Convex functions

Superadditivity inequalities

“…This inequality is an extension of the defining inequality (1). Analogues of the inequality (49), for functions of the sort treated in this paper, are given in the two theorems that follow.…”

“…A function that is convex on a left-hand portion of its interval of definition, and concave on the complementary right-hand portion, is said to be convexo-concave [1], Thus both the function / given by (7) and the function F given by (6) Thus it happens that the maximum interval of starshapedness of the function / forms a golden section [7] of the maximum interval of starshapedness of the function F.…”

“…Combination inequalities* If the function / is convex for x e [α, 6], then, by Jensen's inequality [6,1], for any numbers…”

“…l Introduction^ In the theory of analytic inequalities, a principal tool is the notion of convex function [6,1]. A hierarchy of convexity conditions, useful in this theory, can be expressed as follows: Let J£ α (α, b) denote the class of functions p that are positive and continuous on an interval a ^ x gΞ b and such that sign (x) [pix)]* is convex on {a, b] if a Φ 0, and log p{x) is convex on [α, b] if a -0; then for all real a and β with β > a we have K\a, b) c K β (a, b) [8], A different sort of hierarchy has been established by Bruckner and Ostrow [3].…”

“…is starshaped on [0, b] for an arbitrarily large value of b but convex on the average only on [0,1], that the function h defined on [0, oo) by…”