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

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Fn(x) be functions of the class L such that f= (fl' ... ,fn) is in K for all x E E. Then A(f) is in K. 5. 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.

A ~ k r k . By s: = {(AI, ... =1 we denote the standard n-simplex. Let us denote as in Micchelli (1979) J [Xo •... ,xnl f = Jf(AoXo+' .. 54) S" where AD = 1 - Al - ... - An, f is a function defined and integrable over [Xo, ... ,xn ] , Xo, ... ,Xn E ~ k . The following definition can be found in Cavaretta, Micchelli, and Sharma (1980). 51. Definition. Let Xo, ... ,Xn E ~ k . Then the k-dimensional divided difference of fat Xo, ... ,Xn , denoted by [Xo, ... ,xn]f, is a map from en ( ~ k ) into ~N, N= (n +: - 1), whose nth component is given by ([Xo,· ..

65. 5. 71. Theorem. Let f E quo, Ul,"" increasing function on (c, b) with Di-1f(c) = 0 for i 33 un)' If Dnf(x) exists and is an = 0, 1, ... , n; D-1f == f, n ~ 1 and then t Os,f(t) s, Dnf(t) JWn(t, x) dx for every t E (c, b). 72. Theorem. Let f: [a, b ] ~lR be an n times differentiable function on (a, b). ,(c)="'=f(n-l)(c)=O, f(n)(x) >0 for every XE (c, b) and c > a, and iff(n)(x) is an increasing function on (c, b), then O -e2x for x> 1 and (ii) o Jx e t2dt < 3ex> +4xx2 _ 3 for x> O.