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This paper deals with an extension of Beckenbach's classical generalized convexity notion, called F-convexity, in such a manner that much of, “modern”convexities of functions on the Euclidean R n(as log-, ω-, [explicit] quasi- or K- convexicity can be included. General theorems concerning connections between several kinds of convexity, convexity properties of composite functions, first and second order characterizations of differentiable F-convex functions are presented.
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