By Josip E. Pečarić, Frank Proschan and Y.L. Tong (Eds.)
ISBN-10: 0080925227
ISBN-13: 9780080925226
ISBN-10: 0125492502
ISBN-13: 9780125492508
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Extra resources for Convex Functions, Partial Orderings, and Statistical Applications
Example text
Definition. Let S £; ~ n be a convex set. 116) s = 0, thenf(x) is quasiconvex. If these inequalities hold for every x, yES and A = 1/2, then f(x) is said to be strongly I-conuex, strongly I-quasiconvex, I-convex, and I-quasiconvex, respectively. 115) r A method for solving the problem of minimization: minf(x), XES is relaxational if there is a sequence {xd such that x; E Sand f(Xk+l) -::; f(Xk) for k = 0, 1, .... In the following we discuss some inequalities which are useful for proving the convergence and estimating the rate of convergence for relaxational methods.
60). 49. Definition. A function f: I X J ~ ~ is said to be P-convex of order n if it is convex of orders (n, 0), (n - 1, 1), ... , (1, n - 1) and (0, n). 50. Remark. P-convex functions of order 2 are related to functions with increasing increments. D Of course, we can easily generalize (n, m)th divided difference. However, here we give another concept of divided difference for functions of several variables. We first introduce some notation and definitions which will be used in the text. By x, y, ...
F(x) is r-convex iff the function f(x) - r IIxl/ 2 is convex. 83. Theorem (Vial, 1983). A differential function f(x) is r-conuex iff for every x, y E C we have f(y) - f(x) 2:. 84. Theorem (Vial, 1983). Let f(y) be a twice continuously differentiable function on a convex nonempty interior set C. Then f(x) is r-conuex iffthere exists a r e ~ such that (V'l(x)y, y) holds for every x E C and y E ~ k . 2:. 1. Jensen's and Jensen-Steffensen's Inequalities Jensen's Inequality In this section we treat Jensen's inequality in the form of a convex combination of the values.
Convex Functions, Partial Orderings, and Statistical Applications by Josip E. Pečarić, Frank Proschan and Y.L. Tong (Eds.)
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