Generalized Borchardt mean identity for transformed theta invariants

Let $\mathcal{F}_2$ be the fundamental domain used in Proposition~, let $b_i(\Omega):=\theta_i^2(\Omega)/\theta_0^2(\Omega)$, and let $B_2$ denote the Borchardt mean. For each $\tau\in\mathcal{F}_2$ and each $\gamma\in\{(JM_{1,1})^2,(JM_{1,2})^2,(JM_{2,2})^2\}$, the transformed invariants satisfy

$$B_2\bigl(b_1(\gamma\tau),b_2(\gamma\tau),b_3(\gamma\tau)\bigr)=\frac{1}{\theta_0^2(\gamma\tau)}.$$

Generalized Borchardt mean identity. With the notation above, the displayed identity holds for all such $\tau$ and $\gamma$. This extends the preceding Borchardt mean formula from $\tau$ to the specified transformed arguments and is used to recover theta constants numerically; the source does not state a resolution status.