Conditional convexity

[摘要] It is well known that a function f: D --> R Frechet differentiable on an open convex subset D of a real normed linear space is convex, i.e., f(lambdax + (1 - lambda )y) less than or equal to lambdaf(x) + (1 - lambda )f(y) (x, y is an element of D, lambda is an element of (0, 1)) (1) holds if and only if f'(x)(y - x) less than or equal to f(y) - f(x) (x,y is an element of I) (2) is valid [see, e.g., Roberts and Varberg (Convex Functions, Academic Press, New York and London, 1973)]. It is shown that (1) with a fixed y = w (or with fixed lambdax + (1 - lambda )y = w) is equivalent to the inequality (2) with fixed y = w (or with fixed x - w, respectively). Then these results are applied to study some conditional inequalities for deviation means. (C) 2000 Academic Press.

[发布日期] 2000-12-15

[关键词] convexity;deviation mean value;conditional inequalities