Abstract

We prove that all solutions f : R → R of the functional inequality (∗) f (x) f (y) − f (x y) ≤ f (x) + f (y) − f (x + y), which are convex or concave on R and differentiable at 0 are given by f (x) = x and f (x) ≡ c, where 0 ≤ c ≤ 2. Moreover, we show that the only non-constant solution f : R → R of (∗), which is continuous on R and differentiable at 0 with f (0) = 0 is f (x) = x.