g are two integrable functions on [a, b] such that
B.G. Pachpatte
p ≤ f (x) ≤ P,
q ≤ g (x) ≤ Q,
for all x ∈ [a, b], where p, P, q, Q are constants.
A large number of generalizations, extensions and variants of this inequality
have been given by several authors since its discovery, see [1, 2], [4] – [6] and
the references given therein. The main purpose of this paper is to establish new
weighted integral and discrete inequalities of the Grüss type involving functions
of several independent variables. The analysis used in the proofs is elementary
and our results provide new estimates on inequalities of this type.
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2.
Statement of Results
In what follows, R and N denote the set of real and natural numbers respectively.
Let Di [a, b] = {xi : ai < xi < bi } for i = 1, . . . , n, ai , bi ∈ R, D =
n
Q
Di [ai , bi ] and D̄ be the closure of D. For a differentiable function u (x) :
i=1
D̄ → R, we denote the first order partial derivatives by ∂u(x)
(i = 1, . . . , n) and
∂xi
R
u (x) dx the n-fold integral
D
Z b1
Z bn
···
u (x1 , . . . , xn ) dx1 . . . dxn .
a1
If
an
B.G. Pachpatte
∂u ∂u (x) ∂xi = sup
∂xi < ∞,
x∈D
∞
∂u(x)
∂xi
are bounded. Let Ni [0, ai ] =
n
Q
{0, 1, 2, . . . , ai } , ai ∈ N, (i = 1, . . . , n) and B =
Ni [0, ai ]. For a functhen we say that the partial derivatives
i=1
tion z (x) : B → R we define the first order forward difference operators as
∆1 z (x) = z (x1 + 1, x2 , . . . , xn ) − z (x) , . . . , ∆n z (x)
= z (x1 , . . . , xn−1 , xn + 1) − z (x)
and denote the n-fold sum over B with respect to the variable y = (y1 , . . . , yn ) ∈
B by
aX
aX
n −1
1 −1
X
z (y) =
···
z (y1 , . . . , yn ) .
y
New Weighted Multivariate
Grüss Type Inequalities
y1 =0
yn =0
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Clearly,
X
y
z (y) =
X
z (x)
for x, y ∈ B.
x
If k∆i zk∞ = sup |∆i z (x)| < ∞, then we say that the partial differences
x∈B
∆i z (x) are bounded. The notation
x
i −1
X
∆i z (y1 , . . . , yi−1 , ti , xi+1 , . . . , xn ) , xi , yi ∈ Ni [0, ai ] (i = 1, . . . , n) ,
ti =yi
P −1
we mean for i = 1 it is xt11=y
∆1 z (t1 , x2 , . . . , xn ) and so on, and for i = n it is
1
Pxn −1
tn =yn ∆n ×z (y1 , . . . , yn−1 , tn ). We use the usual convention that the empty
sum is taken to be zero. We use the following notations to simplify the details
of presentation.
For continuous functions p, q defined on D̄ and differentiable on D,
R w (x) a
real-valued nonnegative and integrable function for every x ∈ D with D w (x) dx
> 0 and xi , yi ∈ Di [ai , bi ], we set
Z
A [w, p, q] =
w (x) p (x) q (x) dx
D
Z
Z
1
w (x) p (x) dx
w (x) q (x) dx ,
−R
w (x) dx
D
D
D
H [p, xi , yi ] =
New Weighted Multivariate
Grüss Type Inequalities
B.G. Pachpatte
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n X
i=1
∂p ∂xi |xi − yi | .
∞
J. Ineq. Pure and Appl. Math. 4(5) Art. 108, 2003
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For the functions p, q : B → R whose forward differences
P ∆i p, ∆i q exist,
w (x) a real-valued nonnegative function defined on B and
w (x) > 0 and
x
xi , yi ∈ Ni [0, ai ], we set
L [w, p, q]
X
1
=
w (x) p (x) q (x) − P
w (x)
x
x
M [p, xi , yi ] =
n
X
!
X
w (x) p (x)
x
!
X
w (x) q (x) ,
x
New Weighted Multivariate
Grüss Type Inequalities
k∆i pk∞ |xi − yi | .
B.G. Pachpatte
i=1
Our main results on weighted Grüss type integral inequalities involving functions of many independent variables are embodied in the following theorem.
Theorem 2.1. Let f, g be real-valued continuous functions on D̄ and differen∂f ∂g
tiable on D whose derivatives ∂x
,
are bounded. Let w (x) be a real-valued,
i ∂xi
R
nonnegative and integrable function for x ∈ D and D w (x) dx > 0. Then
(2.1)
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|A [w, f, g]|
1
≤ R
2 D w (x) dx
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Z
w (x) |g (x)|
H [f, xi , yi ] w (y) dy
D
D
Z
+ |f (x)|
H [g, xi , yi ] w (y) dy dx,
Z
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D
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(2.2)
|A [w, f, g]|
≤ R
D
w (x) dx
Z
Z
1
2
H [f, xi , yi ] w (y) dy
w (x)
D
D
Z
×
H [g, xi , yi ] w (y) dy dx.
D
Remark 2.1. If we take n = 1 and D = I = {a < x < b} in (2.1), then we get
Z
Z b
Z b
b
1
w (t) f (t) g (t) dt − R b
w (t) f (t) dt
w (t) g (t) dt a
w (t) dt
a
a
a
Z b
Z b
1
≤ Rb
w (t) |g (t)|
kf 0 k∞ |t − s| w (s) ds
2 a w (t) dt a
a
Z b
0
+ |f (t)|
kg k∞ |t − s| w (s) ds dt.
a
Similarly, one can obtain the special version of (2.2). It is easy to see that the
upper bound given on the right side in the above inequality (when w(t) = 1) is
different from those given by Grüss in [3].
New Weighted Multivariate
Grüss Type Inequalities
B.G. Pachpatte
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The next theorem deals with the discrete versions of the inequalities in Theorem 2.1.
Theorem 2.2. Let f, g be real-valued functions defined on B and ∆i f, ∆i g are
bounded. Let w (x) be a real-valued nonnegative function defined on B and
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P
w (x) > 0. Then
x
(2.3)
"
X
X
1
|L [w, f, g]| ≤ P
w (x) |g (x)|
M [f, xi , yi ]w (y)
2 w (x) x
y
x
#
+ |f (x)|
X
M [g, xi , yi ]w (y) ,
y
New Weighted Multivariate
Grüss Type Inequalities
(2.4)
|L (w, f, g)| ≤ 1
2
P
w (x)
X
B.G. Pachpatte
w (x)
x
Title Page
x
!
×
X
y
M [f, xi , yi ] w (y)
!
X
M [g, xi , yi ] w (y) .
y
Remark 2.2. In a recent paper [6] the author gave multidimensional Grüss
type finite difference inequalities whose proofs were based on a certain finite
difference identity. Here we note that the inequalities established in (2.3) and
(2.4) are of more general type and can be considered as the weighted generalizations of the similar inequalities given in [6].
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3.
Proof of Theorem 2.1
Let x = (x1 , . . . , xn ) ∈ D̄, y = (y1 , . . . , yn ) ∈ D. From the n-dimensional
version of the mean value theorem we have (see [7, p. 174])
(3.1)
f (x) − f (y) =
n
X
∂f (c)
i=1
∂xi
(xi − yi )
and
(3.2)
g (x) − g (y) =
n
X
∂g (c)
i=1
∂xi
(xi − yi ) ,
B.G. Pachpatte
where c = (y1 + α (x1 − y1 ) , . . . , yn + α (xn − yn )) (0 < α < 1). Multiplying both sides of (3.1) and (3.2) by g(x) and f (x) respectively and adding we
get
(3.3) 2f (x) g (x) − g (x) f (y) − f (x) g (y)
n
n
X
X
∂f (c)
∂g (c)
= g (x)
(xi − yi ) + f (x)
(xi − yi ) .
∂xi
∂xi
i=1
i=1
Multiplying both sides of (3.3) by w (y) and integrating the resulting identity
with respect to y over D we have
Z
w (y) dy f (x) g (x)
(3.4) 2
D
Z
Z
− g (x)
w (y) f (y) dy − f (x)
w (y) g (y) dy
D
New Weighted Multivariate
Grüss Type Inequalities
D
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= g (x)
Z X
n
∂f (c)
D i=1
∂xi
(xi − yi ) w (y) dy+f (x)
Z X
n
∂g (c)
D i=1
∂xi
(xi − yi ) w (y) dy.
Next, multiplying both sides of (3.4) by w (x) and integrating the resulting identity with respect to x on D we get
Z
Z
(3.5) 2
w (y) dy
w (x) f (x) g (x) dx
D
D
Z
Z
−
w (x) g (x) dx
w (y) f (y) dy
D
D
Z
Z
−
w (x) f (x) dx
w (y) g (y) dy
D
D
!
Z
Z X
n
∂f (c)
=
w (x) g (x)
(xi − yi ) w (y) dy dx
D
D i=1 ∂xi
!
Z
Z X
n
∂g (c)
+
w (x) f (x)
(xi − yi ) w (y) dy dx.
D
D i=1 ∂xi
From (3.5) and using the properties of modulus we have
|A [w, f, g]|
New Weighted Multivariate
Grüss Type Inequalities
B.G. Pachpatte
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!
Z X
n ∂f (c) 1
w (x) |g (x)|
≤ R
∂xi |xi − yi | w (y) dy dx
2 D w (x) dx D
D i=1
! #
Z
Z X
n ∂g (c) +
w (x) |f (x)|
∂xi |xi − yi | w (y) dy dx
D
D i=1
Z
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"
Z X
n ∂f |xi − yi | w (y) dy
w (x) |g (x)|
D
D i=1 ∂xi ∞
#
Z X
n ∂g |xi − yi | w (y) dy dx
+ |f (x)|
D i=1 ∂xi ∞
Z
Z
1
= R
w (x) |g (x)|
H [f, xi , yi ] w (y) dy
2 D w (x) dx D
D
Z
+ |f (x)|
H [g, xi , yi ] w (y) dy dx.
1
≤ R
2 D w (x) dx
Z
D
This is the required inequality in (2.1).
Multiplying both sides of (3.1) and (3.2) by w (y) and integrating the resulting identities with respect to y on D we get
Z
Z
w (y) dy f (x) −
w (y) f (y) dy
(3.6)
D
D
=
Z X
n
∂f (c)
∂xi
D i=1
(xi − yi ) w (y) dy
New Weighted Multivariate
Grüss Type Inequalities
B.G. Pachpatte
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Z
w (y) dy g (x) −
(3.7)
D
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Z
w (y) g (y) dy
Page 11 of 18
D
=
Z X
n
∂g (c)
D i=1
∂xi
(xi − yi ) w (y) dy.
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Multiplying the left sides and right sides of (3.6) and (3.7) we get
Z
2
Z
Z
w (y) dy f (x)
w (y) g (y) dy
w (y) dy f (x) g (x) −
(3.8)
D
D
D
Z
Z
−
w (y) dy g (x)
w (y) f (y) dy
D
D
Z
Z
+
w (y) f (y) dy
w (y) g (y) dy
D
D
!
!
Z X
Z X
n
n
∂f (c)
∂g (c)
=
(xi − yi ) w (y) dy
(xi − yi ) w (y) dy .
D i=1 ∂xi
D i=1 ∂xi
Multiplying both sides of (3.8) by w (x) and integrating the resulting identity
with respect to x on D, by simple calculations we obtain
Z
(3.9)
w (x) f (x) g (x) dx
D
Z
Z
1
−R
w (x) f (x) dx
w (x) g (x) dx
w (y) dy
D
D
D
!
Z
Z X
n
1
∂f (c)
= R
(xi − yi ) w (y) dy
w (x)
2
D
D i=1 ∂xi
w
(y)
dy
D
!
Z X
n
∂g (c)
(xi − yi ) w (y) dy dx.
×
D i=1 ∂xi
From (3.9) and following the proof of the inequality (2.1) with suitable modifications we get the required inequality in (2.2). The proof is complete.
New Weighted Multivariate
Grüss Type Inequalities
B.G. Pachpatte
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Remark 3.1. Multiplying the left sides and right sides of (3.1) and (3.2), then
multiplying the resulting identity by w (y), integrating it with respect to y on D,
again multiplying the resulting identity by w (x), integrating it with respect to x
over D and following the similar arguments as in the proofs of (2.1), (2.2) we
have
(3.10)
|A [w, f, g]|
Z
Z
1
≤ R
w (x)
H [f, xi , yi ] H [g, xi , yi ] w (y) dy dx.
2 D w (x) dx D
D
New Weighted Multivariate
Grüss Type Inequalities
B.G. Pachpatte
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4.
Proof of Theorem 2.2
For x = (x1 , . . . , xn ) , y = (y1 , . . . , yn ) in B, it is easy to observe that the
following identities hold (see [6]):
(4.1)
f (x) − f (y) =
n
X
( x −1
i
X
i=1
ti =yi
)
∆i f (y1 , . . . , yi−1 , ti , xi+1 , . . . , xn )
New Weighted Multivariate
Grüss Type Inequalities
and
(4.2)
g (x) − g (y) =
n
X
( x −1
i
X
i=1
ti =yi
)
B.G. Pachpatte
∆i g (y1 , . . . , yi−1 , ti , xi+1 , . . . , xn ) .
Title Page
Multiplying both sides of (4.1) and (4.2) by g (x) and f (x) respectively, and
adding we obtain
(4.3) 2f (x) g (x) − g (x) f (y) − f (x) g (y)
( x −1
)
n
i
X
X
= g (x)
∆i f (y1 , . . . , yi−1 , ti , xi+1 , . . . , xn )
i=1
II
I
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( x −1
n
i
X
X
i=1
JJ
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ti =yi
+ f (x)
Contents
ti =yi
)
∆i g (y1 , . . . , yi−1 , ti , xi+1 , . . . , xn ) .
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Multiplying both sides of (4.3) by w (y) and summing both sides of the resulting
identity with respect to y over B, we have
(4.4) 2
X
w (y) f (x) g (x) − g (x)
y
X
w (y) f (y) − f (x)
X
y
= g (x)
X
n
X
( x −1
i
X
y
i=1
ti =yi
+ f (x)
n
X X
y
w (y) g (y)
y
)!
∆i f (y1 , . . . , yi−1 , ti , xi+1 , . . . , xn )
( x −1
i
X
w (y)
)!
∆i g (y1 , . . . , yi−1 , ti , xi+1 , . . . , xn )
New Weighted Multivariate
Grüss Type Inequalities
w (y) .
ti =yi
i=1
B.G. Pachpatte
Now, multiplying both sides of (4.4) by w (x) and summing the resulting identity with respect to x on B we have
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!
(4.5) 2
X
X
w (y)
y
x
!
−
X
!
X
w (x) g (x)
x
=
w (x) g (x)
y
"
X
X
+
w (x) f (x)
y
!
X
x
n
X X
x
x
w (y) f (y) −
y
"
X
JJ
J
w (x) f (x) g (x)
i=1
( x −1
i
X
w (x) f (x)
!
X
w (y) g (y)
Go Back
y
)!
∆i f (y1 , . . . , yi−1 , ti , xi+1 , . . . , xn )
#
w (y)
ti =yi
n
X
( x −1
i
X
i=1
ti =yi
)!
∆i g (y1 , . . . , yi−1 , ti , xi+1 , . . . , xn )
II
I
#
w (y) .
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From (4.5) and using the properties of modulus we have
|L (w, f, g)| |
"
X
1
≤ P
w (x) |g (x)|
2 w (x) x
x
( −1
)!
n x
i
X X
X
×
|∆i f (y1 , . . . , yi−1 , ti , xi+1 , . . . , xn )| w (y)
y
ti =yi
i=1
X
+
w (x) |f (x)|
New Weighted Multivariate
Grüss Type Inequalities
B.G. Pachpatte
x
( −1
)!
#
n x
i
X X
X
×
|∆i g (y1 , . . . , yi−1 , ti , xi+1 , . . . , xn )| w (y)
y
ti =yi
i=1
(
)!
"
x
n i −1
X
X X
X
1
w (x) |g (x)|
1 w (y)
≤ P
k∆i f k∞
2 w (x) x
ti =yi
y
i=1
x
!
#
n
X X
+ |f (x)|
k∆i gk∞ |xi − yi | w (y)
y
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i=1
"
!
n
X
X
X X
1
= P
w (x)
w (x) |g (x)|
k∆i f k∞ |xi − yi | w (y)
2 w (x) x
x
y
i=1
x
!
#
n
X X
+ |f (x)|
k∆i gk∞ |xi − yi | w (y)
y
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"
X
X
1
= P
w (x) |g (x)|
M [f, xi , yi ] w(y)
2 w (x) x
y
x
#
X
+ |f (x)|
M [g, xi , yi ] w(y) ,
y
which is the required inequality in (2.3).
The proof of the inequality (2.4) can be completed by following the proof of
(2.3) and closely looking at the proof of (2.2). Here we omit the details.
Remark 4.1. Multiplying the left sides and right sides of (4.1) and (4.2), then
multiplying the resulting identity by w (y), summing it with respect to y over B,
again multiplying the resulting identity by w (x), summing it with respect to x
over B and closely looking at the proof of the inequality (2.3) we get
New Weighted Multivariate
Grüss Type Inequalities
B.G. Pachpatte
Title Page
Contents
(4.6)
|L (w, f, g)|
≤
2
P
x
1
w (x)
!
X
x
w (x)
X
M [f, xi , yi ] M [g, xi , yi ] w (y) .
y
In concluding we note that in [2] Fink has given some Grüss type inequalities for measures other than the Lebesgue measure, including signed measures
which provide different upper bounds. In addition, in [2] new proofs to some
old results are also given. However, the inequalities established here are different and cannot be compared with those of given in [2].
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References
[1] S.S. DRAGOMIR, Some integral inequalities of Grüss type, Indian J. Pure
and Appl. Math., 31 (2000), 379–415.
[2] A.M. FINK, A treatise on Grüss inequality, Analytic and Geometric Inequalities and Applications, T.M. Rassias and H.M. Srivastava (eds.),
Kluwer Academic Publishers, Dordrecht 1999, 93–113.
[3] G. RGRÜSS, Über das maximum
des
Rb
R b absoluten Betrages von
b
1
1
f (x) g (x) dx − (b−a)2 a f (x) dx a g (x) dx, Math. Z., 39
b−a a
(1935), 215–226.
[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, On multidimensional Grüss type inequalities, J. Inequal. Pure and Appl. Math., 3(2) (2002), Art. 27. [ONLINE: http:
//jipam.vu.edu.au]
[6] B.G. PACHPATTE, On multidimensional Ostrowski and Grüss type finite
difference inequalities, J. Inequal. Pure and Appl. Math., 4(2) (2003), Art.
7. [ONLINE: http://jipam.vu.edu.au]
[7] W. RUDIN, Principles of Mathematical Analysis, McGraw-Hill Book Company, Inc. 1953.
New Weighted Multivariate
Grüss Type Inequalities
B.G. Pachpatte
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