Regularity of weakly subquadratic functions

Related to the theory of convex and subadditive functions, we investigate weakly subquadratic mappings, that is, solutions of the inequalityf(x+y)+f(x−y)⩽2f(x)+2f(y)(x,y∈G) for real-valued functions defined on a topological group G=(G,+)G=(G,+). Especially, we study the lower and upper hulls of such functions and we prove Bernstein–Doetsch type theorems for them.