Journal of Inequalities in Pure and
Applied Mathematics
ON MULTIDIMENSIONAL GRÜSS TYPE INEQUALITIES
B.G. PACHPATTE
57, Shri Niketen Colony
Aurangabad - 431 001,
(Maharashtra) India.
EMail: bgpachpatte@hotmail.com
volume 3, issue 2, article 27,
2002.
Received 14 August, 2001.;
accepted 25 January, 2002.
Communicated by: R.N. Mohapatra
Abstract
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Abstract
In this paper we establish some new multidimensional integral inequalities of
the Grüss type by using a fairly elementary analysis.
2000 Mathematics Subject Classification: 26D15, 26D20.
Key words: Multidimensional, Grüss type inequalities, Partial Derivatives, Continuous Functions, Bounded, Identities.
Contents
On Multidimensional Grüss
Type Inequalities
B.G. Pachpatte
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4
Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5
Proof of Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
References
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1.
Introduction
The following inequality is well known in the literature as the Grüss inequality
[2] (see also [4, p. 296]):
Z b
Z b
Z b
1
1
1
f
(x)
g
(x)
dx
−
f
(x)
dx
g
(x)
dx
b − a
b−a
b−a
a
a
a
≤ (M − m) (N − n) ,
provided that f, g : [a, b] → R are integrable on [a, b] and
m ≤ f (x) ≤ M, n ≤ g (x) ≤ N,
for all x ∈ [a, b] , where m, M, n, N are given constants.
Since the appearance of the above inequality in 1935, it has evoked considerable interest and many variants, generalizations and extensions have appeared,
see [1, 4] and the references cited therein. The main purpose of the present
paper is to establish some new integral 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.
On Multidimensional Grüss
Type Inequalities
B.G. Pachpatte
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2.
Statement of Results
In what follows, R denotes the set of real numbers. Let ∆ = [a, b] × [c, d],
∆0 = (a, b) × (c, d), Ω = [a, k] × [b, m] × [c, n], Ω0 = (a, k) × (b, m) × (c, n)
for a, b, c, d, k, m, n in R. For functions α and β defined respectively on ∆ and
2 α(x,y)
3 β(x,y,z)
Ω, the partial derivatives ∂ ∂y∂x
and ∂ ∂z∂y∂x
are denoted by D2 D1 α (x, y) and
D3 D2 D1 β (x, y, z) . Let
D = {x = (x1 , . . . , xn ) : ai < xi < bi (i = 1, 2, . . . , n)}
and D̄ be the closure of D. For a function e defined on D̄ the partial derivatives
∂e(x)
(i = 1, 2, . . . , n) are denoted by Di e (x) .
∂xi
First, we give the following notations used to simplify the details of presentation.
c
x
Z
y
−
Contents
d
D2 D1 f (s, t) dtds
a
Z bZ
y
Z bZ
D2 D1 f (s, t) dtds +
x
c
d
D2 D1 f (s, t) dtds,
x
y
B(D3 D2 D1 f (r, s, t))
= B [a, b, c; r, s, t; k, m, n; D3 D2 D1 f (u, v, w)]
Z rZ sZ t
=
D3 D2 D1 f (u, v, w) dwdvdu
a
b
c
Z rZ sZ n
−
D3 D2 D1 f (u, v, w) dwdvdu
a
b
t
B.G. Pachpatte
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A (D2 D1 f (x, y)) = A [a, c; x, y; b, d; D2 D1 f (s, t)]
Z xZ y
Z
=
D2 D1 f (s, t) dtds −
a
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Z rZ
m
t
Z
−
D3 D2 D1 f (u, v, w) dwdvdu
Z kZ sZ t
−
D3 D2 D1 f (u, v, w) dwdvdu
r
b
c
Z rZ mZ n
+
D3 D2 D1 f (u, v, w) dwdvdu
a
s
t
Z kZ mZ t
+
D3 D2 D1 f (u, v, w) dwdvdu
r
s
c
Z kZ sZ n
+
D3 D2 D1 f (u, v, w) dwdvdu
r
b
t
Z kZ mZ n
−
D3 D2 D1 f (u, v, w) dwdvdu,
a
s
c
r
s
t
On Multidimensional Grüss
Type Inequalities
B.G. Pachpatte
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E (f (x, y)) = E [a, c; x, y; b, d; f ]
1
= [f (x, c) + f (x, d) + f (a, y) + f (b, y)]
2
1
− [f (a, c) + f (a, d) + f (b, c) + f (b, d)] ,
4
L (f (r, s, t)) = L [a, b, c; r, s, t; k, m, n; f ]
1
= [f (a, b, c) + f (k, m, n)]
8
1
− [f (r, b, c) + f (r, m, n) + f (r, m, c) + f (r, b, n)]
4
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1
[f (a, s, c) + f (k, s, n) + f (a, s, n) + f (k, s, c)]
4
1
− [f (a, b, t) + f (k, m, t) + f (k, b, t) + f (a, m, t)]
4
1
1
+ [f (a, s, t) + f (k, s, t)] + [f (r, b, t) + f (r, m, t)]
2
2
1
+ [f (r, s, c) + f (r, s, n)] .
2
−
Our main results are given in the following theorems.
Theorem 2.1. Let f, g : ∆ → R be continuous functions on ∆; D2 D1 f (x, y),
D2 D1 g (x, y) exist on ∆0 and are bounded, i.e.
kD2 D1 f k∞ =
kD2 D1 gk∞ =
sup |D2 D1 f (x, y)| < ∞,
(x,y)∈∆0
sup |D2 D1 g (x, y)| < ∞.
(x,y)∈∆0
Then
(2.1)
Z b Z
On Multidimensional Grüss
Type Inequalities
B.G. Pachpatte
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d
f (x, y) g (x, y) dydx
a
c
Z Z
1 b d
[E (f (x, y)) g (x, y) + E (g (x, y)) f (x, y)] dydx
−
2 a c
Z bZ d
1
≤ (b − a) (d − c)
(|g (x, y)| kD2 D1 f k∞
8
a
c
+ |f (x, y)| kD2 D1 gk∞ ) dydx.
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Theorem 2.2. Let p, q : Ω → R be continuous functions on Ω; D3 D2 D1 p (r, s, t),
D3 D2 D1 q (r, s, t) exist on Ω0 and are bounded, i.e.
kD3 D2 D1 pk∞ =
kD3 D2 D1 qk∞ =
|D3 D2 D1 p (r, s, t)| < ∞,
sup
(r,s,t)∈Ω0
|D3 D2 D1 q (r, s, t)| < ∞.
sup
(r,s,t)∈Ω0
Then
(2.2)
Z
a
k
Z
b
m
n
Z
c
p (r, s, t) q (r, s, t) dtdsdr
Z Z Z
1 k m n
−
[L (p (r, s, t)) q (r, s, t)
2 a b
c
+ L (q (r, s, t)) p (r, s, t)]dtdsdr
1
≤
(k − a) (m − b) (n − c)
16
Z kZ mZ n
×
(|q (r, s, t)| kD3 D2 D1 pk∞
a
b
c
+ |p (r, s, t)| kD3 D2 D1 qk∞ )dtdsdr.
Theorem 2.3. Let f, g : Rn → R be continuous functions on D̄ and differentiable on D whose derivatives Di f (x), Di g (x) are bounded, i.e.,
kDi f k∞ = sup |Di f (x)| < ∞,
On Multidimensional Grüss
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B.G. Pachpatte
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x∈D
kDi gk∞ = sup |Di g (x)| < ∞.
x∈D
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Then
(2.3)
Z
Z
Z
1
1
1
f (x) g (x) dx −
f (x) dx
g (x) dx M
M D
M D
D
Z X
n
1
(|g (x)| kDi f k∞ + |f (x)| kDi gk∞ ) Ei (x) dx,
≤
2M 2 D i=1
where
M =
n
Y
On Multidimensional Grüss
Type Inequalities
Z
(bi − ai ) , Ei (x) =
i=1
dx = dx1 · · · dxn ,
|xi − yi | dy,
D
B.G. Pachpatte
dy = dy1 · · · dyn .
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3.
Proof of Theorem 2.1
From the hypotheses we have the following identities (see [6]):
(3.1)
(3.2)
1
f (x, y) = E (f (x, y)) + A (D2 D1 f (x, y)) ,
4
1
g (x, y) = E (g (x, y)) + A (D2 D1 g (x, y)) ,
4
for (x, y) ∈ ∆. Multiplying (3.1) by g (x, y) and (3.2) by f (x, y) and adding
the resulting identities, then integrating on ∆ and rewriting we have
Z bZ d
(3.3)
f (x, y) g (x, y) dydx
a
c
Z Z
1 b d
=
(E (f (x, y)) g (x, y) + E (g (x, y)) f (x, y)) dydx
2 a c
Z Z
1 b d
+
(A (D2 D1 f (x, y)) g (x, y)
8 a c
+ A (D2 D1 g (x, y)) f (x, y)) dydx.
From the properties of modulus and integrals, it is easy to see that
Z bZ d
(3.4)
|A (D2 D1 f (x, y))| ≤
|D2 D1 f (s, t)| dtds,
a
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Type Inequalities
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Z bZ
(3.5)
|A (D2 D1 g (x, y))| ≤
d
|D2 D1 g (s, t)| dtds.
a
c
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From (3.3) – (3.5) we observe that
Z b Z d
f (x, y) g (x, y) dydx
a
c
Z Z
1 b d
−
(E (f (x, y)) g (x, y) + E (g (x, y)) f (x, y)) dydx
2 a c
Z bZ d
1
≤
(|g (x, y)| |A (D2 D1 f (x, y))|
8 a c
+ |f (x, y)| |A (D2 D1 g (x, y))|) dydx
Z bZ d
Z bZ d
1
≤
|g (x, y)|
|D2 D1 f (s, t)| dtds
8 a c
a
c
Z bZ d
+ |f (x, y)|
|D2 D1 g (s, t)| dtds dydx
a
c
Z bZ d
1
(|g (x, y)| kD2 D1 f k∞
≤ (b − a) (d − c)
8
a
c
+ |f (x, y)| kD2 D1 gk∞ ) dydx,
which is the required inequality in (2.1). The proof is complete.
On Multidimensional Grüss
Type Inequalities
B.G. Pachpatte
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4.
Proof of Theorem 2.2
From the hypotheses we have the following identities (see [5]):
(4.1)
(4.2)
1
p (r, s, t) = L (p (r, s, t)) + B (D3 D2 D1 p (r, s, t)) ,
8
1
q (r, s, t) = L (q (r, s, t)) + B (D3 D2 D1 q (r, s, t)) ,
8
for (r, s, t) ∈ Ω. Multiplying (4.1) by q (r, s, t) and (4.2) by p (r, s, t) and
adding the resulting identities, then integrating on Ω and rewriting we have
Z kZ mZ n
(4.3)
p (r, s, t) q (r, s, t) dtdsdr
a
b
c
Z Z Z
1 k m n
=
[L (p (r, s, t)) q (r, s, t)
2 a b
c
+ L (q (r, s, t)) p (r, s, t)]dtdsdr
Z kZ mZ n
1
+
(B (D3 D2 D1 p (r, s, t)) q (r, s, t)
16 a b
c
+ B (D3 D2 D1 q (r, s, t)) p (r, s, t)) dtdsdr.
From the properties of modulus and integrals, we observe that
Z kZ mZ n
(4.4) |B (D3 D2 D1 p (r, s, t))| ≤
|D3 D2 D1 p (u, v, w)| dwdvdu,
a
b
c
Z kZ mZ n
(4.5) |B (D3 D2 D1 q (r, s, t))| ≤
|D3 D2 D1 q (u, v, w)| dwdvdu.
a
b
c
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Now, from (4.3) – (4.5) and following the same arguments as in the proof of
Theorem 2.1 with suitable changes, we get the required inequality in (2.2).
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B.G. Pachpatte
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5.
Proof of Theorem 2.3
Let x ∈ D̄, y ∈ D. From the n−dimensional version of the mean value theorem, we have (see [3]):
f (x) − f (y) =
(5.1)
n
X
Di f (c) (xi − yi ) ,
i=1
where c = (y1 + δ (x1 − y1 ) , . . . , yn + δ (xn − yn )) , 0 < δ < 1.
Integrating (5.1) with respect to y, we obtain
Z
n Z
X
(5.2)
f (x) mesD =
f (y) dy +
Di f (c) (xi − yi ) dy,
D
Qn
i=1
D
i=1
B.G. Pachpatte
D
(bi − ai ) = M. Similarly, we obtain
Z
n Z
X
g (y) dy +
Di g (c) (xi − yi ) dy.
g (x) mesD =
where mesD =
(5.3)
i=1
On Multidimensional Grüss
Type Inequalities
D
Multiplying (5.2) by g (x) and (5.3) by f (x) and adding the resulting identities,
then integrating on D and noting that mes > 0, we have
Z
(5.4)
f (x) g (x) dx
D
Z
Z
1
=
f (x) dx
g (x) dx
M
D
D
"Z
!
n Z
X
1
+
g (x)
Di f (c) (xi − yi ) dy dx
2M
D
i=1 D
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Z
+
f (x)
D
n Z
X
i=1
!
#
Di g (c) (xi − yi ) dy dx .
D
From (5.4) we observe that
Z
Z
Z
1
1
1
f
(x)
g
(x)
dx
−
f
(x)
dx
g
(x)
dx
M
M D
M D
D
Z X
n
1
≤
(|g (x)| kDi f k∞ + |f (x)| kDi gk∞ ) Ei (x) dx.
2M 2 D i=1
This is the required inequality in (2.3). The proof is complete.
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Type Inequalities
B.G. Pachpatte
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References
[1] S.S. DRAGOMIR, Some integral inequalities of Grüss type, Indian J. Pure
& Appl. Math., 31(4) (2000), 379–415. ONLINE: RGMIA Research Report
Collection, 1(2) (1998), 97–113.
[2] 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.
[3] G.V. MILOVANOVIĆ, On some integral inequalities, Univ. Beograd Publ.
Elek. Fak. Ser. Mat. Fiz., No498–No541 (1975), 112–124.
[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 an inequality of Ostrowski type in three independent
variables, J. Math. Anal. Appl., 249 (2000), 583–591.
[6] B.G. PACHPATTE, On a new Ostrowski type inequality in two independent
variables, Tamkang J. Math., 32 (2001), 45–49.
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Type Inequalities
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