Question 5
Let $\mathbb{R}^{+}$ be the set of all positive real numbers. Find all functions $f\colon \mathbb{R}^{+} \longrightarrow \mathbb{R}^{+}$ that satisfy the following conditions:
(i) $$ f(x y z)+f(x)+f(y)+f(z)=f(\sqrt{x y})f(\sqrt{y z})f(\sqrt{z x}){\text{for all }}x,y,z\in\mathbb{R}^{+} $$
(ii) $ f(x) \lt f(y) $ for all $ 1 \leq x \lt y $.
(i) $$ f(x y z)+f(x)+f(y)+f(z)=f(\sqrt{x y})f(\sqrt{y z})f(\sqrt{z x}){\text{for all }}x,y,z\in\mathbb{R}^{+} $$
(ii) $ f(x) \lt f(y) $ for all $ 1 \leq x \lt y $.