AN INEQUALITY FOR DIVIDED DIFFERENCES IN
HIGH DIMENSIONS
Divided Differences
AIMIN XU AND ZHONGDI CEN
vol. 10, iss. 4, art. 103, 2009
Institute of Mathematics
Zhejiang Wanli University
Ningbo, 315100, China
EMail: xuaimin1009@yahoo.com.cn
Aimin Xu and Zhongdi Cen
czdningbo@tom.com
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Contents
Received:
26 November, 2008
Accepted:
05 November, 2009
Communicated by:
JJ
II
J. Pečarić
J
I
2000 AMS Sub. Class.:
26D15.
Page 1 of 13
Key words:
Mixed partial divided difference, Convex hull.
Abstract:
This paper is devoted to an inequality for divided differences in the multivariate
case which is similar to the inequality obtained by [J. Pečarić, and M. Rodić
Lipanović, On an inequality for divided differences, Asian-European Journal of
Mathematics, Vol. 1, No. 1 (2008), 113-120].
Acknowledgements:
The work is supported by the Education Department of Zhejiang Province of
China (Grant No. Y200806015).
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Contents
1
Introduction
3
2
Notations and Definitions
4
3
Main Result
6
Divided Differences
Aimin Xu and Zhongdi Cen
vol. 10, iss. 4, art. 103, 2009
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1.
Introduction
Recently, Pečarić and Lipanović [3] have proved the following inequality for divided
differences.
Theorem 1.1. Let f, g be two n−1 times continuously differentiable functions on the
interval I ⊆ R and n times differentiable on the interior I ◦ of I, with the properties
(n) (x)
that g (n) (x) > 0 on I ◦ , and that the function fg(n) (x)
is bounded on I ◦ . Then for
P
xi , yi ∈ I (i = 1, 2, . . . , n) such that xi ≥ yi for all i = 1, 2, . . . , n and ni=1 (xi −
yi ) 6= 0, the following estimation holds true:
f (n) (x)
f (n) (x)
[x1 , . . . , xn ]f − [y1 , . . . , yn ]f
inf
≤
≤ sup (n)
.
x∈I ◦ g (n) (x)
[x1 , . . . , xn ]g − [y1 , . . . , yn ]g
(x)
x∈I ◦ g
Divided Differences
Aimin Xu and Zhongdi Cen
vol. 10, iss. 4, art. 103, 2009
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This theorem generalized the following result obtained by [2].
Corollary 1.2. Let f, g be two continuously differentiable functions on [a, b] and
twice differentiable on (a, b),with the properties that g 00 > 0 on (a, b), and that the
00
function fg00 is bounded on (a, b). Then for a < c ≤ d < b, the following estimation
holds:
00
f (x)
≤
x∈(a,b) g 00 (x)
inf
f (b)−f (d)
b−d
g(b)−g(d)
b−d
−
−
f (c)−f (a)
c−a
g(c)−g(a)
c−a
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00
f (x)
.
00
x∈(a,b) g (x)
≤ sup
It is worth noting that the technique of the proof for Theorem 1.1 in [3] is very
natural and useful. In this paper, using the technique and following the definition of
mixed partial divided difference proposed by [1], we present a similar inequality for
divided differences in the multivariate case.
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2.
Notations and Definitions
The following notations will be used in this paper.
We denote by Rm the m−dimensional Euclidean space. Let x ∈ Rm be a vector
denoted by (x1 , x2 , . . . , xm ). Let N0 be the set of nonnegative integers. Then it is
m
i
m
obvious that Nm
0 ⊆ R . Denote by e ∈ N0 a unit vector whose jth component is
δij , where
0,
j 6= i;
δij =
1,
j = i.
Let 00 = 1. For α = (α1 , α2 , . . . , αm ) ∈ Nm
xα = xα1 1 xα2 2 · · · xαmm ,
0 , we define
P
Q
i
m
m
and then we have xi = xe . Define |α| = m
i=1 αi , α! =
i=1 αi !. For x, y ∈ R ,
we denote x ≥ y, if xi ≥ yi , i = 1, 2, . . . , m.
Further, let
α 1 α 2
α m
∂
∂
∂
α
···
D =
∂x1
∂x2
∂xm
be a mixed partial differential operator of order |α|.
For x0 , x1 , . . . , xn ∈ Rm , we denote by
(
!
)
n
n
X
X
x 0 , x1 , . . . , x n =
1−
tj x0 + t1 x1 + · · · + tn xn | tj ≥ 0,
tj ≤ 1
j=1
j=1
the convex hull of x0 , x1 , . . . , xn ∈ Rm . Then according to the Hermite-Genocchi
formula for univariate divided difference, the multivariate divided difference (or
mixed partial divided difference) of order n can be defined by the following formula.
Divided Differences
Aimin Xu and Zhongdi Cen
vol. 10, iss. 4, art. 103, 2009
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0
1
n
Definition 2.1 ([1], see also [4, 5]). Let α ∈ Nm
0 with |α| = n, and x , x , . . . , x ∈
m
R . Then the mixed partial divided difference of order n of f is defined by
[x0 , x1 , . . . , xn ]α f
Z
Dα f
=
Sn
1−
n
X
!
!
tj
x0 + t1 x1 + · · · + tn xn dt1 dt2 . . . dtn ,
j=1
Divided Differences
where
Aimin Xu and Zhongdi Cen
(
n
S =
(t1 , t2 , . . . , tn )| tj ≥ 0, j = 1, 2, . . . , n;
n
X
)
tj ≤ 1 .
j=1
It is easy to see that if we let m = 1, then [x0 , x1 , . . . , xn ]α f is the ordinary divided difference in the univariate case. By the definition of the mixed partial divided
difference, we also conclude that
[xσ0 , xσ1 , . . . , xσn ]α f = [x0 , x1 , . . . , xn ]α f
if (σ0 , σ1 , . . . , σn ) is a permutation of (0, 1, . . . , n). Finally, we give another definition to end this section.
0
1
n
m
Definition 2.2 ([4, 5]). Let α ∈ Nm
0 with |α| = n, and x , x , . . . , x ∈ R . Then
the Newton fundamental functions are defined by
(
1,
n = 0,
P
Q
ij
j n−1
n
j−1
e
ωα (x, {x }j=0 ) =
) ,
n > 0.
j=1 (x − x
ei1 +···+ein =α
vol. 10, iss. 4, art. 103, 2009
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3.
Main Result
We start this section with two lemmas. Using the definition of the mixed partial
divided difference of f we have the following lemma.
n
0
1
n
Lemma 3.1 (cf. [4, 5]). Let α ∈ Nm
0 with |α| = n. If f ∈ C (hx , x , . . . , x i), then
there exists a point ξ ∈ hx0 , x1 , . . . , xn i such that
Divided Differences
1
[x , x , . . . , x ]α f = Dα f (ξ).
n!
0
1
n
Aimin Xu and Zhongdi Cen
vol. 10, iss. 4, art. 103, 2009
Also using the definition, we have the recurrence relations of the divided differences.
Lemma 3.2. For β ∈
Nm
0
with |β| = n − 1, we have
[x1 , x2 , . . . , xn ]β f − [x0 , x1 , . . . , xn−1 ]β f =
m
X
Contents
i
(xn − x0 )e [x0 , x1 , . . . , xn ]β+ei f.
i=1
Proof. By the chain rule for the derivative of the composite function, we have
∂ β
D f
∂tn
1−
n
X
!
tj
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x + · · · + tn x
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!
0
JJ
n
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j=1
=
m
X
i=1
D
β+ei
f
1−
n
X
j=1
!
tj
Close
!
0
x + · · · + tn x
n
n
0 ei
(x − x ) .
Hence,
Z
1−
Pn−1
j=1
tj
0
m
X
D
β+ei
1−
f
n
X
i=1
!
tj
!
i
x0 + · · · + tn xn (xn − x0 )e dtn
j=1
= Dβ f
t1 x1 + · · · +
n−1
X
1−
!
!
xn
tj
Divided Differences
j=1
β
−D f
1−
n−1
X
!
0
x + · · · + tn−1 x
tj
Aimin Xu and Zhongdi Cen
!
n−1
.
vol. 10, iss. 4, art. 103, 2009
j=1
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Thus,
Z
m
X
D
β+ei
1−
f
n
X
S n i=1
tj
Contents
!
!
0
x + · · · + tn x
0 ei
n
n
(x − x ) dtn . . . dt1
j=1
Z
β
=
1
t1 x + · · · +
D f
1−
S n−1
!
!
tj
x
n
dt1 . . . dtn−1
β
D f
S n−1
1−
n−1
X
!
tj
!
0
x + · · · + tn−1 x
n−1
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dt1 . . . dtn−1 ,
j=1
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which implies
m
X
II
j=1
Z
−
n−1
X
JJ
i
(xn − x0 )e [x0 , x1 , . . . , xn ]β+ei f = [x1 , x2 , . . . , xn ]β f − [x0 , x1 , . . . , xn−1 ]β f.
i=1
This completes the proof.
Let hx0 , . . . , xn , y 0 , . . . , y n i be the convex hull of x0 , x1 , . . . , xn , y 0 , y 1 , . . . , y n .
Now, we state our main theorem as follows.
Theorem 3.3. Let f, g ∈ C n+1 (hx0 , . . . , xn , y 0 , . . . , y n i), α ∈ Nm
0 , and |α| = n. If
i
for all z ∈ hx0 , . . . , xn , y 0 , . . . , y n i we have Dα+e g(z) > 0 (i = 1, 2, . . . , m) and
P P
j
j ei
for all xj , y j (j = 0, 1, . . . , n) we have xj ≥ y j and nj=0 m
6= 0,
i=1 (x − y )
then
[x0 , x1 , . . . , xn ]α f − [y 0 , y 1 , . . . , y n ]α f
L≤ 0 1
≤ U,
[x , x , . . . , xn ]α g − [y 0 , y 1 , . . . , y n ]α g
Divided Differences
Aimin Xu and Zhongdi Cen
vol. 10, iss. 4, art. 103, 2009
where
i
L = min
Dα+e f (z)
,
i
,y 0 ,...,y n i D α+e g(z)
inf
n
1≤i≤m z∈hx0 ,...,x
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i
Dα+e f (z)
U = max
sup
.
i
1≤i≤m z∈hx0 ,...,xn ,y 0 ,...,y n i D α+e g(z)
Proof. It is evident that
Dα+e f (z)
i
,y 0 ,...,y n i D α+e g(z)
inf
n
z∈hx0 ,...,x
≤
Since D
(3.1)
α+ei
D
f (z)
D
f (z)
≤
sup
≤ U.
i
i
α+e
α+e
D
g(z) z∈hx0 ,...,xn ,y0 ,...,yn i D
g(z)
g(z) > 0, 1 ≤ i ≤ m, then
i
i
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α+ei
α+ei
II
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i
L≤
JJ
i
LDα+e g(z) ≤ Dα+e f (z) ≤ U Dα+e g(z).
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Let
x̄ =
1−
n
X
!
x0 + t1 x1 + · · · + tn xn ,
tj
j=1
ȳ =
1−
n
X
!
tj
y 0 + t1 y 1 + · · · + tn y n .
j=1
Divided Differences
i
i
Since f, g ∈ C n+1 (hx0 , . . . , xn , y 0 , . . . , y n i), Dα+e g(z) and Dα+e f (z) are continuous on each of the contours between the points ȳ and x̄. Then we can find three line
integrals satisfying
Z x̄ X
Z x̄ X
m
m
i
α+ei
L
D
g(z)dzi ≤
Dα+e f (z)dzi
ȳ
i=1
ȳ
Z
≤U
i=1
m
x̄ X
ȳ
D
α+ei
g(z)dzi .
i=1
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This implies that
(3.2)
Aimin Xu and Zhongdi Cen
vol. 10, iss. 4, art. 103, 2009
L[Dα g(x̄) − Dα g(ȳ)] ≤ Dα f (x̄) − Dα f (ȳ) ≤ U [Dα g(x̄) − Dα g(ȳ)].
Integrating (3.2) with respect to t1 , t2 , . . . , tn over the n-dimensional simplex Sn as
defined in the previous section, we arrive at
L([x0 , x1 , . . . ,xn ]α g − [y 0 , y 1 , . . . , y n ]α g)
≤ [x0 , x1 , . . . , xn ]α f − [y 0 , y 1 , . . . , y n ]α f
≤ U ([x0 , x1 , . . . , xn ]α g − [y 0 , y 1 , . . . , y n ]α g).
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Using Lemmas 3.1 and 3.2, we have
[x0 , x1 , . . . , xn ]α g − [y 0 , y 1 , . . . , y n ]α g
n X
m
X
i
=
(xj − y j )e [y 0 , . . . , y j , xj , . . . , xn ]α+ei g
j=0 i=1
n
m
XX
1
i
i
(xj − y j )e Dα+e g(ξi,j ),
=
(n + 1)! j=0 i=1
P P
j
j ei
where ξi,j ∈ hx0 , . . . , xn , y 0 , . . . , y n i. Since xj ≥ y j , nj=0 m
6= 0
i=1 (x − y )
α+ei
and D
g(z) > 0, we have
[x0 , x1 , . . . , xn ]α g − [y 0 , y 1 , . . . , y n ]α g > 0.
Divided Differences
Aimin Xu and Zhongdi Cen
vol. 10, iss. 4, art. 103, 2009
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Thus,
[x0 , x1 , . . . , xn ]α f − [y 0 , y 1 , . . . , y n ]α f
L≤ 0 1
≤ U.
[x , x , . . . , xn ]α g − [y 0 , y 1 , . . . , y n ]α g
This completes the proof.
P
i
Considering pi (z) = ei0 +ei1 +···+ein =α+ei z α+e , i = 1, 2, . . . , m, we can obtain
that
pi (z) = ωα+ei (z, {0}nj=0 ).
Further, let
m
X
1
pi (z).
p(z) =
(n + 1)! i=1
By calculating, we have
p(z) =
m
X
i=1
1
i
z α+e .
i
(α + e )!
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This implies that, for 1 ≤ i ≤ m,
i
Dα+e p(z) = 1 > 0,
and
α
D p(z) =
m
X
i
ze .
i=1
Then
Divided Differences
[x0 , x1 , . . . , xn ]α p
Z
=
Dα p
Aimin Xu and Zhongdi Cen
vol. 10, iss. 4, art. 103, 2009
1−
Sn
=
=
=
1−
n
X
Sn
m Z
X
i=1
!
!
tj
x0 + t1 x1 + · · · + tn xn dt1 dt2 · · · dtn
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j=1
m Z
X
i=1
n
X
Sn
1
(n + 1)!
!ei
!
x0 + t1 x1 + · · · + tn xn
tj
Contents
dt1 dt2 · · · dtn
j=1
1−
n
X
!
tj
!
i
i
i
(x0 )e + t1 (x1 )e + · · · + tn (xn )e
dt1 dt2 · · · dtn
j=1
n
m
XX
j ei
(x ) .
j=0 i=1
Therefore, if we take g(z) = p(z) in Theorem 3.3, we have the following corollary.
Corollary 3.4. Let f ∈ C n+1 (hx0 , . . . , xn , y 0 , . . . , y n i), α ∈ Nm
, and |α| = n. If
P P 0 j
j ei
for all xj , y j (j = 0, 1, . . . , n) we have xj ≥ y j and nj=0 m
6= 0,
i=1 (x − y )
then
L0 ≤ [x0 , x1 , . . . , xn ]α f − [y 0 , y 1 , . . . , y n ]α f ≤ U 0 ,
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where
L0 =
n X
m
X
1
α+ei
j
j ei
min
inf
D
f
(z)
(x
−
y
) ,
(n + 1)! 1≤i≤m z∈hx0 ,...,xn ,y0 ,...,yn i
j=0 i=1
U0 =
n X
m
X
1
i
i
max
sup
Dα+e f (z)
(xj − y j )e .
(n + 1)! 1≤i≤m z∈hx0 ,...,xn ,y0 ,...,yn i
j=0 i=1
In fact, from the procedure of the proof of Theorem 3.3, it is not difficult to find
that
of the corollary
If we replace xj ≥ y j and
Pn the
Pconditions
Pn can
Pmbe weakened.
m
j
j ei
j
j ei
j=0
i=1 (x − y ) 6= 0 by
j=0
i=1 (x − y ) > 0, the corollary holds true
as well.
Divided Differences
Aimin Xu and Zhongdi Cen
vol. 10, iss. 4, art. 103, 2009
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References
[1] A. CAVARETTA, C. MICCHELLI AND A. SHARMA, Multivariate interpolation and the Radon transform, Math. Z., 174A (1980), 263–279.
[2] A.I. KECHRINIOTIS AND N.D. ASSIMAKIS, On the inequality of the difference of two integral means and applications for pdfs, J. Inequal. Pure and
Appl. Math., 8(1) (2007), Art. 10. [ONLINE: http://jipam.vu.edu.au/
article.php?sid=839].
[3] J. PEČARIĆ, AND M. RODIĆ LIPANOVIĆ, On an inequality for divided differences, Asian-European J. Math., 1(1) (2008), 113–120.
[4] X. WANG AND M. LAI, On multivariate newtonian interpolation, Scientia
Sinica, 29 (1986), 23–32.
[5] X. WANG AND A. XU, On the divided difference form of Faà di Bruno formula
II, J. Comput. Math., 25(6) (2007), 697–704.
Divided Differences
Aimin Xu and Zhongdi Cen
vol. 10, iss. 4, art. 103, 2009
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