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 ) , 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 ) = c x 2 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.