Characterization of quadratic mappings through a functional inequality

We provide conditions under which every solution (f,φ) of the functional inequality f (x +y) − f (2x) +f (2y) 4 φ(x, y) satisfies f (x) = 12 φ(x, x) and φ being biadditive and symmetric. © 2005 Elsevier Inc. All rights reserved.