Let $y_a(t):U \to \mathcal{C}$ be any comformal mappitn taking $U \in \mathcal{C}$ (simply conected domeain ) into a unit disc $D$ and the a point $a\in U$ into its center, $y_a(a)=0$.
Now define a relative diameter of a compact set $K$ with respect to set $U$ as \begin{align} R(U,K)= \max_{a,t \in K} |y_a(t)|. \end{align}
My question: Let $U$ be a disc of radius $R$ centered at the orign and $K$ a disc of radius $r$ centered at the origin where $r<R$. What is the value of $R(U,K)$?
I think $R(U,K)$ is equal to $ \frac{2 r}{R}$. However, I am not sure how to start this problem. Specifically, I am not sure how to characterize the mapping $y_a(t)$.
Edit: I would also be happy with an upper bound on $R(U,K)$.