Titlebook: An Introduction to the Theory of Functional Equations and Inequalities; Cauchy‘s Equation an Marek Kuczma,Attila Gilányi Textbook 2009Lates

貪婪的人

Radiation

Cabinet

AlgebraLet . be a field (cf. 4.7), and let . be a set endowed with two operations: the addition of elements of ., and the multiplication of elements of . by elements of . such that ., +) is a commutative group (i.e., fulfils conditions (2.9.1)–(2.9.4); cf. 4.5), and moreover . for every .,. for every .,. for every ..

國家明智

袋鼠

Elementary Properties of Convex FunctionsIn this chapter we discuss some properties of convex functions connected with their boundedness and continuity. We start with the following Lemma 6.1.1. . ? ?. .→ ? . . . ∈ . ∈ ?. . ∈ ? . 0 < . < . ± . ∈ ..

傻瓜

Continuous Convex FunctionsLet . ? ?. be a convex and open set. In 5.3 we saw that a convex function f : . → ? fulfills the inequality . for all . ∈ . and all λ ∈ ? ∩ [0, 1]. It was also pointed out that if, moreover, . is continuous, then inequality (7.1.1) holds actually for all real λ ∈ [0, 1].

FLUSH

InequalitiesSince the convex functions are defined by a functional inequality, it is not surprising that this notion will lead to a number of interesting and important inequalities. Some inequalities connected with the notion of convexity will be presented in this chapter.

cringe

Further Properties of Additive Functions and Convex FunctionsLet . ? ?. be a convex and open set, and let f : . → ? be a convex function. Let . be the lower hull of f (cf. 6.3). By Theorem 6.3.1 either . . = -∞ for all . ∈ ., or . : . → ∝ is a continuous and convex function.

離開

fixed-joint

Derivations and AutomorphismsIn this chapter we will deal with functions satisfying the Cauchy equation (5.2.1) and also, simultaneously, another equations of a similar type.