MEMO 2023 I1

Let $\mathbb{R}$ denote the set of all real numbers. For each pair $(\alpha, \beta)$ of nonnegative real numbers subject to $\alpha + \beta \geq 2$, determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying

$$f(x)f(y) \leq f(xy) + \alpha x + \beta y$$

for all real numbers $x$ and $y$.