A twist in your normal functional equations

Algebra Level 5

Let \( \mathfrak{G} \) be the set of all functions \( f \) from the set of positive reals to the set of positive reals such that

\( f(3x) \geq f(f(2x)) + x, \ ∀ x \in \mathbb{R^{+}} \).

Find the maximum real number \( \mathfrak{k} \) such that for all such functions \( f \in \mathfrak{G} \), the following holds:

\( f(x) \geq \mathfrak{k} x \).

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