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functional inequality

Source: Romania National Olympiad 2014, Grade X, Problem 2

March 2, 2019
function

Problem Statement

Let be a function f:N⟶N f:\mathbb{N}\longrightarrow\mathbb{N} satisfying (i)f(1)=1 \text{(i)} f(1)=1 (ii)f(p)=1+f(p−1), \text{(ii)} f(p)=1+f(p-1), for any prime p p (iii)f(p1p2⋯pu)=f(p1)+f(p2)+⋯f(pu), \text{(iii)} f(p_1p_2\cdots p_u)=f(p_1)+f(p_2)+\cdots f(p_u), for any natural number u u and any primes p1,p2,…,pu. p_1,p_2,\ldots ,p_u.
Show that 2f(n)≤n3≤3f(n), 2^{f(n)}\le n^3\le 3^{f(n)}, for any natural n≥2. n\ge 2.
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