For a given function $f(z)$ on an annulus $r<|z|<\frac{1}{r}$, I am trying to define a dilation function $f_t(z)$. That is, for $0< r<s< 1$, how to define a function $f_t(z)$ on annulus such that $f_t(r)=f(s)$ and $f_t(\frac{1}{r})=f(\frac{1}{s})$?
This can be done for a unit disk $\mathbb{D}$, where for fixed $t$, where $0< t< 1$, we can define $f_t(z)=f(tz)$ so that $f_t(0)=f(0)$ and $f_t(1)=f(t)$. That is, points from $|z|=0$ are mapped to $f(0)$ and points from $|z|=1$ are mapped to $f(t)$. But I don't know how to do this for annulus with the conditions described above.