On the Functional Inequality f(x)f(y)-f(xy)≤ f(x)+f(y)-f(x+y)

We prove that all solutions f: ℝ→ℝ of the functional inequality (*) f(x)f(y)-f(xy)≤ 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 .