  For any bounded domains $\Omega$ in $\mathbb{C}^{n}$, Deng, Guan and Zhang
introduced the squeezing function $S_\Omega (z)$ which is a biholomorphic
invariant of bounded domains. We show that for $n=1$, the squeezing function on
an annulus $A_r = \lbrace z \in \mathbb{C} : r <|z| <1 \rbrace$ is given by
$S_{A_r}(z)= \max \left\lbrace |z| ,\frac{r}{|z|} \right\rbrace$ for all
$0<r<1$. This disproves the conjectured formula for the squeezing function
proposed by Deng, Guan and Zhang and establishes (up to biholomorphisms) the
squeezing function for all doubly-connected domains in $\mathbb{C}$ other than
the punctured plane. It provides the first non-trivial formula for the
squeezing function for a wide class of plane domains and answers a question of
Wold. Our main tools used to prove this result are the Schottky-Klein prime
function (following the work of Crowdy) and a version of the Loewner
differential equation on annuli due to Komatu. We also show that these results
can be used to obtain lower bounds on the squeezing function for certain
product domains in $\mathbb{C}^{n}$.
