Characterization of quadratic mappings through a functional inequality

We provide conditions under which every solution (f,ϕ)(f,ϕ) of the functional inequalityf(x+y)−f(2x)+f(2y)4⩾ϕ(x,y) satisfies f(x)=12ϕ(x,x) and ϕ being biadditive and symmetric.