On some relative convexities

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)f=h(g), where h   is strictly increasing and convex, and denote it by f≻(1)gf≻(1)g. Similarly, if f is convex relative to g   in the sense studied in Rajba (2011), that is if the function f−gf−g is convex then we denote it by f≻(2)gf≻(2)g. The relative convexity relation ≻(2)≻(2) of a function f   with respect to the function g(x)=cx2g(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.