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- W3141522919 abstract "Using the axiomatic method,abstract concepts such as abstract mean, abstract convex function and abstract majorization are proposed. They are the generalizations of concepts of mean, convex function and majorization, respectively. Through the logical deduction, the fundamental theorems about abstract majorization inequalities are established as follows: for arbitrary abstract mean Σ and Σ , and abstract Σ → Σ strict convex function f(x) on the interval I, if xi, yi ∈ I (i = 1, 2, . . . , n) satisfy that (x1, x2, . . . , xn) nΣ (y1, y2, . . . , yn), then Σ {f(x1), f(x2), . . . , f(xn)} ≥Σ {f(y1), f(y2), . . . , f(yn)}. This class of inequalities extends and generalizes the fundamental theorem of majorization inequalities. Moreover, concepts such as abstract vector mean are proposed, the fundamental theorems about abstract majorization inequalities are generalized to n-dimensional vector space. The fundamental theorem of majorization inequalities about the abstract vector mean are established as follows: for arbitrary symmetrical convex set S Rn, and n-variable abstract symmetrical Σ → Σ strict convex function φ() on S, if , ■∈S satisfy nΣ ■, then φ() 〈(■); if vector group i, ■i ∈ S (i = 1, 2, . . . , m) satisfy {1, 2, . . . , m} 〈Σn {■1, ■2, . . . , ■m}, then Σ {φ(1), φ(2), . . . , φ(m)} Σ {φ(■1), φ(■2), . . . , φ(■m)}." @default.
- W3141522919 created "2021-04-13" @default.
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- W3141522919 date "2009-01-01" @default.
- W3141522919 modified "2023-09-24" @default.
- W3141522919 title "The fundamental theory of abstract majorization inequalities" @default.
- W3141522919 hasPublicationYear "2009" @default.
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