parmenides51 wrote: Find all the functions $f : C \to R^+$ such that they fulfill simultaneously the following conditions: $$(i) \ \ f(uv) = f(u)f(v) \ \ \forall u, v \in C$$$$(ii) \ \ f(u) = |a | f(u) \ \ \forall a \in R, u \in C$$$$(iii) \ \ f(u) + f(v) \le |u| + |v| \ \ \forall u, v \in C$$ Uhhh ? If your $\mathbb R_+$ is $\mathbb R_{\ge 0}$ (and not, as usual on this forum, $\mathbb R_{>0}$), then (ii) implies $\boxed{f(u)=0\quad\forall u\in\mathbb C}$, which indeed fits.

I do not know what notations they have in Cuba, but in the official files. they find another one solution

parmenides51 wrote: I do not know what notations they have in Cuba, but in the official files. they find another one f(u)=|u| Which obviously is not a solution, whatever are notations : Just choose $a=2$ and $u=1$ and (ii) is wrong

Last sentence says

: It is easily verified that this function satisfies the characteristics of the statement.

My source is:

Attachments:

condition b) of original problem is different from condition (ii) of your posted problem.

oops, I missed that , I shall update and hide the solution, thanks for noticing

Answer is only f(x) = |x|, pm me if I forget to post sol