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In this paper, for given positive integer m and real functions k and h , we prove (k,h;m) -convexity of the mapping p→ φ(p) f ( Φ(p) φ (p) ) with a convex (increasing) function f and a (k,h;m) -convex mapping Φ and a positive (k,h;m) -concave mapping φ . As application, we establish a subadditivity result for completely monotone and Bernstein functions. We also show monotonity of the mappings p → Φ(p) φ (p) and p → f ( Φ(p) φ (p) ) with respect to a group majorization combined with other preorders. Mathematics subject classification (2010): 39B62, 26D15, 52A40, 06F20.

On (k,h;m) -convex mappings and applications