Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 4, Issue 3, Article 58, 2003
AN INTEGRAL APPROXIMATION IN THREE VARIABLES
A. SOFO
S CHOOL OF C OMPUTER S CIENCE AND M ATHEMATICS
V ICTORIA U NIVERSITY OF T ECHNOLOGY
PO B OX 14428, MCMC 8001,
V ICTORIA , AUSTRALIA .
sofo@csm.vu.edu.au
URL: http://rgmia.vu.edu.au/sofo
Received 15 November, 2002; accepted 25 August, 2003
Communicated by C.E.M. Pearce
A BSTRACT. 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.
Key words and phrases: Ostrowski inequality, Three independent variables, Partial derivatives.
2000 Mathematics Subject Classification. Primary 26D15; Secondary 41A55.
1. I NTRODUCTION
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
(1.1)
M (f ) :=
f (t) dt,
b−a a
(1.2)
ISSN (electronic): 1443-5756
c 2003 Victoria University. All rights reserved.
125-02
IT (f ) :=
f (b) + f (a)
2
2
A. S OFO
and
IM (f ) := f
(1.3)
a+b
2
.
The Ostrowski result is given by:
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∞
2
4
(b − a)
for all x ∈ [a, b] .
The constant 14 is the best possible.
Improvements of the result (1.4) has also been obtained by Dedić, Matić and Pearce [2],
Pearce, Pečarić,
Ujević
and Varošanec [11], Dragomir [3] and Sofo [12]. For a symmetrical
point x ∈ a, a+b
,
very
recently Guessab and Schmeisser [4] studied the more general quad2
rature formula
f (x) + f (a + b − x)
M (f ) −
= E (f ; x)
2
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
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 = {(x1 , . . . , xm ) |ai ≤
∂f xi ≤ bi , (i = 1, . . . , m)} and let ∂xi ≤ Mi (Mi > 0, 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
M
p
(y)
f
(y)
dy
i D p (y) |xi − yi | dy
i=1
D
≤
f (x) − R
R
(1.5)
.
p
(y)
dy
p (y) dy
D
D
In 2001, Barnett and Dragomir [1] obtained the following Ostrowski type inequality for double integrals.
00
Theorem 1.3. Let f : [a, b] × [c, d] → R be continuous on [a, b] × [c, d] , fx,y
=
(a, b) × (c, d) and is bounded, that is,
2
00 ∂ f (x, y) fs,t :=
sup
∂x∂y < ∞,
∞
(x,y)∈(a,b)×(c,d)
J. Inequal. Pure and Appl. Math., 4(3) Art. 58, 2003
∂2f
∂x∂y
exist on
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A N I NTEGRAL A PPROXIMATION
then we have the inequality:
Z b Z d
Z
f (s, t) dsdt − (b − a)
(1.6) a
c
3
d
f (x, t) dt
c
b
− (d − c)
f (s, y) ds + (d − c) (b − a) f (x, y)
a
"
2 # "
2 #
2
2
00 (b − a)
a+b
(d − c)
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).
2. T RIPLE I NTEGRALS
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 differen∂
∂
tiable on ∆. Denote the partial derivatives by D1 f (r, s, t) = ∂r
f (r, s, t) ; D2 f (r, s, t) = ∂s
,
3
∂ f
∂
D3 f (r, s, t) = ∂t and D3 D2 D1 f = ∂t∂s∂r . Let 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
a
Z m
1
+ (k − a) (n − c)
[f (a, s, c) + f (k, s, n) + f (a, s, n) + f (k, s, c)] ds
4
b
Z n
1
+ (k − a) (m − b)
[f (a, b, t) + f (k, m, t) + f (k, b, t) + f (a, m, t)] dt
4
c
Z mZ 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
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.
J. Inequal. Pure and Appl. Math., 4(3) Art. 58, 2003
http://jipam.vu.edu.au/
4
A. S OFO
Theorem 2.2. Let f : [a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ] → R be a continuous mapping such that the
i+j+k (·,·,·)
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

(r − a1 )n


; r ∈ [a1 , x),


n!
(2.2)
Pn (x, r) :=

 (r − b1 )n


; r ∈ [x, b1 ] ,
n!

m

 (s − a2 ) ; s ∈ [a2 , y),


m!
(2.3)
Qm (y, s) :=
m



 (s − b2 ) ; s ∈ [y, b2 ] ,
m!
and

(t − a3 )p



; t ∈ [a3 , z),


p!
(2.4)
Sp (z, t) :=


(t − b3 )p


; s ∈ [z, b3 ] ,

p!
then for all (x, y, z) ∈ [a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ] we have the identity
Z b1 Z b2 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
m
+ (−1)
b3
Z
p−1
n−1 X
X
+ (−1)
p−1
m−1
XX
Xi (x) Zk (z)
Qm (y, s)
a2
Z
n−1
X
− (−1)
Pn (x, r)
a1
m−1
X
− (−1)
= − (−1)n+m+p
a2
a3
b1 Z
b3
Yj (y)
k=0
b1
Z
Pn (x, r) Sp (z, t)
a1
Z
a3
b1
Z
∂ n+j+p f (r, y, t)
dtdr
∂rn ∂y j ∂tp
Pn (x, r) Qm (y, s)
a1
b2
∂ i+m+p f (x, s, t)
dtds
∂xi ∂sm ∂tp
b2
Zk (z)
Z
∂ n+j+k f (r, y, z)
dr
∂rn ∂y j ∂z k
Qm (y, s) Sp (z, t)
Z
p−1
X
∂ i+m+k f (x, s, z)
ds
∂xi ∂sm ∂z k
b3
Z
Xi (x)
j=0
n+m
b2
Z
i=0
− (−1)n+p
b1
Yj (y) Zk (z)
j=0 k=0
m+p
∂ i+j+p f (x, y, t)
Sp (z, t)
dt
∂xi ∂y j ∂tp
b2
Z
i=0 k=0
n
∂ i+j+k f (x, y, z)
∂xi ∂y j ∂z k
Z
a2
∂ n+m+k f (r, s, z)
dsdr
∂rn ∂sm ∂z k
b3
Pn (x, r) Qm (y, s) Sp (z, t)
a1
a2
J. Inequal. Pure and Appl. Math., 4(3) Art. 58, 2003
a3
∂ n+m+p f (r, s, t)
dtdsdr,
∂rn ∂sm ∂tp
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A N I NTEGRAL A PPROXIMATION
5
where
(2.6)
Xi (x) :=
(b1 − x)i+1 + (−1)i (x − a1 )i+1
,
(i + 1)!
(2.7)
Yj (y) :=
(b2 − y)j+1 + (−1)j (y − a2 )j+1
,
(j + 1)!
and
(b3 − z)k+1 + (−1)k (z − a3 )k+1
Zk (z) :=
.
(k + 1)!
(2.8)
Proof. We have an identity, see [5]
b1
Z
g (r) dr =
(2.9)
a1
n−1
X
Xi (x) g
(i)
n
b1
Z
Pn (x, r) g (n) (r) dr.
(x) + (−1)
a1
i=0
Now for the partial mapping f (·, s, t) , s ∈ [a2 , b2 ] , we have
b1
Z
f (r, s, t) dr =
(2.10)
a1
n−1
X
i=0
∂ if
Xi (x) i + (−1)n
∂x
b1
a1
∂ nf
Pn (x, r) n dr
∂r
Z
b1
Z
for every r ∈ [a1 , b1 ] , s ∈ [a2 , b2 ] and t ∈ [a3 , b3 ] .
Now integrate over s ∈ [a2 , b2 ]
Z
b1
Z
b2
f (r, s, t) dsdr
(2.11)
a1
a2
=
n−1
X
Z
Xi (x)
a2
i=0
for all x ∈ [a1 , b1 ] .
From (2.9) for the partial mapping
Z
b2
(2.12)
a2
b2
∂if
∂xi
∂ if
ds + (−1)n
i
∂x
Z
b2
Pn (x, r)
a1
a2
∂ nf
ds dt
∂rn
on [a2 , b2 ] we have,
∂i
f (x, s, t) ds
∂xi
=
m−1
X
j=0
=
m−1
X
j=0
∂j
Yj (y) j
∂y
∂ if
∂xi
m
Z
b2
+ (−1)
∂ i+j f
Yj (y) i j + (−1)m
∂x ∂y
a2
Z
∂m
Qm (y, s) m
∂s
b2
Qm (y, s)
a2
∂ if
∂xi
ds
∂ i+m f
ds.
∂xi ∂sm
Also, from (2.8)
Z
b2
(2.13)
a2
Z b2
m−1
X
∂ nf
∂ 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
J. Inequal. Pure and Appl. Math., 4(3) Art. 58, 2003
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6
A. S OFO
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)
Yj (y) i j + (−1)m
Qm (y, s) i m ds
∂x
∂y
∂x ∂s
a
2
i=0
j=0
"m−1
Z b1
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
Z b2
n−1
m−1
n−1
X
X
X
∂ i+j f
∂ i+m f
m
=
Xi (x)
Qm (y, s) i m ds
Xi (x)
Yj (y) i j + (−1)
∂x ∂y
∂x ∂s
a2
i=0
j=0
i=0
Z b1
m−1
X
∂ j+n f
n
+ (−1)
Yj (y)
Pn (x, r) j n
∂y ∂r
a1
j=0
Z b1 Z b2
∂ n+m f
n+m
+ (−1)
Pn (x, r) Qm (y, s) m n dsdr
∂s ∂r
a1
a2
Now integrate (2.14) for t ∈ [a3 , b3 ]
Z
b1
b2
Z
b3
f (r, s, t) dtdsdr =
(2.15)
a1
a2
n−1 m−1
X
X
b3
∂ i+j f
dt
i ∂y j
∂x
a3
a
3
i=0 j=0
Z b3 i+m
Z
n−1
b2
X
∂
f
m
+ (−1)
Xi (x)
Qm (y, s)
dt ds
i
m
a2
a3 ∂x ∂s
i=0
Z b3 j+n
Z b1
m−1
X
∂
n
+ (−1)
Yj (y)
Pn (x, r)
dt dr
j ∂r n
∂y
a
a
2
3
j=0
Z b3 n+m
Z b1 Z b2
∂
f
n+m
+ (−1)
Pn (x, r) Qm (y, s)
dt dsdr.
m
n
a1
a2
a3 ∂s ∂r
Z
Z
Xi (x) Yj (y)
From (2.9),
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
p
Z
b3
+ (−1)
a3
Z
b3
(2.17)
a3
p−1
X
∂ i+m f
∂k
dt
=
Z
(z)
k
∂xi ∂sm
∂z k
k=0
J. Inequal. Pure and Appl. Math., 4(3) Art. 58, 2003
∂p
Sp (z, t) p
∂t
∂ i+j f
∂xi ∂y j
dt,
∂ i+m f
∂xi ∂sm
i+m Z b3
∂p
∂
f
p
+ (−1)
Sp (z, t) p
dt,
i
∂t
∂x ∂sm
a3
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A N I NTEGRAL A PPROXIMATION
Z
b3
a3
p−1
X
∂ j+n f
∂k
Z
(z)
dt
=
k
∂y j ∂rn
∂z k
k=0
∂ j+n f
∂y j ∂rn
X
∂k
∂ n+m f
dt
=
Z
(z)
k
∂sm ∂rn
∂z k
k=0
p
7
Z
b3
+ (−1)
a3
∂p
Sp (z, t) p
∂t
∂ j+n f
∂y j ∂rn
dt,
and
Z
b3
(2.18)
a3
p−1
∂ n+m f
∂rn ∂sm
n+m Z b3
∂p
∂
f
p
+ (−1)
Sp (z, t) p
dt.
∂t
∂rn ∂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
a1 + b 1
a2 + b 2
a3 + b 3
, ȳ =
, z̄ =
2
2
2
x̄ =
we have the following corollary.
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)
Sp (z̄, t)
a3
i=0 j=0
m
p−1
n−1 X
X
p−1
m−1
XX
Xi (x̄) Zk (z̄)
Qm (ȳ, s)
a2
Z
n−1
X
− (−1)
Pn (x̄, r)
a1
− (−1)n+p
− (−1)
n+m+p
a2
a3
b1 Z
b3
Yj (ȳ)
k=0
b1
Z
Pn (x̄, r) Sp (z̄, t)
a1
Z
b1
Z
a1
b2
a2
∂ i+m+p f (x̄, s, t)
dtds
∂xi ∂sm ∂tp
∂ n+j+p f (r, ȳ, t)
dtdr
∂rn ∂y j ∂tp
b2
Pn (x̄, r) Qm (ȳ, s)
Z
b3
= − (−1)
a1
a3
Zk (z̄)
Z
∂ n+j+k f (r, ȳ, z̄)
dr
∂rn ∂y j ∂z k
Qm (ȳ, s) Sp (z̄, t)
Z
p−1
X
∂ i+m+k f (x̄, s, z̄)
ds
∂xi ∂sm ∂z k
b3
Z
Xi (x̄)
j=0
n+m
b2
Z
i=0
m−1
X
b1
Yj (ȳ) Zk (z̄)
j=0 k=0
m+p
∂ i+j+p f (x̄, ȳ, t)
dt
∂xi ∂y j ∂tp
b2
Z
i=0 k=0
+ (−1)n
∂ i+j+k f (x̄, ȳ, z̄)
∂xi ∂y j ∂z k
a3
a2
∂ n+m+k f (r, s, z̄)
dsdr
∂rn ∂sm ∂z k
∂ n+m+p f (r, s, t)
Pn (x̄, r) Qm (ȳ, s) Sp (z̄, t)
dtdsdr.
∂rn ∂sm ∂tp
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.
J. Inequal. Pure and Appl. Math., 4(3) Art. 58, 2003
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8
A. S OFO
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
#"
#
 "
n+1
n+1
m+1
m+1

(x
−
a
)
+
(b
−
x)
(y
−
a
)
+
(b
−
y)
1
1
2
2




(n
+
1)!
(m
+
1)!


#
"


p+1
p+1 
∂ n+m+p f 
(z
−
a
)
+
(b
−
z)
3
3


×

∂rn ∂sm ∂tp 
(p
+
1)!

∞



n+m+p

∂
f


if n m p ∈ L∞ ([a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ]) ;


∂r ∂s ∂t





"
#1 "
#1


mβ+1
mβ+1 β
nβ+1
nβ+1 β


1
(y
−
a
)
+
(b
−
y)
(x
−
a
)
+
(b
−
x)
1
1
2
2




n!m!p!
nβ + 1
mβ + 1



"
#1
pβ+1
pβ+1 β |V | ≤
∂ n+m+p f (z
−
a
)
+
(b
−
z)
3
3

×

∂rn ∂sm ∂tp 
pβ
+
1


α


n+m+p

∂
f


if n m p ∈ Lα ([a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ]) , α > 1, α−1 + β −1 = 1;


∂r ∂s ∂t







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|]




n+m+p f 
p
p
p
p ∂

× [(z − a3 ) + (b3 − z) + |(z − a3 ) − (b3 − z) |] n m p 


∂r ∂s ∂t 1


n+m+p

∂
f


if n m p ∈ L1 ([a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ]) ;
∂r ∂s ∂t
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)
n+m+p Z
∂
f =
∂rn ∂sm ∂tp α
b1
a1
Z
b2
a2
Z
b3
a3
n+m+p α
α1
∂
f < ∞.
∂rn ∂sm ∂tp dtdsdr
Proof.
Z b1 Z b2 Z b3
n+m+p
∂
f
(r,
s,
t)
|V | = Pn (x, r) Qm (y, s) Sp (z, t)
dtdsdr
∂rn ∂sm ∂tp
a1
a2
a3
n+m+p
Z b1 Z b2 Z b3
∂
f (r, s, t) ≤
|Pn (x, r) Qm (y, s) Sp (z, t)| dtdsdr.
∂rn ∂sm ∂tp a1
a2
a3
Using Hölder’s inequality and property of the modulus and integral, then we have that
Z
b1
Z
b2
Z
b3
(2.21)
a1
a2
a3
n+m+p
∂
f
(r,
s,
t)
dtdsdr
|Pn (x, r) Qm (y, s) Sp (z, t)| ∂rn ∂sm ∂tp J. Inequal. Pure and Appl. Math., 4(3) Art. 58, 2003
http://jipam.vu.edu.au/
A N I NTEGRAL A PPROXIMATION











≤










9
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,
∞
n+m+p R R R
β1
∂
b1 b2 b3
β
f |Pn (x, r) Qm (y, s) Sp (z, t)| dtdsdr ,
∂rn ∂sm ∂tp a1 a2 a3
α
α > 1, α−1 + β −1 = 1;
n+m+p ∂
sup
|Pn (x, r) Qm (y, s) Sp (z, t)| .
∂rn ∂sm ∂tfp 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
b2
Z
|Pn (x, r)| dr
=
a1
Z
b3
|Qm (y, s)| ds
a2
|Sp (z, t)| dt
a3
Z y
Z b1
Z b2
(r − a1 )n
(b1 − r)n
(s − a2 )m
(b2 − s)m
=
dr +
dr
ds +
ds
n!
n!
m!
m!
a1
x
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)!
(z − a3 )p+1 + (b3 − z)p+1
×
(p + 1)!
Z
x
giving the first inequality in (2.20).
Now, if we again use (2.21) we have
Z b1 Z b2 Z b3
β1
β
|Pn (x, r) Qm (y, s) Sp (z, t)| dtdsdr
a1
a2
a3
Z
b1
=
|Pn (x, r)|β dr
β1 Z
1
n!m!p!
Z
×
Z
x
(r − a1 )nβ dr +
Z
mβ
(s − a2 )
Z
(b1 − r)nβ dr
b2
mβ
(b2 − s)
ds +
β1
β1
ds
y
z
pβ
Z
b3
(t − a3 ) dt +
a3
"
β1
|Sp (z, t)|β dt
x
y
×
b3
a3
b1
Z
a1
a2
=
β1 Z
|Qm (y, s)|β ds
a2
a1
=
b2
β1
(b3 − t) dt
pβ
z
nβ+1
nβ+1
1
(x − a1 )
+ (b1 − x)
n!m!p!
nβ + 1
"
#1
pβ+1
pβ+1 β
(z − a3 )
+ (b3 − z)
×
pβ + 1
# β1 "
mβ+1
(y − a2 )
mβ+1
+ (b2 − y)
mβ + 1
# β1
producing the second inequality in (2.20).
J. Inequal. Pure and Appl. Math., 4(3) Art. 58, 2003
http://jipam.vu.edu.au/
10
A. S OFO
Finally, we have
|Pn (x, r) Qm (y, s) Sp (z, t)|
sup
(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 ]
n
t∈[a3 ,b3 ]
(x − a1 ) (b1 − x)
(y − a2 )m (b2 − y)m
= max
,
max
,
n!
n!
m!
m!
p
p
(z − a3 ) (b3 − z)
× max
,
p!
p!
n
n
(x − a1 )n − (b1 − x)n 1
(x − a1 ) + (b1 − x)
=
+ n!m!p!
2
2
(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
n
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.
The following corollary is a consequence of Theorem 2.4.
Corollary 2.5. Under the assumptions of Corollary 2.3, we have the inequality
#
 "
n+1
m+1
p+1 n+m+p
(b
−
a
)
(b
−
a
)
(b
−
a
)
∂
f

1
1
2
2
3
3



∂rn ∂sm ∂tp ,
n+m+p (n + 1)! (m + 1)! (p + 1)!

2

∞






"
#1


nβ+1
mβ+1
pβ+1 β

(b
−
a
)
(b
−
a
)
(b
−
a
)
1

1
1
2
2
3
3
 n+m+p
V̄ ≤
2
n!m!p!
(nβ + 1) (mβ + 1) (pβ + 1)
n+m+p 

∂
f 

,

×

n
m
p

∂r
∂s
∂t

α




n+m+p 


1
f n
m
p ∂

 n+m+p
(b1 − a1 ) (b2 − a2 ) (b3 − a3 ) n m p ,
2
n!m!p!
∂r ∂s ∂t 1
where k·kα (α ∈ [1, ∞)) are the Lebesgue norms on [a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ] .
The following two corollaries concern the estimation of V at the end points.
J. Inequal. Pure and Appl. Math., 4(3) Art. 58, 2003
http://jipam.vu.edu.au/
A N I NTEGRAL A PPROXIMATION
11
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
a2
p−1
n−1 m−1
X
XX
b3
f (r, s, t) dtdsdr −
a3
+ (−1)p
i=0 j=0 k=0
n−1 m−1
X
X
+ (−1)
b3
Z
Xi (a1 ) Yj (a2 )
S̄p (a3 , t)
a3
i=0 j=0
m
p−1
n−1 X
X
Xi (a1 ) Zk (a3 )
+ (−1)
p−1
m−1
XX
Q̄m (a2 , s)
a2
Z
P̄n (a1 , r)
a1
j=0 k=0
m+p
− (−1)
n−1
X
− (−1)
m−1
X
j=0
b2
Z
∂ i+m+k f
ds
∂xi ∂sm ∂z k
∂ n+j+k f
dr
∂rn ∂y j ∂z k
b3
Z
Xi (a1 )
Q̄m (a2 , s) S̄p (a3 , t)
a2
i=0
n+p
b1
Yj (a2 ) Zk (a3 )
b1
Z
a3
b3
Z
Yj (a2 )
P̄n (a1 , r) S̄p (a3 , t)
a1
Z
p−1
X
n+m
− (−1)
Zk (a3 )
k=0
a3
b1
a1
Z
b2
a2
∂ i+j+k f
∂xi ∂y j ∂z k
∂ i+j+p f
dt
∂xi ∂y j ∂tp
b2
Z
i=0 k=0
n
Xi (a1 ) Yj (a2 ) Zk (a3 )
∂ i+m+p f
dtds
∂xi ∂sm ∂tp
∂ n+j+p f
dtdr
∂rn ∂y j ∂tp
∂
f
P̄n (a1 , r) Q̄m (a2 , s) n m k dsdr
∂r ∂s ∂z
n+m+k

n+1
m+1
p+1 ∂ n+m+p f 
(b
(b
(b
1 − a1 )
2 − a2 )
3 − a3 )



∂rn ∂sm ∂tp ,

(n
+
1)!
(m
+
1)!
(p
+
1)!

∞







∂ n+m+p f


if
∈ L∞ ([a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ]) ;


∂rn ∂sm ∂tp






p+ β1 m+ β1
n+ β1

∂ n+m+p f 
(b
(b
(b
3 − a3 )
2 − a2 )
1 − a1 )

,


1 1 ·
1 ·
n ∂sm ∂tp 
∂r
β
β
β
p! (pβ + 1)
m! (mβ + 1)
n! (nβ + 1)
α
≤




∂ n+m+p f


∈ Lα ([a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ]) , α > 1, α−1 + β −1 = 1;
if

n ∂sm ∂tp

∂r




n+m+p 


(b1 − a1 )n (b2 − a2 )m (b3 − a3 )p f 
∂


∂rn ∂sm ∂tp ,

n!m!p!

1







∂ n+m+p f


if n m p ∈ L1 ([a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ]) ,
∂r ∂s ∂t
where
Xi (a1 ) :=
(b1 − a1 )i+1
(b2 − a2 )j+1
(b3 − a3 )k+1
, Yj (a2 ) :=
, Zk (a3 ) :=
.
(i + 1)!
(j + 1)!
(k + 1)!
(r − b1 )n
(s − b2 )m
P̄n (a1 , r) =
, r ∈ [a1 , b1 ] ; Q̄m (a2 , s) =
, s ∈ [a2 , b2 ]
n!
m!
J. Inequal. Pure and Appl. Math., 4(3) Art. 58, 2003
http://jipam.vu.edu.au/
12
A. S OFO
and
(t − b3 )p
S̄p (a3 , t) =
; t ∈ [a3 , b3 ] .
p!
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
b3
Z
f (r, s, t) dtdsdr −
a2
p−1
n−1 m−1
XX
X
a3
p
+ (−1)
i=0 j=0 k=0
n−1 m−1
X
X
+ (−1)
b3
Z
Xi (b1 ) Yj (b2 )
a3
i=0 j=0
m
p−1
n−1 X
X
Xi (b1 ) Zk (b3 )
+ (−1)
p−1
m−1
XX
Q̄m (b2 , s)
a2
Z
P̄n (b1 , r)
a1
j=0 k=0
m+p
− (−1)
n−1
X
m−1
X
b2
Z
∂ n+j+k f
dr
∂rn ∂y j ∂z k
Q̄m (b2 , s) S̄p (b3 , t)
a2
a3
b1 Z
b3
Z
Yj (b2 )
j=0
∂ i+m+k f
ds
∂xi ∂sm ∂z k
b3
Z
Xi (b1 )
i=0
− (−1)n+p
b1
Yj (b2 ) Zk (b3 )
P̄n (b1 , r) S̄p (b3 , t)
a1
a3
∂ i+j+k f
∂xi ∂y j ∂z k
∂ i+j+p f
S̄p (b3 , t) i j p dt
∂x ∂y ∂t
b2
Z
i=0 k=0
n
Xi (a1 ) Yj (a2 ) Zk (a3 )
∂ i+m+p f
dtds
∂xi ∂sm ∂tp
∂ n+j+p f
dtdr
∂rn ∂y j ∂tp
∂ n+m+k f
P̄n (b1 , r) Q̄m (b2 , s) n m k dsdr
∂r ∂s ∂z
a1
a2
k=0
n+m+p (b1 − a1 )n+1 (b2 − a2 )m+1 (b3 − a3 )p+1 f ∂
∂rn ∂sm ∂tp ,
(n + 1)! (m + 1)! (p + 1)!
∞
Z
p−1
X
n+m
− (−1)
Zk (b3 )
b1
Z
b2














∂ n+m+p f


if
∈ L∞ ([a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ]) ;

n ∂sm ∂tp

∂r






p+ β1 m+ β1
n+ β1

∂ n+m+p f 
(b
(b
(b
3 − a3 )
2 − a2 )
1 − a1 )

,


1 1 ·
1 ·
n
m p

n! (nβ + 1) β m! (mβ + 1) β p! (pβ + 1) β ∂r ∂s ∂t α
≤




∂ n+m+p f


if
∈ Lα ([a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ]) , α > 1, α−1 + β −1 = 1;

n ∂sm ∂tp

∂r




n+m+p 


(b1 − a1 )n (b2 − a2 )m (b3 − a3 )p f 
∂


∂rn ∂sm ∂tp ,

n!m!p!

1







∂ n+m+p f


if n m p ∈ L1 ([a1 , b1 ] × [a2 , b2 ] × [a3 , b3 ]) ,
∂r ∂s ∂t
where
(−1)i (b1 − a1 )i+1
(−1)j (b2 − a2 )j+1
(−1)k (b3 − a3 )k+1
Xi (b1 ) :=
, Yj (b2 ) :=
, Zk (b3 ) :=
.
(i + 1)!
(j + 1)!
(k + 1)!
J. Inequal. Pure and Appl. Math., 4(3) Art. 58, 2003
http://jipam.vu.edu.au/
A N I NTEGRAL A PPROXIMATION
13
(r − a1 )n
P̄n (b1 , r) =
; r ∈ [a1 , b1 ] ,
n!
(s − a2 )m
; s ∈ [a2 , b2 ]
Q̄m (b2 , s) =
m!
and
(t − a3 )p
S̄p (b3 , t) =
; t ∈ [a3 , b3 ] .
p!
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