  For $A\subseteq \mathbb{R}$, let $A+A=\{a+b: a,b\in A\}$ and $AA=\{ab: a,b\in
A\}$. For $k\in \mathbb{N}$, let $SP(k)$ denote the minimum value of
$\max\{|A+A|, |AA|\}$ over all $A\subseteq \mathbb{N}$ with $|A|=k$. Here we
establish $SP(k)=3k-3$ for $2\leq k \leq 7$, the $k=7$ case achieved for
example by $\{1,2,3,4,6,8,12\}$, while $SP(k)=3k-2$ for $k=8,9$, the $k=9$ case
achieved for example by $\{1,2,3,4,6,8,9,12,16\}$. For $4\leq k \leq 7$, we
provide two proofs using different applications of Freiman's $3k-4$ theorem;
one of the proofs includes extensive case analysis on the product sets of
$k$-element subsets of $(2k-3)$-term arithmetic progressions. For $k=8,9$, we
apply Freiman's $3k-3$ theorem for product sets, and investigate the sumset of
the union of two geometric progressions with the same common ratio $r>1$, with
separate treatments of the overlapping cases $r\neq 2$ and $r\geq 2$.
