Show, for $r > 0$, that $$r < \rho(f) \Rightarrow \exists a > 0 \text{ such that } f \prec \frac{a}{r - z} \Rightarrow r \leqslant \rho(\hat{f})$$ deduce in particular that $\rho(\hat{f}) = \rho(f)$.
Show, for $r > 0$, that
$$r < \rho(f) \Rightarrow \exists a > 0 \text{ such that } f \prec \frac{a}{r - z} \Rightarrow r \leqslant \rho(\hat{f})$$
deduce in particular that $\rho(\hat{f}) = \rho(f)$.