By Josip E. Pečarić, Frank Proschan and Y.L. Tong (Eds.)
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Extra info for Convex Functions, Partial Orderings, and Statistical Applications
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7. Remarks. The following are some examples from McShane's (1937) paper. In the first example the conditions on the convergence or integrability are too obvious to be formally stated. ,m) or (1,2, ... ), so that f(x) is a (finite (a) The range of x is (1, or infinite) sequence {al' az, }, and A(f) = I: c;aJI: C;, where c, 2= 0 and 0 < I: c, < 00. (b) E is in the interval (0, 1), L is the class of all bounded functions on E, A(f) is the Banach integral of f over (0,1). (c) E is the set of all real numbers, L the class of all uniformly almost periodic functions, A(f) is the mean value of f.
105) is given in Djordjevic and Milovanovic (1976). Further generalizations are given in Djordjevic, Milovanovic, and Pecaric (1979), where some related results are also given. 75. Theorem. Let f E C[a - h, a h) (k = 1, ... , n -1). Then llf(nlll r ~ + h] and f(k)(a - b) (n - 1)' (nr _l)(r-l llr 2hn' r -1 If(a + h) - = (-l)kf(kl(a + f(a - h)l, where Ilgllr = (l/(b - a» f ~ Igl r dt)lIr with g defined on [a, b]. (For a function defined on [a - h, a + h], we replace a and b by a - hand b + h, respectively.
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. Let fI(x), ... 1. Jensen's Inequality (fl(X), ... ,fn(x)) is in K for all x