Question 182
Let $\mathbb{R}_{\gt 0}$ be the set of positive real numbers. Determine all functions $f: \mathbb{R}_{\gt 0} \rightarrow \mathbb{R}_{\gt 0}$ such that
$$
x(f(x)+f(y)) \geqslant (f(f(x))+y) f(y)
$$
for every $x, y \in \mathbb{R}_{\gt 0}$.
$$
x(f(x)+f(y)) \geqslant (f(f(x))+y) f(y)
$$
for every $x, y \in \mathbb{R}_{\gt 0}$.