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For any bounded domains $\Omega$ in $\mathbb{C}^{n}$, Deng, Guan and Zhangintroduced the squeezing function $S_\Omega (z)$ which is a biholomorphicinvariant of bounded domains. We show that for $n=1$, the squeezing function onan 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 functionproposed by Deng, Guan and Zhang and establishes (up to biholomorphisms) thesqueezing function for all doubly-connected domains in $\mathbb{C}$ other thanthe punctured plane. It provides the first non-trivial formula for thesqueezing function for a wide class of plane domains and answers a question ofWold. Our main tools used to prove this result are the Schottky-Klein primefunction (following the work of Crowdy) and a version of the Loewnerdifferential equation on annuli due to Komatu. We also show that these resultscan be used to obtain lower bounds on the squeezing function for certainproduct domains in $\mathbb{C}^{n}$.

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