Journal of Inequalities in Pure and
Applied Mathematics
AN INTEGRAL APPROXIMATION IN THREE VARIABLES
A. SOFO
School of Computer Science and Mathematics
Victoria University of Technology
PO Box 14428, MCMC 8001,
Victoria, Australia.
EMail: sofo@csm.vu.edu.au
URL: http://rgmia.vu.edu.au/sofo
volume 4, issue 3, article 58,
2003.
Received 15 November, 2002;
accepted 25 August, 2003.
Communicated by: C.E.M. Pearce
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
125-02
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Abstract
In this paper we will investigate a method of approximating an integral in three
independent variables. The Ostrowski type inequality is established by the use
of Peano kernels and provides a generalisation of a result given by Pachpatte.
2000 Mathematics Subject Classification: Primary 26D15; Secondary 41A55.
Key words: Ostrowski inequality, Three independent variables, Partial derivatives.
An Integral Approximation in
Three Variables
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
A. Sofo
3
7
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1.
Introduction
The numerical estimation of the integral, or multiple integral of a function over
some specified interval is important in many scientific applications. Generally
speaking, the error bound for the midpoint rule is about one half of the trapezoidal rule and Stewart [14] has a nice geometrical explanation of this generality. The speed of convergence of an integral is also important and Weideman [15] has some pertinent examples illustrating perfect, algebraic, geometric,
super-geometric and sub-geometric convergence for periodic functions.
In particular, we shall establish an Ostrowski type inequality for a triple integral which provides a generalisation or extension of a result given by Pachpatte
[10].
In 1938 Ostrowski [7] obtained a bound for the absolute value of the difference of a function to its average over a finite interval. The following definitions
will be used in this exposition
Z b
1
f (t) dt,
(1.1)
M (f ) :=
b−a a
(1.2)
f (b) + f (a)
IT (f ) :=
2
A. Sofo
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(1.3)
An Integral Approximation in
Three Variables
IM (f ) := f
a+b
2
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.
J. Ineq. Pure and Appl. Math. 4(3) Art. 58, 2003
The Ostrowski result is given by:
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Theorem 1.1. Let f : [a, b] → R be a differentiable mapping on (a, b) whose
derivative f 0 : (a, b) → R is bounded on (a, b) , that is,
kf 0 k∞ := sup |f 0 (t)| < ∞.
t∈(a,b)
Then we have the inequality
(1.4)
|f (x) − M (f )| ≤
2 !
x − a+b
1
2
(b − a) kf 0 k∞
+
4
(b − a)2
An Integral Approximation in
Three Variables
A. Sofo
for all x ∈ [a, b] .
The constant 41 is the best possible.
Improvements of the result (1.4) has also been obtained by Dedić, Matić
and Pearce [2], Pearce, Pečarić, Ujević and
[11], Dragomir [3] and
Varošanec
,
very
recently Guessab and
Sofo [12]. For a symmetrical point x ∈ a, a+b
2
Schmeisser [4] studied the more general quadrature formula
f (x) + f (a + b − x)
M (f ) −
= E (f ; x)
2
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where E (f ; x) is the remainder.
For x = a+b
and f defined on [a, b] with Lipschitz constant M, then
2
|M (f ) − IM (f )| ≤
M (b − a)
.
4
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For x = a, then
M (b − a)
.
4
The following result, which is a generalisation of Theorem 1.1, was given by
Milovanović [6, p. 468] in 1975 concerning a function, f, of several variables.
|M (f ) − IT (f )| ≤
Theorem 1.2. Let f : Rn → R be a differentiable function
defined on D =
∂f {(x1 , . . . , xm ) |ai ≤ xi ≤ bi , (i = 1, . . . , m)} and let ∂x
≤ Mi (Mi > 0,
i
i = 1, . . . , m) in D. Furthermore, let x 7→ p (x) be integrable and p (x) > 0
for every x ∈ D. Then for every x ∈ D, we have the inequality:
R
R
Pm
p (y) f (y) dy i=1 Mi D p (y) |xi − yi | dy
DR
R
(1.5)
≤
.
f (x) −
p (y) dy p (y) dy
D
D
In 2001, Barnett and Dragomir [1] obtained the following Ostrowski type
inequality for double integrals.
Theorem 1.3. Let f : [a, b] × [c, d] → R be continuous on [a, b] × [c, d] ,
∂2f
00
fx,y
= ∂x∂y
exist on (a, b) × (c, d) and is bounded, that is,
00 fs,t :=
∞
sup
(x,y)∈(a,b)×(c,d)
2
∂ f (x, y) ∂x∂y < ∞,
then we have the inequality:
(1.6)
Z b Z
a
c
d
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Three Variables
A. Sofo
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Z
f (s, t) dsdt − (b − a)
d
f (x, t) dt
c
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b
− (d − c)
f (s, y) ds + (d − c) (b − a) f (x, y)
a
"
2 # "
2 #
2
00 (b − a)
(d − c)2
a+b
c+d
fs,t ≤
+ x−
+ y−
∞
4
2
4
2
Z
for all (x, y) ∈ [a, b] × [c, d] .
Pachpatte [8], obtained an inequality in the vein of (1.6) but used elementary
analysis in his proof.
Pachpatte [9] also obtains a discrete version of an inequality with two independent variables. Hanna, Dragomir and Cerone [5] obtained a further complementary result to (1.6) and Sofo [13] further improved the result (1.6).
An Integral Approximation in
Three Variables
A. Sofo
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2.
Triple Integrals
In three independent variables Pachpatte obtains several results. For discrete
variables he obtains a result in [9] and in [10] for continuous variables he obtained the following.
Theorem 2.1. Let ∆ := [a, k] × [b, m] × [c, n] for a, b, c, k, m, n ∈ R+ and
f (r, s, t) be differentiable on ∆. Denote the partial derivatives by D1 f (r, s, t) =
∂3f
∂
∂
∂
f
(r,
s,
t)
;
D
f
(r,
s,
t)
=
,
D
f
(r,
s,
t)
=
and
D
D
D
f
=
. Let
2
3
3
2
1
∂r
∂s
∂t
∂t∂s∂r
F (∆) be the clan of continuous functions f : ∆ → R for which D1 f, D2 f,
D3 f, D3 D2 D1 f exist and are continuous on ∆. For f ∈ F (∆) we have
Z k Z m Z n
(2.1) f (r, s, t) dtdsdr
a
b
c
1
− (k − a) (m − b) (n − c) [f (a, b, c) + f (k, m, n)]
8
Z k
1
+ (m−b)(n−c) [f (r, b, c)+f (r, m, n)+f (r, m, c)+f (r, b, n)]dr
4
Z am
1
+ (k−a)(n−c) [f (a, s, c)+f (k, s, n)+f (a, s, n)+f (k, s, c)]ds
4
Zb n
1
+ (k−a)(m−b) [f (a, b, t)+f (k, m, t)+f (k, b, t)+f (a, m, t)]dt
4
Z m Zc n
1
− (k − a)
[f (a, s, t) + f (k, s, t)] dtds
2
b
c
Z kZ n
1
− (m − b)
[f (r, b, t) + f (r, m, t)] dtdr
2
a
c
An Integral Approximation in
Three Variables
A. Sofo
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Z kZ m
1
− (n − c)
[f (r, s, c) + f (r, s, n)] dsdr
2
a
b
Z kZ mZ n
≤
|D3 D2 D1 f (r, s, t)| dtdsdr.
a
b
c
The following theorem establishes an Ostrowski type identity for an integral
in three independent variables.
Theorem 2.2. Let f : [a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ] → R be a continuous mapping
i+j+k (·,·,·)
such that the following partial derivatives ∂ ∂xi ∂yfj ∂z
k ; i = 0, . . . , n − 1, j =
0, . . . , m − 1; k = 0, . . . , p − 1 exist and are continuous on [a1 , b1 ] × [a2 , b2 ] ×
[a3 , b3 ] . Also, let

n
1)
 (r−a
; r ∈ [a1 , x),
n!
(2.2)
Pn (x, r) :=
 (r−b1 )n
; r ∈ [x, b1 ] ,
n!
(2.3)
Qm (y, s) :=


(s−a2 )m
;
m!
s ∈ [a2 , y),

(s−b2 )m
;
m!
s ∈ [y, b2 ] ,
A. Sofo
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(2.4)
An Integral Approximation in
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Sp (z, t) :=


(t−a3 )p
;
p!
t ∈ [a3 , z),

(t−b3 )p
;
p!
s ∈ [z, b3 ] ,
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then for all (x, y, z) ∈ [a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ] we have the identity
Z
b1
b2
Z
Z
b3
(2.5) V :=
−
f (r, s, t) dtdsdr
a1
a2
a3
p−1
n−1
m−1
XXX
Xi (x) Yj (y) Zk (z)
i=0 j=0 k=0
p
+ (−1)
n−1 m−1
X
X
Xi (x) Yj (y)
a3
i=0 j=0
p−1
n−1 X
X
m
+ (−1)
+ (−1)
XX
Xi (x) Zk (z)
m+p
− (−1)
n−1
X
Qm (y, s)
a2
Z
b1
Yj (y) Zk (z)
a1
j=0 k=0
Z
b2
Z
∂ i+j+p f (x, y, t)
Sp (z, t)
dt
∂xi ∂y j ∂tp
b2
Z
i=0 k=0
m−1 p−1
n
b3
Z
∂ i+j+k f (x, y, z)
∂xi ∂y j ∂z k
∂ i+m+k f (x, s, z)
ds
∂xi ∂sm ∂z k
∂ n+j+k f (r, y, z)
Pn (x, r)
dr
∂rn ∂y j ∂z k
b3
Xi (x)
Qm (y, s) Sp (z, t)
a2
i=0
i+m+p
An Integral Approximation in
Three Variables
a3
A. Sofo
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f (x, s, t)
dtds
i
∂x ∂sm ∂tp
Z b1 Z b3
m−1
X
n+p
− (−1)
Yj (y)
Pn (x, r) Sp (z, t)
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a1
a3
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J. Ineq. Pure and Appl. Math. 4(3) Art. 58, 2003
×
∂
j=0
×
∂
n+j+p
f (r, y, t)
∂rn ∂y j ∂tp
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n+m
− (−1)
p−1
X
Z
b1
Z
b2
Zk (z)
Pn (x, r) Qm (y, s)
a1
k=0
a2
∂ n+m+k f (r, s, z)
×
dsdr
∂rn ∂sm ∂z k
Z b1 Z b2 Z b3
n+m+p
= − (−1)
Pn (x, r) Qm (y, s) Sp (z, t)
a1
a2
a3
∂ n+m+p f (r, s, t)
×
dtdsdr,
∂rn ∂sm ∂tp
An Integral Approximation in
Three Variables
where
A. Sofo
(2.6)
(b1 − x)i+1 + (−1)i (x − a1 )i+1
Xi (x) :=
,
(i + 1)!
(2.7)
Yj (y) :=
(b2 − y)j+1 + (−1)j (y − a2 )j+1
,
(j + 1)!
Zk (z) :=
(b3 − z)k+1 + (−1)k (z − a3 )k+1
.
(k + 1)!
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Proof. We have an identity, see [5]
Z
b1
g (r) dr =
(2.9)
a1
n−1
X
i=0
Xi (x) g
II
I
(i)
n
Z
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b1
(x) + (−1)
Pn (x, r) g
a1
(n)
(r) dr.
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Now for the partial mapping f (·, s, t) , s ∈ [a2 , b2 ] , we have
Z
b1
f (r, s, t) dr =
(2.10)
a1
n−1
X
i=0
∂ if
Xi (x) i + (−1)n
∂x
Z
b1
Pn (x, r)
a1
∂ nf
dr
∂rn
for every r ∈ [a1 , b1 ] , s ∈ [a2 , b2 ] and t ∈ [a3 , b3 ] .
Now integrate over s ∈ [a2 , b2 ]
Z
b1
Z
b2
An Integral Approximation in
Three Variables
f (r, s, t) dsdr
(2.11)
a1
=
a2
n−1
X
i=0
Z
b2
Xi (x)
a2
∂ if
ds + (−1)n
∂xi
for all x ∈ [a1 , b1 ] .
From (2.9) for the partial mapping
b2
Z
b1
Z
b2
Pn (x, r)
a1
a2
∂ nf
ds dt
∂rn
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∂if
∂xi
on [a2 , b2 ] we have,
∂i
(2.12)
f (x, s, t) ds
i
a2 ∂x
Z b2
m−1
X
∂ j ∂ if
∂ m ∂ if
m
=
Yj (y) j
+ (−1)
Qm (y, s) m
ds
∂y
∂xi
∂s
∂xi
a2
j=0
Z b2
m−1
X
∂ i+j f
∂ i+m f
m
=
Yj (y) i j + (−1)
Qm (y, s) i m ds.
∂x ∂y
∂x ∂s
a2
j=0
Z
A. Sofo
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Also, from (2.8)
Z
Z b2
m−1
X
∂ f
∂ j+n f
∂ m ∂ nf
m
ds =
Yj (y) j n +(−1)
Qm (y, s) m
ds.
∂rn
∂y ∂r
∂s ∂rn
a2
j=0
b2 n
(2.13)
a2
From (2.11) substitute (2.12) and (2.13), so that
Z b1 Z b2
(2.14)
f (r, s, t) dsdr
a1
a2
"m−1
#
Z b2
n−1
i+m
X
X
∂ i+j f
∂
f
=
Xi (x)
Qm (y, s) i m ds
Yj (y) i j +(−1)m
∂x ∂y
∂x ∂s
a2
i=0
j=0
"
Z b1
m−1
X
∂ j+n f
n
+ (−1)
Pn (x, r)
Yj (y) j n
∂y ∂r
a1
j=0
Z b2
∂ m ∂ nf
m
+ (−1)
Qm (y, s) m
ds dt
∂s
∂rn
a2
n−1
m−1
X
X
∂ i+j f
=
Xi (x)
Yj (y) i j
∂x ∂y
i=0
j=0
Z b2
n−1
X
∂ i+m f
m
+ (−1)
Xi (x)
Qm (y, s) i m ds
∂x ∂s
a2
i=0
Z b1
m−1
X
∂ j+n f
n
+ (−1)
Yj (y)
Pn (x, r) j n
∂y ∂r
a1
j=0
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A. Sofo
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n+m
Z
b1
Z
b2
+ (−1)
Pn (x, r) Qm (y, s)
a1
a2
∂ n+m f
dsdr
∂sm ∂rn
Now integrate (2.14) for t ∈ [a3 , b3 ]
Z
b1
Z
b2
Z
b3
f (r, s, t) dtdsdr
(2.15)
a1
a2
a3
m−1
n−1
XX
b3
∂ i+j f
dt
i ∂y j
∂x
a
3
i=0 j=0
Z
Z b2
n−1
X
m
+ (−1)
Xi (x)
Qm (y, s)
=
Z
Xi (x) Yj (y)
An Integral Approximation in
Three Variables
A. Sofo
b3
∂ i+m f
dt ds
i
m
a2
a3 ∂x ∂s
i=0
Z b3 j+n
Z b1
m−1
X
∂
n
+ (−1)
Yj (y)
dt dr
Pn (x, r)
j
n
a2
a3 ∂y ∂r
j=0
Z b3 n+m
Z b1 Z b2
∂
f
n+m
dt dsdr.
+ (−1)
Pn (x, r) Qm (y, s)
m
n
a1
a2
a3 ∂s ∂r
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From (2.9),
Close
Z
b3
(2.16)
a3
p−1
X
∂ i+j f
∂k
dt
=
Z
(z)
k
∂xi ∂y j
∂z k
k=0
∂ i+j f
∂xi ∂y j
i+j Z b3
∂p
∂ f
p
+ (−1)
Sp (z, t) p
dt,
∂t
∂xi ∂y j
a3
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Z
b3
(2.17)
a3
Z
b3
a3
p−1
X
∂k
∂ i+m f
dt
=
Z
(z)
k
∂xi ∂sm
∂z k
k=0
∂ i+m f
∂xi ∂sm
i+m Z b3
∂
f
∂p
p
+ (−1)
dt,
Sp (z, t) p
∂t
∂xi ∂sm
a3
∂ j+n f
dt
∂y j ∂rn
p−1
=
X
Zk (z)
k=0
∂k
∂z k
∂ j+n f
∂y j ∂rn
+ (−1)p
Z
b3
Sp (z, t)
a3
∂p
∂tp
∂ j+n f
∂y j ∂rn
An Integral Approximation in
Three Variables
dt,
and
A. Sofo
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Z
b3
(2.18)
a3
p−1
X
∂ n+m f
∂k
dt =
Zk (z) k
∂sm ∂rn
∂z
k=0
∂ n+m f
∂rn ∂sm
n+m Z b3
∂p
∂
f
p
+ (−1)
Sp (z, t) p
dt.
n
∂t
∂r ∂sm
a3
Putting (2.16), (2.17) and (2.18) into (2.15) we arrive at the identity (2.5).
At the midpoint of the interval
x̄ =
a2 + b 2
a3 + b 3
a1 + b 1
, ȳ =
, z̄ =
2
2
2
we have the following corollary.
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Corollary 2.3. Under the assumptions of Theorem 2.2, we have the identity
Z b1 Z b2 Z b3
(2.19) V̄ :=
f (r, s, t) dtdsdr
a1
−
a2
a3
p−1
n−1
m−1
XXX
Xi (x̄) Yj (ȳ) Zk (z̄)
i=0 j=0 k=0
p
+ (−1)
n−1 m−1
X
X
+ (−1)
b3
Z
Xi (x̄) Yj (ȳ)
p−1
n−1 X
X
Sp (z̄, t)
a3
i=0 j=0
m
∂ i+j+k f (x̄, ȳ, z̄)
∂xi ∂y j ∂z k
Z
∂ i+j+p f (x̄, ȳ, t)
dt
∂xi ∂y j ∂tp
b2
Xi (x̄) Zk (z̄)
A. Sofo
Qm (ȳ, s)
a2
i=0 k=0
i+m+k
f (x̄, s, z̄)
ds
∂xi ∂sm ∂z k
Z b1
p−1
m−1
XX
n
+ (−1)
Yj (ȳ) Zk (z̄)
Pn (x̄, r)
×
∂
a1
j=0 k=0
∂
i=0
i+m+p
×
∂
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n+j+k
f (r, ȳ, z̄)
dr
∂rn ∂y j ∂z k
Z b2 Z b3
n−1
X
m+p
− (−1)
Xi (x̄)
Qm (ȳ, s) Sp (z̄, t)
×
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a3
f (x̄, s, t)
dtds
i
∂x ∂sm ∂tp
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n+p
− (−1)
m−1
X
b1
Z
Z
b3
Yj (ȳ)
Pn (x̄, r) Sp (z̄, t)
a1
j=0
a3
∂ n+j+p f (r, ȳ, t)
dtdr
∂rn ∂y j ∂tp
Z b1 Z b2
p−1
X
n+m
− (−1)
Zk (z̄)
Pn (x̄, r) Qm (ȳ, s)
×
k=0
n+m+k
×
n+m+p
∂
a1
f (r, s, z̄)
∂rn ∂sm ∂z k
Z
b1
Z
b2
Z
dsdr
A. Sofo
Pn (x̄, r) Qm (ȳ, s) Sp (z̄, t)
a1
a2
n+m+p
∂
a3
f (r, s, t)
∂rn ∂sm ∂tp
dtdsdr.
The identity (2.5) will now be utilised to establish an inequality for a function
of three independent variables which will furnish a refinement for the inequality
(2.1) given by Pachpatte.
Theorem 2.4. Let f : [a1 , b1 ]×[a2 , b2 ]×[a3 , b3 ] → R be continuous on (a1 , b1 )×
(a2 , b2 ) × (a3 , b3 ) and the conditions of Theorem 2.2 apply. Then we have the
inequality
|V | ≤
An Integral Approximation in
Three Variables
b3
= − (−1)
×
a2
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ih
i
 h
(y−a2 )m+1 +(b2 −y)m+1
(x−a1 )n+1 +(b1 −x)n+1


(n+1)!
(m+1)!

i n+m+p h

p+1
p+1

∂
(z−a
)
+(b

3
3 −z)

×
∂rn ∂sm ∂tfp 
(p+1)!

∞


∂ n+m+p f


if
∈
L
([a
,
b
]
×
[a2 , b2 ] × [a3 , b3 ]) ;
∞
1 1

∂rn ∂sm ∂tp





h
i1 h
i1


(x−a1 )nβ+1 +(b1 −x)nβ+1 β (y−a2 )mβ+1 +(b2 −y)mβ+1 β
1



n!m!p!
nβ+1
mβ+1


h
i1 

(z−a3 )pβ+1 +(b3 −z)pβ+1 β ∂ n+m+p f ×
∂rn ∂sm ∂tp pβ+1
≤
α
n+m+p

∂
f

∈
L
([a
,
b
]
× [a2 , b2 ] × [a3 , b3 ]) ,
if

α
1
1
n
m
p

∂r ∂s ∂t



α > 1, α−1 + β −1 = 1;






1


[(x − a1 )n + (b1 − x)n + |(x − a1 )n − (b1 − x)n |]

8n!m!p!



× [(y − a2 )m + (b2 − y)m + |(y − a2 )m − (b2 − y)m|]



p
p
p
p ∂ n+m+p f 

×
[(z
−
a
)
+
(b
−
z)
+
|(z
−
a
)
−
(b
−
z)
|]

3
3
3
3
∂rn ∂sm ∂tp 
1

n+m+p

if ∂r∂ n ∂sm ∂tfp ∈ L1 ([a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ]) ;
for all (x, y, z) ∈ [a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ] , where
n+m+p n+m+p ∂
∂
f f =
sup
∂rn ∂sm ∂tp n m p < ∞,
(r,s,t)∈[a1 ,b1 ]×[a2 ,b2 ]×[a3 ,b3 ] ∂r ∂s ∂t
∞
and
(2.20)
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n+m+p Z
∂
f =
∂rn ∂sm ∂tp α
b1 Z
a1
b2 Z
a2
n+m+p α
α1
b3 f ∂
< ∞.
∂rn ∂sm ∂tp dtdsdr
a3
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Proof.
Z b1 Z b2 Z b3
∂ n+m+p f (r, s, t)
dtdsdr
|V | = Pn (x, r) Qm (y, s) Sp (z, t)
n
m
p
∂r ∂s ∂t
a1
a2
a3
n+m+p
Z b1 Z b2 Z b3
∂
f
(r,
s,
t)
dtdsdr.
≤
|Pn (x, r) Qm (y, s) Sp (z, t)| ∂rn ∂sm ∂tp a1
a2
a3
Using Hölder’s inequality and property of the modulus and integral, then we
have that
n+m+p
Z b1 Z b2 Z b3
∂
f
(r,
s,
t)
dtdsdr
(2.21)
|Pn (x, r) Qm (y, s) Sp (z, t)| ∂rn ∂sm ∂tp a1
a2
a3
 n+m+p R R R
∂
b
b
b

∂rn ∂sm ∂tfp a11 a22 a33 |Pn (x, r) Qm (y, s) Sp (z, t)| dtdsdr,


∞




R R R
β1


b1 b2 b3
β
∂ n+m+p f  ,
|P
(x,
r)
Q
(y,
s)
S
(z,
t)|
dtdsdr
∂rn ∂sm ∂tp n
m
p
a1 a2 a3
α
≤

α > 1, α−1 + β −1 = 1;




n+m+p 

∂

f 
sup
|Pn (x, r) Qm (y, s) Sp (z, t)| .
 ∂rn ∂sm ∂tp 1 (r,s,t)∈[a1 ,b1 ]×[a2 ,b2 ]×[a3 ,b3 ]
From (2.21) and using (2.2), (2.3) and (2.4)
Z b1 Z b2 Z b3
|Pn (x, r) Qm (y, s) Sp (z, t)| dtdsdr
a1
a2
Z
a3
b1
Z
b2
|Pn (x, r)| dr
=
a1
Z
A. Sofo
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b3
|Qm (y, s)| ds
a2
An Integral Approximation in
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|Sp (z, t)| dt
a3
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Z b1
(r − a1 )n
(b1 − r)n
=
dr +
dr
n!
n!
a1
x
Z y
Z b2
(s − a2 )m
(b2 − s)m
×
ds +
ds
m!
m!
a2
y
Z z
Z b3
(t − a3 )p
(b3 − t)p
×
dt +
dt
p!
p!
a3
z
(x − a1 )n+1 + (b1 − x)n+1 (y − a2 )m+1 + (b2 − y)m+1
=
(n + 1)! (m + 1)!
p+1
(z − a3 )
+ (b3 − z)p+1
×
(p + 1)!
Z
x
An Integral Approximation in
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A. Sofo
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giving the first inequality in (2.20).
Now, if we again use (2.21) we have
Z
b1
Z
b2
Contents
b3
Z
JJ
J
β1
β
|Pn (x, r) Qm (y, s) Sp (z, t)| dtdsdr
a1
a2
a3
Z
b1
β
b2
|Pn (x, r)| dr
=
a1
=
β1 Z
1
n!m!p!
β
β1 Z
|Qm (y, s)| ds
a2
Z
x
(r − a1 )nβ dr +
a1
a3
b1
Z
(b1 − r)nβ dr
β1
y
×
a2
(s − a2 )mβ ds +
β1
|Sp (z, t)| dt
β
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x
Z
b3
II
I
Z
y
b2
β1
mβ
(b2 − s) ds
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Z
z
×
pβ
b3
Z
(t − a3 ) dt +
a3
β1
(b3 − t) dt
pβ
z
# β1
(x − a1 )nβ+1 + (b1 − x)nβ+1
1
=
n!m!p!
nβ + 1
"
# β1
(y − a2 )mβ+1 + (b2 − y)mβ+1
×
mβ + 1
# β1
"
(z − a3 )pβ+1 + (b3 − z)pβ+1
×
pβ + 1
"
producing the second inequality in (2.20).
Finally, we have
Contents
(r,s,t)∈[a1 ,b1 ]×[a2 ,b2 ]×[a3 ,b3 ]
= sup |Pn (x, r)| sup |Qm (y, s)| sup |Sp (z, t)|
r∈[a1 ,b1 ]
s∈[a2 ,b2 ]
t∈[a3 ,b3 ]
(y − a2 )m (b2 − y)m
(x − a1 ) (b1 − x)
,
max
,
= max
n!
n!
m!
m!
p
p
(z − a3 ) (b3 − z)
,
× max
p!
p!
n
n
n
n (x − a1 ) + (b1 − x)
(x
−
a
)
−
(b
−
x)
1
1
1
+ =
n!m!p!
2
2
n
n
A. Sofo
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|Pn (x, r) Qm (y, s) Sp (z, t)|
sup
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J. Ineq. Pure and Appl. Math. 4(3) Art. 58, 2003
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(y − a2 )m + (b2 − y)m (y − a2 )m − (b2 − y)m ×
+
2
2
(z − a3 )p + (b3 − z)p (z − a3 )p − (b3 − z)p ×
+
,
2
2
giving us the third inequality in (2.20) and we have used the fact that for A > 0,
B > 0 then
A + B A − B max {A, B} =
+
.
2
2 Hence the theorem is completely solved.
An Integral Approximation in
Three Variables
A. Sofo
The following corollary is a consequence of Theorem 2.4.
Corollary 2.5. Under the assumptions of Corollary 2.3, we have the inequality
 h
i
(b1 −a1 )n+1 (b2 −a2 )m+1 (b3 −a3 )p+1 ∂ n+m+p f 

,

2n+m+p (n+1)!(m+1)!(p+1)!
∂rn ∂sm ∂tp 
∞




h
i1 
(b1 −a1 )nβ+1 (b2 −a2 )mβ+1 (b3 −a3 )pβ+1 β ∂ n+m+p f V̄ ≤
1
,
2n+m+p n!m!p!
(nβ+1)(mβ+1)(pβ+1)
∂rn ∂sm ∂tp 
α




n+m+p 


 2n+m+p1 n!m!p! (b1 − a1 )n (b2 − a2 )m (b3 − a3 )p ∂r∂ n ∂sm ∂tfp ,
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where k·kα (α ∈ [1, ∞)) are the Lebesgue norms on [a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ] .
Page 21 of 27
The following two corollaries concern the estimation of V at the end points.
J. Ineq. Pure and Appl. Math. 4(3) Art. 58, 2003
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Corollary 2.6. Under the assumptions of Theorem 2.4 we have, for x = a1 ,
y = a2 and z = a3 , the inequality
|V (a1 , a2 , a3 )|
Z b1 Z b2 Z
:= a1
−
b3
f (r, s, t) dtdsdr
a2
a3
p−1
n−1
m−1
XXX
Xi (a1 ) Yj (a2 ) Zk (a3 )
i=0 j=0 k=0
+ (−1)p
n−1 m−1
X
X
+ (−1)
Xi (a1 ) Yj (a2 )
+ (−1)n
p−1
n−1 X
X
i=0 k=0
p−1
m−1
XX
n−1
X
− (−1)
− (−1)n+p
i=0
m−1
X
a3
Xi (a1 ) Zk (a3 )
a2
b1
Yj (a2 ) Zk (a3 )
b2
∂ i+m+k f
ds
∂xi ∂sm ∂z k
n+j+k
P̄n (a1 , r)
a1
∂
f
dr
n
j
∂r ∂y ∂z k
b3
Z
Xi (a1 )
Q̄m (a2 , s) S̄p (a3 , t)
a2
b1
Z
a3
b3
Z
Yj (a2 )
Z
a3
b1
Z
b2
Zk (a3 )
a1
a2
∂ i+m+p f
dtds
∂xi ∂sm ∂tp
n+j+p
P̄n (a1 , r) S̄p (a3 , t)
a1
p−1
k=0
Q̄m (a2 , s)
Z
Z
X
∂
f
dt
∂xi ∂y j ∂tp
b2
Z
j=0
− (−1)n+m
i+j+p
S̄p (a3 , t)
j=0 k=0
m+p
b3
Z
i=0 j=0
m
∂ i+j+k f
∂xi ∂y j ∂z k
∂
f
dtdr
n
j
∂r ∂y ∂tp
∂
f
P̄n (a1 , r) Q̄m (a2 , s) n m k dsdr
∂r ∂s ∂z
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n+m+k
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≤







































(b1 −a1 )n+1 (b2 −a2 )m+1 (b3 −a3 )p+1
(n+1)!(m+1)!(p+1)!
if
n+m+p ∂
∂rn ∂sm ∂tfp ,
∞
∂ n+m+p f
∂rn ∂sm ∂tp
(b1 −a1 )
n+ 1
β
1
∈ L∞ ([a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ]) ;
m+ 1
p+ 1 n+m+p
β
β · (b2 −a2 ) 1 · (b3 −a3 ) 1 ∂r∂ n ∂sm ∂tfp ,
n!(nβ+1) β
m!(mβ+1) β
p!(pβ+1) β
n+m+p
if ∂r∂ n ∂sm ∂tfp ∈ Lα ([a1 , b1 ] × [a2 , b2 ]
−1
(b1 −a1 )n (b2 −a2 )m (b3 −a3 )p
n!m!p!
if
∂ n+m+p f
∂rn ∂sm ∂tp
α
× [a3 , b3 ]) ,
α > 1, α + β −1 = 1;
n+m+p ∂
∂rn ∂sm ∂tfp ,
An Integral Approximation in
Three Variables
A. Sofo
1
∈ L1 ([a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ]) ,
Title Page
where
Contents
(b2 − a2 )j+1
(b3 − a3 )k+1
(b1 − a1 )i+1
Xi (a1 ) :=
, Yj (a2 ) :=
, Zk (a3 ) :=
.
(i + 1)!
(j + 1)!
(k + 1)!
(r − b1 )n
P̄n (a1 , r) =
, r ∈ [a1 , b1 ] ;
n!
(s − b2 )m
Q̄m (a2 , s) =
, s ∈ [a2 , b2 ]
m!
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and
S̄p (a3 , t) =
(t − b3 )p
; t ∈ [a3 , b3 ] .
p!
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Corollary 2.7. Under the assumptions of Theorem 2.4 we have, for x = b1 ,
y = b2 and z = b3 , the inequality
|V (b1 , b2 , b3 )|
Z b1 Z
:= a1
−
b2
Z
b3
f (r, s, t) dtdsdr
a2
a3
p−1
n−1
m−1
XXX
Xi (a1 ) Yj (a2 ) Zk (a3 )
i=0 j=0 k=0
+ (−1)p
n−1 m−1
X
X
+ (−1)
Xi (b1 ) Yj (b2 )
S̄p (b3 , t)
a3
p−1
n−1 X
X
Xi (b1 ) Zk (b3 )
XX
a2
n−1
X
− (−1)
− (−1)n+p
i=0
m−1
X
P̄n (b1 , r)
a1
∂ i+m+k f
ds
∂xi ∂sm ∂z k
∂ n+j+k f
dr
∂rn ∂y j ∂z k
b3
Z
Xi (b1 )
Q̄m (b2 , s) S̄p (b3 , t)
a2
a3
b1 Z
b3
Z
Yj (b2 )
p−1
X
k=0
b2
Z
P̄n (b1 , r) S̄p (b3 , t)
a1
j=0
− (−1)n+m
b1
Yj (b2 ) Zk (b3 )
j=0 k=0
m+p
Q̄m (b2 , s)
Z
An Integral Approximation in
Three Variables
∂ i+j+p f
dt
∂xi ∂y j ∂tp
b2
Z
i=0 k=0
m−1 p−1
+ (−1)n
b3
Z
i=0 j=0
m
∂ i+j+k f
∂xi ∂y j ∂z k
Z
a3
∂ i+m+p f
dtds
∂xi ∂sm ∂tp
∂ n+j+p f
dtdr
∂rn ∂y j ∂tp
b2
∂ n+m+k f
P̄n (b1 , r) Q̄m (b2 , s) n m k dsdr
∂r
∂s
∂z
a2
A. Sofo
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b1 Z
Zk (b3 )
a1
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≤



































(b1 −a1 )n+1 (b2 −a2 )m+1 (b3 −a3 )p+1
(n+1)!(m+1)!(p+1)!
if
(b1 −a1 )
n+ 1
β
1
·
(b2 −a2 )
n+m+p ∂
∂rn ∂sm ∂tfp ,
∞
∂ n+m+p f
∈ L∞ ([a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ]) ;
p+ 1 (b3 −a3 ) β ∂ n+m+p f ·
1 ∂r n ∂sm ∂tp ,
∂rn ∂sm ∂tp
m+ 1
β
1
n!(nβ+1) β
m!(mβ+1) β
p!(pβ+1) β
n+m+p
if ∂r∂ n ∂sm ∂tfp ∈ Lα ([a1 , b1 ] × [a2 , b2 ]
(b1 −a1 )n (b2 −a2 )m (b3 −a3 )p
n!m!p!
if
α
× [a3 , b3 ]) , α > 1,
1
α
+
1
β
= 1;
An Integral Approximation in
Three Variables
n+m+p ∂
∂rn ∂sm ∂tfp ,
1
∂ n+m+p f
∂rn ∂sm ∂tp
A. Sofo
∈ L1 ([a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ]) ,
where
Title Page
i
Xi (b1 ) :=
i+1
(−1) (b1 − a1 )
(i + 1)!
j
,
Yj (b2 ) :=
(−1) (b2 − a2 )
(j + 1)!
(−1)k (b3 − a3 )k+1
Zk (b3 ) :=
.
(k + 1)!
(r − a1 )n
P̄n (b1 , r) =
; r ∈ [a1 , b1 ] ,
n!
(s − a2 )m
Q̄m (b2 , s) =
; s ∈ [a2 , b2 ]
m!
and
j+1
,
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p
S̄p (b3 , t) =
(t − a3 )
; t ∈ [a3 , b3 ] .
p!
J. Ineq. Pure and Appl. Math. 4(3) Art. 58, 2003
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References
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[8] B.G. PACHPATTE, On a new Ostrowski type inequality in two independent variables, Tamkang J. Math., 32(1) (2001), 45–49.
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[14] J. STEWART, Calculus: Early Transcendentals, 3rd Edition. Brooks/Cole,
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