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AbstractLet X be a real normed space, V be a subset of X and α: [0, ∞) → [0, ∞] be a nondecreasing function. We say that a function f : V → [−∞, ∞] is conditionally α-convex if for each convex combination ∑i=0ntivi$\sum\nolimits_{i = 0}^n {t_i v_i }$of elements from V such that ∑i=0ntivi∈V$\sum\nolimits_{i = 0}^n {t_i v_i \in V}$, the following inequality holds truef(∑i=0ntivi)≤∑i=0ntif(vi)+α(maxi∈{0,…,n}ti‖vi−∑i=0ntivi‖).$$f\left( {\sum\limits_{i = 0}^n {t_i v_i } } \right) \le \sum\limits_{i = 0}^n {t_i f(v_i )} + \alpha (\mathop {\max }\limits_{i \in \{ 0, \ldots ,n\} } \left. {t_i } \right\|v_i - \sum\limits_{i = 0}^n {t_i v_i } \left\| ) \right..$$We present some necessary and some suffcient conditions for f to be conditionally α-convex.
Demonstratio Mathematica – de Gruyter
Published: Mar 1, 2016
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