n-CONVEXITY AND MAJORIZATION 1. n-Convexity ... - Project Euclid
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n-CONVEXITY AND MAJORIZATION 1. n-Convexity ... - Project Euclid
Sep 3, 1986 - ABSTRACT. The fact that the nt h order divided difference of an (n + 2)-convex function is a symmetric, convex function of its arguments, and is ...
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 19. Number 1. Winter 1989
n-CONVEXITY A N D MAJORIZATION J.E. PECARIC AND D. ZWICK ABSTRACT. The fact that the n t h order divided difference of an (n + 2)-convex function is a symmetric, convex function of its arguments, and is therefore Schur convex, allows us to apply the theory of Majorization in order to derive inequalities for such functions. Several consequences of this result are presented. In a separate section the theory of majorization is used to compute bounds on the derivatives of polynomials.
1. n-Convexity and Schur convexity. The first two definitions are given in [2]. DEFINITION 1. Let x,y € R n + 1 be given. We say that y is majorized by x(y -< x) if and only if Y17=o Xi = 537=0 ^ a n d k
(1)
k x
^2 [i] i=0
> ^2v[rh fc = 0 , . . . , n - l , i=0
where X[0] > • • • > X[nj denotes a decreasing rearrangement of ^o,... ,xn. Numerous example of majorization are given in [2]. DEFINITION 2. Let x,y e R n + 1 be given. A function ip : R n + 1 -»- R is called Schur convex if and only if x -< y =$> ip(x) < tp(y). The next definition can be found in [4]. DEFINITION 3. A function / is (n + 2)-convex on (a, b) if and only if, for all a < x 0 < • • • < £ n + 2 < b, the divided differences [ x 0 , . . . , x n + 2 ] / a
are nonnegative. In particular, a 2-convex function is convex in the classical sense. We note that / is (n + 2)-convex on (a, 6) if and only if f^ continuous and convex there.
