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

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Additional info for Convex Functions, Partial Orderings, and Statistical Applications

Sample text

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.

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