On the Functional Inequality $$f(x)f(y)-f(xy)\le f(x)+f(y)-f(x+y)$$

Abstract

We prove that all solutions $$f: {\mathbb {R}} \rightarrow {\mathbb {R}}$$ of the functional inequality $$\begin{aligned} (*) \quad f(x)f(y)-f(xy)\le f(x)+f(y)-f(x+y), \end{aligned}$$ which are convex or concave on $${\mathbb {R}}$$ and differentiable at 0 are given by $$\begin{aligned} f(x)=x\quad \text{ and } \quad f(x)\equiv c, \quad \text{ where } \quad 0\le c\le 2. \end{aligned}$$ Moreover, we show that the only non-constant solution $$f: {\mathbb {R}} \rightarrow {\mathbb {R}}$$ of $$(*)$$ , which is continuous on $${\mathbb {R}}$$ and differentiable at 0 with $$f(0)=0$$ is $$f(x)=x$$ .