Abstract

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)=xandf(x)≡c,where0≤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.