It suffices to prove the growth estimate for $z_{1},z_{2},z_{0}$. By elementary calculation $|\frac{\partial\alpha_{1}}{\partial\eta}|\leq\frac{Ca_{22}|\eta|}{(\mu_{1}^{2}+a_{22}|\eta|^{2})^{3/2}}$, so upon integration

$$\int_{\mu_{1}}^{\infty}|\frac{\partial\alpha_{1}}{\partial\eta}|d\mu_{1}\leq\frac{C}{|\eta|}(\frac{\mu_{1}}{\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}-1)\leq\frac{C}{|\eta|},$$

hence $|\beta_{i}|\leq\frac{C}{|\eta|}$ by the integral definition of $\beta_{i}$. From

$$d\log|z_{1}|=d(a_{11}\mu_{1}+a_{12}\mu_{2})+\alpha_{1}d\mu_{1}+\alpha_{3}d(\mu_{1}-\mu_{2})+\text{Re}(\beta_{1}d\eta),$$

we integrate to obtain the growth bound on $z_{1}$ for $A^{1/4}|\vec{\mu}|_{a}\geq 1$. But $|z_{1}|$ is a continuous function, so the bound holds also near the origin. Similarly we can bound $|z_{0}|$ and $|z_{2}|$. ∎