Functional inequation

Question

Is it possible to find all smooth functions $f : \mathbb{R}_+ \to \mathbb{R}$ such that $f(1)=1$ and for all $x,y \in \mathbb{R}_+$ there holds $f(x)f(y)< f(xy)$?

Answer 1

Yes; there are no such functions, because your conditions imply that for all $x\in\Bbb{R}_+$ you have $$f(x)=f(x)f(1)<f(x\cdot1)=f(x).$$

Edit: The simple argument above doesn't hold for the updated question; clearly $f\equiv1$ is a solution, as is $f(x)=x$.