The squeezing function on doubly-connected domains via the Loewner differential equation

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 $\mathbb{C}^{n}$.