By Josip E. Pečarić, Frank Proschan and Y.L. Tong (Eds.)

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41, (b) can be found in Pecaric (1980e), and (c) can be found in Milovanovic (1977). 44. Theorem. Let the function Fm be of the form m Fm(x) = I1n- l(x) + 2. 3. Convex Functions of Higher Order 17 where II n is a polynomial of degree n, = (t - wn(t, c) C),;--l =( t - c + It 2 - c1)n-l ' Cm are real constants, and a :s; Xl < ... < X m :s; b. (a) A necessary and sufficient condition for Fm to be convex of order n is that Cj;=:: (j = 1, ... ,m). (b) Every continuous n-convex function on [a, b] is the uniform limit of the sequence of functions E; (m = 1, 2, ...

55. Definition. N' is said to be convex of order (n, m) if iln,maij;:::o (n, m ;:::0, i,j= 1, 2, ... ). 22 1. Convex Functions Similarly, we can consider functions defined on m + 1 points + 1. Letting I = {XO, Xl, . 56. Definition. A function f: I -IR is said to be a discrete n-convex function if i = 0, 1, ... 57. Remarks. , .. , constants A o , ... 64) I, then there exist nonnegative I:7:= on Ai = 1 and m-n L [ x ~ ,. 65) Ai[xi, ... 39 (instead of [a, b J we only need I). (b) Similarly, we can give definitions of discrete (n, m)-convex functions and discrete P-convex functions of order n.

Theorem. Proof. Let t(z) + c = 0 be a hyperplane in IR n such that K is entirely to one side of the hyperplane, say, t(z) + c 2= 0 for z in K. Then t(f) + c 2= 0 for all x, and 0 ~ A(t(f) + c) = A(t(f» + A(c) = t(A(f» + c, so that A(f) lies on the same side of the hyperplane as K does. That is, no hyperplane separates A(f) from K. This is possible only if A(f) is in K. 6. Theorem. Let Ll, L2 and AI-A3 be satisfied. Let K be a closed convex point set in IR n and Ij>(z) be continuous and convex on K.