贪婪的人
发表于 2025-3-26 23:48:46
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Radiation
发表于 2025-3-27 04:52:55
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Cabinet
发表于 2025-3-27 07:32:51
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 ..
国家明智
发表于 2025-3-27 09:31:41
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袋鼠
发表于 2025-3-27 17:05:11
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. . ⊂ ℝ. .→ ℝ . . . ∈ . ∈ ℝ. . ∈ ℕ . 0 < . < . ± . ∈ ..
傻瓜
发表于 2025-3-27 18:34:45
Continuous Convex FunctionsLet . ⊂ ℝ. be a convex and open set. In 5.3 we saw that a convex function f : . → ℝ fulfills the inequality . for all . ∈ . and all λ ∈ ℚ ∩ . It was also pointed out that if, moreover, . is continuous, then inequality (7.1.1) holds actually for all real λ ∈ .
FLUSH
发表于 2025-3-28 00:01:10
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
发表于 2025-3-28 02:11:01
Further Properties of Additive Functions and Convex FunctionsLet . ⊂ ℝ. be a convex and open set, and let f : . → ℝ 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.
离开
发表于 2025-3-28 07:31:12
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fixed-joint
发表于 2025-3-28 12:53:41
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.