Volume 10 (2009), Issue 4, Article 103, 6 pp. AN INEQUALITY FOR DIVIDED DIFFERENCES IN HIGH DIMENSIONS AIMIN XU AND ZHONGDI CEN I NSTITUTE OF M ATHEMATICS Z HEJIANG WANLI U NIVERSITY N INGBO , 315100, C HINA xuaimin1009@yahoo.com.cn czdningbo@tom.com Received 26 November, 2008; accepted 05 November, 2009 Communicated by J. Pečarić A BSTRACT. 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]. Key words and phrases: Mixed partial divided difference, Convex hull. 2000 Mathematics Subject Classification. 26D15. 1. I NTRODUCTION 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 that g (n) (x) > 0 on (n) (x) I ◦ , and that the function fg(n) (x) is bounded on I ◦ . Then for xi , yi ∈ I (i = 1, 2, . . . , n) such that P xi ≥ yi for all i = 1, 2, . . . , n and ni=1 (xi − yi ) 6= 0, the following estimation holds true: f (n) (x) [x1 , . . . , xn ]f − [y1 , . . . , yn ]f f (n) (x) ≤ ≤ sup . (n) (x) x∈I g (n) (x) [x1 , . . . , xn ]g − [y1 , . . . , yn ]g x∈I ◦ g This theorem generalized the following result obtained by [2]. inf◦ Corollary 1.2. Let f, g be two continuously differentiable functions on [a, b] and twice differ00 entiable on (a, b),with the properties that g 00 > 0 on (a, b), and that the function fg00 is bounded on (a, b). Then for a < c ≤ d < b, the following estimation holds: f 00 (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 f 00 (x) . 00 x∈(a,b) g (x) ≤ sup The work is supported by the Education Department of Zhejiang Province of China (Grant No. Y200806015). 321-08 2 A IMIN X U AND Z HONGDI C EN 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. N OTATIONS AND D EFINITIONS 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 m (x1 , x2 , . . . , xm ). Let N0 be the set of nonnegative integers. Then it is obvious that Nm 0 ⊆ R . i m Denote by e ∈ N0 a unit vector whose jth component is δij , where 0, j 6= i; δij = 1, j = i. α1 α2 α αm Let 00 = 1. For α = (α1 , α2 , . . . , αm ) ∈ Nm 0Q, we define x = x1 x2 · · · xm , and then we P i m m 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 0 1 x , x , . . . , xn = 1− tj x0 + t1 x1 + · · · + tn xn | tj ≥ 0, tj ≤ 1 j=1 0 1 n j=1 m the convex hull of x , x , . . . , x ∈ R . 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. 0 1 n m Definition 2.1 ([1], see also [4, 5]). Let α ∈ Nm 0 with |α| = n, and x , x , . . . , x ∈ R . Then the mixed partial divided difference of order n of f is defined by ! ! Z n X [x0 , x1 , . . . , xn ]α f = Dα f 1− tj x0 + t1 x1 + · · · + tn xn dt1 dt2 . . . dtn , Sn where j=1 ( 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 Qn j−1 eij ωα (x, {xj }n−1 ) = j=0 ) , n > 0. j=1 (x − x ei1 +···+ein =α J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 103, 6 pp. http://jipam.vu.edu.au/ A N I NEQUALITY FOR D IVIDED D IFFERENCES IN H IGH D IMENSIONS 3 3. M AIN R ESULT 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 1 [x0 , x1 , . . . , xn ]α f = Dα f (ξ). n! 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 1 2 n 0 1 [x , x , . . . , x ]β f − [x , x , . . . , x n−1 ]β f = m X 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 ! ! n X ∂ β 0 n D f 1− tj x + · · · + tn x ∂tn j=1 ! ! m n X X i i = Dβ+e f 1− tj x0 + · · · + tn xn (xn − x0 )e . i=1 Hence, Z 1−Pn−1 m j=1 tj X 0 D β+ei 1− f n X i=1 j=1 ! tj ! i x0 + · · · + tn xn (xn − x0 )e dtn j=1 = Dβ f t1 x1 + · · · + 1− n−1 X ! ! tj xn j=1 − Dβ f 1− n−1 X ! tj ! x0 + · · · + tn−1 xn−1 . j=1 Thus, Z X m D β+ei 1− f S n i=1 n X ! ! i x0 + · · · + tn xn (xn − x0 )e dtn . . . dt1 tj j=1 Z = Dβ f t1 x1 + · · · + 1− S n−1 n−1 X ! tj ! xn dt1 . . . dtn−1 j=1 Z − Dβ f S n−1 1− n−1 X ! tj ! x0 + · · · + tn−1 xn−1 dt1 . . . dtn−1 , j=1 which implies m X 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. J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 103, 6 pp. http://jipam.vu.edu.au/ 4 A IMIN X U AND Z HONGDI C EN 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 for all 0 n 0 n α+ei z ∈ hx , . . . , x , y , . . . , y i we have D g(z) > 0 (i = 1, 2, . . . , m) and for all xj , y j P P n j j ei (j = 0, 1, . . . , n) we have xj ≥ y j and j=0 m i=1 (x − y ) 6= 0, 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 where i Dα+e f (z) L = min inf , i 1≤i≤m z∈hx0 ,...,xn ,y 0 ,...,y n i D α+e g(z) i Dα+e f (z) . i 1≤i≤m z∈hx0 ,...,xn ,y 0 ,...,y n i D α+e g(z) U = max sup Proof. It is evident that i Dα+e f (z) L≤ inf i z∈hx0 ,...,xn ,y 0 ,...,y n i D α+e g(z) i ≤ i Dα+e f (z) Dα+e f (z) ≤ sup ≤ U. Dα+ei g(z) z∈hx0 ,...,xn ,y0 ,...,yn i Dα+ei g(z) i Since Dα+e g(z) > 0, 1 ≤ i ≤ m, then i i i LDα+e g(z) ≤ Dα+e f (z) ≤ U Dα+e g(z). (3.1) 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 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 Z x̄ X m m m i α+ei α+ei L D g(z)dzi ≤ D f (z)dzi ≤ U Dα+e g(z)dzi . ȳ i=1 ȳ i=1 ȳ i=1 This implies that (3.2) 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). J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 103, 6 pp. http://jipam.vu.edu.au/ A N I NEQUALITY D IVIDED D IFFERENCES IN H IGH D IMENSIONS FOR 5 Using Lemmas 3.1 and 3.2, we have 0 1 0 n 1 n X m X n [x , x , . . . , x ]α g − [y , y , . . . , y ]α g = 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 α+ei where ξi,j ∈ hx0 , . . . , xn , y 0 , . . . , y n i. Since xj ≥ y j , nj=0 m g(z) > i=1 (x −y ) 6= 0 and D 0, we have [x0 , x1 , . . . , xn ]α g − [y 0 , y 1 , . . . , y n ]α g > 0. Thus, L≤ [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 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 p(z) = pi (z). (n + 1)! i=1 By calculating, we have p(z) = m X i=1 1 i z α+e . i (α + e )! This implies that, for 1 ≤ i ≤ m, i Dα+e p(z) = 1 > 0, and Dα p(z) = m X i ze . i=1 Then [x0 , x1 , . . . , xn ]α p = Z Dα p 1− Sn = = 1− Sn tj ! x0 + t1 x1 + · · · + tn xn dt1 dt2 · · · dtn n X !ei ! x0 + t1 x1 + · · · + tn xn tj dt1 dt2 · · · dtn j=1 m Z X i=1 ! j=1 m Z X i=1 n X 1− Sn n X ! ! tj ei ei ei (x0 ) + t1 (x1 ) + · · · + tn (xn ) dt1 dt2 · · · dtn j=1 n m XX 1 i = (xj )e . (n + 1)! j=0 i=1 Therefore, if we take g(z) = p(z) in Theorem 3.3, we have the following corollary. J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 103, 6 pp. http://jipam.vu.edu.au/ 6 A IMIN X U AND Z HONGDI C EN Corollary 3.4. Let f ∈ C n+1 (hx0 , . . . , xn , y 0 , . . . , y n i), α ∈ Nm 0 , and |α| = n. If for all Pn Pm j j j j j j ei x , y (j = 0, 1, . . . , n) we have x ≥ y and j=0 i=1 (x − y ) 6= 0, then L0 ≤ [x0 , x1 , . . . , xn ]α f − [y 0 , y 1 , . . . , y n ]α f ≤ U 0 , 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 P difficult that the P to find j j ei conditions of the corollary can be weakened. If we replace xj ≥ y j and nj=0 m (x −y ) 6= i=1 Pn Pm j j ei 0 by j=0 i=1 (x − y ) > 0, the corollary holds true as well. R EFERENCES [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. 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