For a finite nonempty subset $A$ of a ring $X=(X,+,\cdot)$, let us denote the set $\{a \cdot b \colon a, b \in X\}$.
If $X=\mathbb{Z}$, it is not difficult to show that
$$|AA| \leq \frac{|A|^{2}+|A|}{2}.$$
Do you know of a similar bound for $|AA|$ when $X$ is a finite field (say, of prime order)?