On some relative convexities |
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Authors: | Teresa Rajba |
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Affiliation: | Department of Mathematics and Computer Science, University of Bielsko-Bia?a, ul. Willowa 2, 43-309 Bielsko-Bia?a, Poland |
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Abstract: | We study two types of relative convexities of convex functions f and g. We say that f is convex relative to g in the sense of Palmer (2002, 2003), if f=h(g), where h is strictly increasing and convex, and denote it by f?(1)g. Similarly, if f is convex relative to g in the sense studied in Rajba (2011), that is if the function f−g is convex then we denote it by f?(2)g. The relative convexity relation ?(2) of a function f with respect to the function g(x)=cx2 means the strong convexity of f. We analyze the relationships between these two types of relative convexities. We characterize them in terms of right derivatives of functions f and g, as well as in terms of distributional derivatives, without any additional assumptions of twice differentiability. We also obtain some probabilistic characterizations. We give a generalization of strong convexity of functions and obtain some Jensen-type inequalities. |
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Keywords: | Convexity Relative convexity Jensen gap of a function Jensen-type inequality Strong convexity |
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