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I want to prove the following.

We have a function $f: \mathbb{Z} \to\mathbb{R}$ s.t.

(1) $f(mn) = f(m)f(n)$

(2) $f(m+n) \leq f(m) + f(n)$

(3) $0 \leq f(x) \leq 1$

then $f(m+n) \leq \max\big(f(m), f(n)\big)$?

I tried to use $f(m+n)^k$ and use (1)-(3), but I could not show the inequality.

Please help me.

  • 0
    Maybe of use: The function $f(n) = \frac{1}{|n|}$ satisfies all your conditions.2012-11-17

2 Answers 2