Suppose that $X$ and $Y$ are connected hyperbolic Riemann surfaces, with $X\subsetneq Y$. Let $\rho_{X,Y}$ be the density of the hyperbolic metric of $X$ with respect to that of $Y$; then $\rho_{X,Y} > 1$ by the Schwarz-Pick theorem.

Estimates for $\rho_{X,Y}$ can be very useful, perhaps particularly in holomorphic dynamics. Here one frequently has the situation where there is a holomorphic covering map $f\colon X\to Y$ between surfaces as above. Then $\rho_{X,Y}$ is precisely the derivative of $f$ with respect to the hyperbolic metric on $Y$. Thus estimates on this quotient tell us how strongly the function is expanding.

A very useful estimate gives bounds in terms of the distance $$\delta(z) := e^{\operatorname{dist}_Y(z,\partial X)}$$ (where $\operatorname{dist}_Y$ denotes hyperbolic distance in $Y$) as follows.

Proposition.For all $z\in X$, $$1 < \frac{2 \delta(z) }{(\delta(z)^2-1)\cdot \log\left(1+\frac{2}{\delta(z)-1}\right)}\leq \rho_{X,Y}(z)\leq 1 + \frac{2}{\delta(z)-1}.$$

The estimate is easy to prove by assuming wlog that $Y$ is the unit disc, and considering the extreme cases of $X$ being a hyperbolic disc in $Y$ (for the upper bound) and $Y$ with one point removed (for the lower bound). In particular, the bounds are sharp. See Proposition 3.4 of my paper with Mihaljevic-Brandt (**Absence of wandering domains for some real entire functions with bounded singular sets**, Math. Ann. 357 (2013), no. 4, 1577-1604, DOI: 10.1007/s00208-013-0936-z; Proposition 3.4).

Rather than the specific estimates, the key point is of course that the difference between the two metrics is large when the distance to the boundary is small, and small when this distance is large (as one might expect).

I have seen versions of this proposition in other recent papers (by Manabu Ito, and also by Asli Deniz), but it is clearly more classical than this. Indeed, as mentioned in our paper, it turns out that the statement essentially appears as Proposition 1 in Adam Epstein’s CUNY thesis from 1993 (**Towers of finite type complex analytic maps**, p. 7).

However, the claim has such a classical feel to it that I would expect it to appear somewhere in the literature well before then. Given its utility, it would be good to have a suitable classical reference.

QUESTION.What is the earliest occurrence of the above Proposition in the published literature?

**EDIT.** Prof. Minda made the very nice observation that the Proposition can be described concisely as giving "a quantitative refinement of ... the monotonicity property of the hyperbolic metric". I have decided to update the question title accordingly.