Key words and phrases: Mixed partial divided difference, Convex hull

AN INEQUALITY FOR DIVIDED DIFFERENCES IN HIGH DIMENSIONS

AIMIN XU AND ZHONGDI CEN INSTITUTE OFMATHEMATICS

ZHEJIANGWANLIUNIVERSITY

NINGBO, 315100, CHINA

xuaimin1009@yahoo.com.cn czdningbo@tom.com

Received 26 November, 2008; accepted 05 November, 2009 Communicated by J. Peˇcari´c

ABSTRACT. This paper is devoted to an inequality for divided differences in the multivariate case which is similar to the inequality obtained by [J. Peˇcari´c, and M. Rodi´c Lipanovi´c, On an inequality for divided differences, Asian-European Journal of Mathematics, Vol. 1, No. 1 (2008), 113-120].

Key words and phrases: Mixed partial divided difference, Convex hull.

2000 Mathematics Subject Classification. 26D15.

1. INTRODUCTION

Recently, Peˇcari´c and Lipanovi´c [3] have proved the following inequality for divided differ- ences.

Theorem 1.1. Letf, gbe twon−1times continuously differentiable functions on the interval I ⊆Randntimes differentiable on the interiorI^◦ofI, with the properties thatg⁽ⁿ⁾(x)>0on I^◦, and that the function ^f_g⁽ⁿ⁾(n)(x)^(x) is bounded onI^◦. Then forxi, yi ∈I(i= 1,2, . . . , n)such that x_i ≥y_ifor alli= 1,2, . . . , nandPn

i=1(x_i−y_i)6= 0, the following estimation holds true:

x∈Iinf^◦

f⁽ⁿ⁾(x)

g⁽ⁿ⁾(x) ≤ [x₁, . . . , x_n]f−[y₁, . . . , y_n]f

[x₁, . . . , x_n]g−[y₁, . . . , y_n]g ≤ sup

x∈I^◦

f⁽ⁿ⁾(x) g⁽ⁿ⁾(x). This theorem generalized the following result obtained by [2].

Corollary 1.2. Letf, g be two continuously differentiable functions on[a, b] and twice differ- entiable on(a, b),with the properties thatg⁰⁰ >0on(a, b), and that the function ^f_g⁰⁰00 is bounded on(a, b). Then fora < c ≤d < b, the following estimation holds:

x∈(a,b)inf f⁰⁰(x) g⁰⁰(x) ≤

f(b)−f(d)

b−d −^f(c)−f(a)_c−a

g(b)−g(d)

b−d −^g(c)−g(a)_c−a ≤ sup

x∈(a,b)

f⁰⁰(x) g⁰⁰(x).

The work is supported by the Education Department of Zhejiang Province of China (Grant No. Y200806015).

321-08

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.

2. NOTATIONS AND DEFINITIONS

The following notations will be used in this paper.

We denote byR^m them−dimensional Euclidean space. Letx∈ R^mbe a vector denoted by (x₁, x₂, . . . , x_m). LetN0 be the set of nonnegative integers. Then it is obvious thatN^m0 ⊆ R^m. Denote byeⁱ ∈N^m0 a unit vector whosejth component isδ_ij,where

δij =

0, j 6=i;

1, j =i.

Let0⁰ = 1. Forα = (α1, α2, . . . , αm) ∈ N^m0 , we definex^α = x^α₁¹x^α₂²· · ·x^α_m^m, and then we have x_i = x^ei. Define|α| = Pm

i=1α_i, α! = Qm

i=1α_i!. For x, y ∈ R^m, we denote x ≥ y, if x_i ≥y_i,i= 1,2, . . . , m.

Further, let

D^α = ∂

∂x₁ α1

∂

∂x₂ α2

· · · ∂

∂x_m αm

be a mixed partial differential operator of order|α|.

Forx⁰, x¹, . . . , xⁿ ∈R^m, we denote by x⁰, x¹, . . . , xⁿ

= (

1−

n

X

j=1

t_j

!

x⁰+t₁x¹+· · ·+t_nxⁿ|t_j ≥0,

n

X

j=1

t_j ≤1 )

the convex hull of x⁰, x¹, . . . , xⁿ ∈ R^m. Then according to the Hermite-Genocchi formula for univariate divided difference, the multivariate divided difference (or mixed partial divided difference) of orderncan be defined by the following formula.

Definition 2.1 ([1], see also [4, 5]). Letα ∈N^m0 with|α| =n, andx⁰, x¹, . . . , xⁿ ∈R^m. Then the mixed partial divided difference of ordernoff is defined by

[x⁰, x¹, . . . , xⁿ]αf = Z

Sⁿ

D^αf

1−

n

X

j=1

tj

!

x⁰+t1x¹+· · ·+tnxⁿ

!

dt1dt2. . . dtn,

where

Sⁿ = (

(t₁, t₂, . . . , t_n)|t_j ≥0, j = 1,2, . . . , n;

n

X

j=1

t_j ≤1 )

.

It is easy to see that if we letm= 1, then[x⁰, x¹, . . . , xⁿ]_αf is the ordinary divided difference in the univariate case. By the definition of the mixed partial divided difference, we also conclude that

[x^σ0, x^σ1, . . . , x^σn]_αf = [x⁰, x¹, . . . , xⁿ]_αf

if(σ₀, σ₁, . . . , σ_n)is a permutation of(0,1, . . . , n). Finally, we give another definition to end this section.

Definition 2.2 ([4, 5]). Letα∈ N^m0 with|α|=n, andx⁰, x¹, . . . , xⁿ ∈ R^m. Then the Newton fundamental functions are defined by

ω_α(x,{x^j}ⁿ⁻¹_j=0) =

( 1, n = 0,

P

eⁱ¹+···+eⁱⁿ=α

Qn

j=1(x−x^j−1)^eij, n > 0.

3. MAINRESULT

We start this section with two lemmas. Using the definition of the mixed partial divided difference off we have the following lemma.

Lemma 3.1 (cf. [4, 5]). Let α ∈ N^m0 with|α| = n. If f ∈ Cⁿ(hx⁰, x¹, . . . , xⁿi), then there exists a pointξ∈ hx⁰, x¹, . . . , xⁿisuch that

[x⁰, x¹, . . . , xⁿ]αf = 1

n!D^αf(ξ).

Also using the definition, we have the recurrence relations of the divided differences.

Lemma 3.2. Forβ ∈N^m0 with|β|=n−1, we have [x¹, x², . . . , xⁿ]_βf −[x⁰, x¹, . . . , xⁿ⁻¹]_βf =

m

X

i=1

(xⁿ−x⁰)^ei[x⁰, x¹, . . . , xⁿ]_β+eⁱf.

Proof. By the chain rule for the derivative of the composite function, we have

∂

∂t_nD^βf

1−

n

X

j=1

tj

!

x⁰+· · ·+tnxⁿ

!

=

m

X

i=1

D^β+eif

1−

n

X

j=1

tj

!

x⁰+· · ·+tnxⁿ

!

(xⁿ−x⁰)^ei.

Hence, Z 1−Pn−1

j=1tj

0

m

X

i=1

D^β+eif

1−

n

X

j=1

tj

!

x⁰+· · ·+tnxⁿ

!

(xⁿ−x⁰)^eidtn

=D^βf t₁x¹ +· · ·+ 1−

n−1

X

j=1

t_j

! xⁿ

!

−D^βf

1−

n−1

X

j=1

tj

!

x⁰+· · ·+tn−1xⁿ⁻¹

! .

Thus, Z

Sⁿ m

X

i=1

D^β+eif

1−

n

X

j=1

t_j

!

x⁰+· · ·+t_nxⁿ

!

(xⁿ−x⁰)^eidt_n. . . dt₁

= Z

Sⁿ⁻¹

D^βf t1x¹+· · ·+ 1−

n−1

X

j=1

tj

! xⁿ

!

dt1. . . dtn−1

− Z

Sⁿ⁻¹

D^βf

1−

n−1

X

j=1

t_j

!

x⁰+· · ·+tn−1xⁿ⁻¹

!

dt₁. . . dtn−1,

which implies

m

X

i=1

(xⁿ−x⁰)^ei[x⁰, x¹, . . . , xⁿ]_β+eⁱf = [x¹, x², . . . , xⁿ]_βf −[x⁰, x¹, . . . , xⁿ⁻¹]_βf.

This completes the proof.

Let hx⁰, . . . , xⁿ, y⁰, . . . , yⁿi be the convex hull of x⁰, x¹, . . . , xⁿ, y⁰, y¹, . . . , yⁿ. Now, we state our main theorem as follows.

Theorem 3.3. Let f, g ∈ Cⁿ⁺¹(hx⁰, . . . , xⁿ, y⁰, . . . , yⁿi), α ∈ N^m0 , and |α| = n. If for all z ∈ hx⁰, . . . , xⁿ, y⁰, . . . , yⁿi we have D^α+eig(z) > 0 (i = 1,2, . . . , m) and for all x^j, y^j (j = 0,1, . . . , n)we havex^j ≥y^j andPn

j=0

Pm

i=1(x^j−y^j)^ei 6= 0, then L≤ [x⁰, x¹, . . . , xⁿ]_αf −[y⁰, y¹, . . . , yⁿ]_αf

[x⁰, x¹, . . . , xⁿ]_αg−[y⁰, y¹, . . . , yⁿ]_αg ≤U, where

L= min

1≤i≤m inf

z∈hx⁰,...,xⁿ,y⁰,...,yⁿi

D^α+eif(z) D^α+eig(z), U = max

1≤i≤m sup

z∈hx⁰,...,xⁿ,y⁰,...,yⁿi

D^α+eif(z) D^α+eig(z). Proof. It is evident that

L≤ inf

z∈hx⁰,...,xⁿ,y⁰,...,yⁿi

D^α+eif(z) D^α+eig(z)

≤ D^α+eif(z)

D^α+eig(z) ≤ sup

z∈hx⁰,...,xⁿ,y⁰,...,yⁿi

D^α+eif(z) D^α+eig(z) ≤U.

SinceD^α+eig(z)>0, 1≤i≤m, then

(3.1) LD^α+eig(z)≤D^α+eif(z)≤U D^α+eig(z).

Let

¯

x= 1−

n

X

j=1

tj

!

x⁰+t1x¹+· · ·+tnxⁿ,

¯

y = 1−

n

X

j=1

tj

!

y⁰+t1y¹+· · ·+tnyⁿ.

Sincef, g ∈Cⁿ⁺¹(hx⁰, . . . , xⁿ, y⁰, . . . , yⁿi),D^α+eig(z)andD^α+eif(z)are continuous on each of the contours between the pointsy¯andx. Then we can find three line integrals satisfying¯

L Z ¯x

¯ y

m

X

i=1

D^α+eig(z)dz_i ≤ Z x¯

¯ y

m

X

i=1

D^α+eif(z)dz_i ≤U Z x¯

¯ y

m

X

i=1

D^α+eig(z)dz_i.

This implies that

(3.2) L[D^αg(¯x)−D^αg(¯y)]≤D^αf(¯x)−D^αf(¯y)≤U[D^αg(¯x)−D^αg(¯y)].

Integrating(3.2)with respect tot₁, t₂, . . . , t_nover the n-dimensional simplexS_n as defined in the previous section, we arrive at

L([x⁰, x¹, . . . ,xⁿ]_αg−[y⁰, y¹, . . . , yⁿ]_αg)

≤[x⁰, x¹, . . . , xⁿ]_αf −[y⁰, y¹, . . . , yⁿ]_αf

≤U([x⁰, x¹, . . . , xⁿ]_αg−[y⁰, y¹, . . . , yⁿ]_αg).

Using Lemmas3.1and3.2, we have

[x⁰, x¹, . . . , xⁿ]_αg−[y⁰, y¹, . . . , yⁿ]_αg =

n

X

j=0 m

X

i=1

(x^j−y^j)^ei[y⁰, . . . , y^j, x^j, . . . , xⁿ]_α+eⁱg

= 1

(n+ 1)!

n

X

j=0 m

X

i=1

(x^j−y^j)^eiD^α+eig(ξ_i,j),

whereξi,j ∈ hx⁰, . . . , xⁿ, y⁰, . . . , yⁿi. Sincex^j ≥y^j,Pn j=0

Pm

i=1(x^j−y^j)^ei 6= 0andD^α+eig(z)>

0, we have

[x⁰, x¹, . . . , xⁿ]αg−[y⁰, y¹, . . . , yⁿ]αg >0.

Thus,

L≤ [x⁰, x¹, . . . , xⁿ]_αf −[y⁰, y¹, . . . , yⁿ]_αf [x⁰, x¹, . . . , xⁿ]_αg−[y⁰, y¹, . . . , yⁿ]_αg ≤U.

This completes the proof.

Consideringp_i(z) =P

eⁱ⁰+eⁱ¹+···+eⁱⁿ=α+eⁱz^α+ei,i= 1,2, . . . , m, we can obtain that p_i(z) = ω_α+eⁱ(z,{0}ⁿ_j=0).

Further, let

p(z) = 1

(n+ 1)!

m

X

i=1

pi(z).

By calculating, we have

p(z) =

m

X

i=1

1

(α+eⁱ)!z^α+ei.

This implies that, for1≤i≤m,

D^α+eip(z) = 1 >0,

and

D^αp(z) =

m

X

i=1

z^ei.

Then

[x⁰, x¹, . . . , xⁿ]_αp= Z

Sⁿ

D^αp

1−

n

X

j=1

t_j

!

x⁰ +t₁x¹+· · ·+t_nxⁿ

!

dt₁dt₂· · ·dt_n

=

m

X

i=1

Z

Sⁿ

1−

n

X

j=1

t_j

!

x⁰+t₁x¹+· · ·+t_nxⁿ

!eⁱ

dt₁dt₂· · ·dt_n

=

m

X

i=1

Z

Sⁿ

1−

n

X

j=1

t_j

!

(x⁰)^ei +t₁(x¹)^ei +· · ·+t_n(xⁿ)^ei

!

dt₁dt₂· · ·dt_n

= 1

(n+ 1)!

n

X

j=0 m

X

i=1

(x^j)^ei.

Therefore, if we takeg(z) = p(z)in Theorem 3.3, we have the following corollary.

Corollary 3.4. Let f ∈ Cⁿ⁺¹ (hx⁰, . . . , xⁿ, y⁰, . . . , yⁿi), α ∈ N^m0 , and |α| = n. If for all x^j, y^j (j = 0,1, . . . , n)we havex^j ≥y^j andPn

j=0

Pm

i=1(x^j−y^j)^ei 6= 0, then L⁰ ≤[x⁰, x¹, . . . , xⁿ]_αf −[y⁰, y¹, . . . , yⁿ]_αf ≤U⁰, where

L⁰ = 1

(n+ 1)! min

1≤i≤m inf

z∈hx⁰,...,xⁿ,y⁰,...,yⁿiD^α+eif(z)

n

X

j=0 m

X

i=1

(x^j−y^j)^ei,

U⁰ = 1

(n+ 1)! max

1≤i≤m sup

z∈hx⁰,...,xⁿ,y⁰,...,yⁿi

D^α+eif(z)

n

X

j=0 m

X

i=1

(x^j−y^j)^ei.

In fact, from the procedure of the proof of Theorem 3.3, it is not difficult to find that the conditions of the corollary can be weakened. If we replacex^j ≥y^jandPn

j=0

Pm

i=1(x^j−y^j)^ei 6=

0byPn j=0

Pm

i=1(x^j −y^j)^ei >0, the corollary holds true as well.

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 ANDN.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. PE ˇCARI ´C, AND M. RODI ´C LIPANOVI ´C, On an inequality for divided differences, Asian- European J. Math., 1(1) (2008), 113–120.

[4] X. WANGANDM. 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.