# Quantifying the monotonicity property of the hyperbolic metric

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.

• I agree that everything going into this estimate has been known for at least 80 years. Still, my experience with the literature of those days is that important auxiliary lemmas were not always given the treatment they might be nowadays. What might an application have been. Ahlfors knew all of this, but was it relevant to his theory of covering surfaces? I think this is a great his historical question, I am hoping Alex will opine. – Adam Epstein Apr 19 '16 at 15:55
• @AdamEpstein I agree that it's not at all clear how one would search for this in the literature, as it is most likely to appear somewhere as an auxiliary result (as it does in our paper, and your thesis). So hopefully someone might remember having seen this in the classical literature. – Lasse Rempe-Gillen Apr 20 '16 at 7:54

Prof. David Minda has kindly pointed out that his 1983 paper, "Refinements of the Schwarz-Pick lemma" (Bull. Inst. Math. Academia Sinica, 11 (1983), 167-179), is relevant. Indeed, while it does not contain the Proposition above, but the results are in a very similar spirit and indeed it is possible to derive the proposition from them.

As noted in the question, we may assume that $Y$ is the unit disc $\newcommand{\D}{\mathbb{D}}\D$. Minda gives estimates on the hyperbolic derivative of a holomorphic function $f\colon \D\to\D$, in terms of the largest hyperbolic disc contained in the image of $f$, and in terms of the largest such disc to which the corresponding branch of $f^{-1}$ can be extended.

In our setting, we can let $f$ be a universal covering map $f\colon \D\to X\subset\D$, and Minda's estimates translate precisely to the above proposition.

This certainly provides a classical reference, although it would be interesting to know whether the same or similar estimates appear earlier.