Convexity properties of functions defined on metric Abelian groups
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Aequationes Mathematicae
Convexity properties of functions defined on metric Abelian groups ´les Wlodzimierz Fechner and Zsolt Pa
Abstract. The notions of quasiconvexity, Wright convexity and convexity for functions defined on a metric Abelian group are introduced. Various characterizations of such functions, the structural properties of the functions classes so obtained are established and several wellknown results are extended to this new setting. Mathematics Subject Classification. Primary 26A51, 26B25, 52A01, Secondary 22A10. Keywords. Convex function, Quasiconvex function, Wright convex function, Metric Abelian group.
1. Introduction Let X be a linear space and let t ∈ [0, 1]. A subset D ⊆ X is termed t-convex if, for all x, y ∈ D, tx + (1 − t)y ∈ D. Analogously, a function f : D → R is called t-quasiconvex, t-Wright convex, and t-convex if D is a t-convex set and, for all x, y ∈ D, the respective inequality f (tx + (1 − t)y) ≤ max(f (x), f (y)), f (tx + (1 − t)y) + f ((1 − t)x + ty) ≤ f (x) + f (y), f (tx + (1 − t)y) ≤ tf (x) + (1 − t)f (y),
(1)
holds. 12 -convex sets are said to be midpoint convex and 12 -convex functions are usually called Jensen convex. The structure and properties of t-convex sets and t-quasiconvex, t-Wright convex, and t-convex functions and their generalizations have been investigated in a large number of recent papers, see e.g. [1,6,7,9,10,12–23,25,27–35,39].
´les W. Fechner, Z. Pa
AEM
As a consequence of a result by Dar´oczy and P´ ales [4], every t-convex function (where t ∈ ]0, 1[ ) is automatically Jensen convex and hence Q-convex, i.e., it is t-convex for all rational numbers t ∈ [0, 1] (cf. [11]). The following more general result about t-convexity was established by Kuhn [12]. Theorem A. If D contains at least two points and f : D → R is a t-convex function for some t ∈ ]0, 1[ , then f is s-convex for all s ∈ Q(t) ∩ [0, 1], where Q(t) denotes the smallest subfield of R containing t. Furthermore, for every subfield F of R, there exists a function f : D → R which is t-convex if and only if t ∈ F ∩ [0, 1]. The following result is the multivariable extension of the t-convexity property. Theorem B. Let F be a subfield of R and f : D → R be t-convex for all t ∈ F ∩ [0, 1]. Then, for all n ∈ N, x1 , . . . , xn ∈ D, t1 , . . . , tn ∈ F ∩ [0, 1] with t1 + · · · + tn = 1, the following inequality holds: n n f ti xi ≤ ti f (xi ). i=1
i=1
Another classical theorem is due to Bernstein and Doetsch [2] (see also [11]). Theorem C. Let D be an open convex subset of a normed linear space and let f : D → R be a Jensen convex function which is bounded from above on a nonvoid open subset of D. Then f is continuous and convex, that is, t-convex for all t ∈ [0, 1]. In the paper [16] the question whether t-Wright convexity implies Jensen convexity was investigated and an affirmative answer was proved if t is a rational number. It was also shown that, for a transcendental t, this implication is not true. Furthermore, it turned out that for some second degree algebraic numbers the answer is po
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