Regularity of weakly subquadratic functions

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