Note that for any $m\in\Delta_{\mathbb{Z}}\backslash\{0\}$, if $\langle m,\log|z|\rangle+\log|a|\leq-\log|\Delta_{\mathbb{Z}}|$, then $|a_{m}z^{m}|\leq\frac{1}{|\Delta_{\mathbb{Z}}|}$. Hence the equation $\sum_{m\in\Delta_{\mathbb{Z}}\backslash\{0\}}a_{m}z^{m}=1$ cannot have solutions for $x\in\log^{-1}(\Delta^{\vee}_{\lambda-c})$. ∎