On a functional inequality connected with quadratic functionals

In the present paper we investigate some functional inequalities which are closely connected with quadratic functionals. In particular, we are interested in inequalities of the typeD(f)(x,y)⩾ϕ(x,y)(for allx,y), where D is given and f and ϕ are unknown. One of the theorems presented provides a characterization of the inner product in a real inner product space of dimension greater than two.