Question 59
Let $\mathbb{R}_{\gt 0}$ be the set of positive real numbers. Determine all functions $f : \mathbb{R}_{\gt 0} \to \mathbb{R}_{\gt 0}$ such that
$$
x(f(x) + f(y)) \geq (f(f(x)) + y)f(y)
$$
for every $x, y \in \mathbb{R}_{\gt 0}$.
$$
x(f(x) + f(y)) \geq (f(f(x)) + y)f(y)
$$
for every $x, y \in \mathbb{R}_{\gt 0}$.