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f(uv) = f(u)f(v), f(au) = |a | f(u), f(u) + f(v) <=|u| + |v|, f : C -> R^+

Source: 2003 Cuba MO 2.8

September 15, 2024
algebracomplex numbersfunctional equation

Problem Statement

Find all the functions f:C→R+f : C \to R^+ such that they fulfill simultaneously the following conditions: (i)  f(uv)=f(u)f(v)  ∀u,v∈C(i) \ \ f(uv) = f(u)f(v) \ \ \forall u, v \in C (ii)  f(au)=∣a∣f(u)  ∀a∈R,u∈C(ii) \ \ f(au) = |a | f(u) \ \ \forall a \in R, u \in C (iii)  f(u)+f(v)≤∣u∣+∣v∣  ∀u,v∈C(iii) \ \ f(u) + f(v) \le |u| + |v| \ \ \forall u, v \in C
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