A question for hyperbolic metric in the proof for Bohr's lemma

Question

Recently I was reading an interesting proof for Bohr's lemma by the tool of hyperbolic metric, however I have a following question:

Given a holomorphic map $f$ on $D$, $f(0)=0$, and $|f|<1$ on $D_{r}=\{|z|<r\}$ and $|f|=1$ somewhere on $\partial D_{r}$. Denote $\rho_{U}$ as the hyperbolic metric in $U$

We define $U=f(D)$ and let

$$R=\sup\{{T:\{|z|=T\}\subset U}\}.$$

I do not know how to deduce the following fact: For any $w\in U$ with $|w|\geq R$, we have the following estimate: $$\rho_{U}(w)\geq \frac{c}{|w|},$$ where $c$ is a constant doesn't depend on choice of $f$.

Any comments and reference will be greatly appreaciated.n

Answer 1

Beardon and Pommerenke give an optimal estimate (up to a fixed multiplicative error) on the density of the hyperbolic metric on a multiply-connected plane domain $W$. It is of the form $$ \rho_W(z) \approx \frac{1}{\operatorname{dist}(z,\partial W)\cdot \max(M,1)},$$ where $M$ is the maximal modulus of an essential round annulus in $W$ whose core curve passes through $z$ (if such an annulus exists.)

See Beardon and Pommerenke, The Poincaré Metric of Plane Domains; J. London Math. Soc. (2) 18, no. 3 (1978): 475-483.

Using this estimate, you can work out the claim that you are looking at. (If there is no essential annulus of the type described, then it is clear from the definitions that $\operatorname{dist}(w,\partial U) \leq 2|w|$. Otherwise, you see that the outer radius of such an annulus cannot be too big, and hence if the modulus is some large value $M$, then you get an estimate on the size of the inner radius, and hence of the distance to the boundary.)

Hope this helps!