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

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Proof. 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, ...

A) The condition that p is a positive n-tuple can be replaced by: "p is a nonnegative n-tuple and P; > O. " Similar remarks can be made for other results in this chapter. 1) can be used as an alternative definition of convexity. 16)). 16) appeared much earlier under different assumptions. 16) in 1889 by assuming that f is twice differentiable on [a, b] and that r(x) ~0 on the interval. If f is twice differentiable, then r(x) ~0 for x E [a, b] is equivalent to f being convex on [a, b]. 1) under the same assumptions imposed by HOlder (1889).

64 can be given by a generalization of Taylor's formula. In their result Wi (i = 0, 1, ... , n) are in the class cn-i[a, b], either positive or negative on [a, b], and the first order differential operator is defined as above. 65. Theorem. Let f: [a, b ] ~ ~ be a real function such that Dnf(x) = (DnD n- 1··· Dof)(x) is continuous on [a, b]. 88) i=O where c E [a, b], a, = Di-1f(c)/wi(c) (i = 0, 1, ... 89) 28 1. Convex Functions for k = 1, ... 91) c c where u E [min(c, t), max(c, t)]. Furthermore, we also have .