Conditions conformal mapping to be expansive

Question

Let $\Omega' \subset \Omega$ be simply connected domains in the complex plane. The Riemann mapping theorem tells us that there a biholomorphism $f: \Omega' \to \Omega$. What conditions guarantee that there exists a biholomorphism so that for each $z \in \Omega'$ we have $|f'(z)| \geq 1$.

Answer 1

Your map is automatically going to be expanding at every point with respect to the hyperbolic metric (assuming $\Omega'\subsetneq\Omega$), and uniformly expanding if $\Omega'$ is compactly contained in $\Omega$.

Expansion with respect to the Euclidean metric is less natural in this context. In particular, the condition will depend not only on the domains, but also on the choice of function $f$. Indeed, whatever your domains, you can easily choose a normalization of $f$ that takes a given hyperbolic ball in $\Omega'$ to a hyperbolic ball of $\Omega$ that is much smaller in the Euclidean metric.

However, if you are really interested in it, then you can use the fact that $f$ is an isometry of the corresponding hyperbolic metrics: we have $$|f'(z)| = \frac{\rho_{\Omega'}(z)}{\rho_{\Omega}(f(z))}.$$ Here $\rho_{\Omega}$ denotes the density of the hyperbolic metric with respect to the Euclidean metric.

Using the standard estimates on the density of the hyperbolic metric, you can then give sufficient conditions (involving both the geometry of the domains and the position of two base points) that will ensure the desired property. However, you cannot expect them to turn out very natural.

EDIT. The standard estimate on the hyperbolic metric in a simply-connected domain is as follows: $$\frac{1}{2\operatorname{dist}(z,\partial\Omega)} \leq \rho_{\Omega}(z) \leq \frac{2}{\operatorname{dist}(z,\partial\Omega)}.$$ (It is proved using the Schwarz Lemma and the Koebe 1/4 theorem.)

So, for example, if you know that $\operatorname{dist}(f(z),\partial\Omega)\geq 4\operatorname{dist}(z,\partial\Omega')$, then $|f'(z)|\geq 1$.

Of course, if you are looking for the property to hold for all $z$, then you need to say something about the behaviour of $f$. However, if your map $f$ has bounded distortion (e.g. because it extends to a conformal map on a larger domain), then by the above you can find a map $f$ with the desired property assuming that $\Omega'$ is sufficiently small.

In a more general situation, by the maximum principle (applied to the derivative), it would seem that the question comes down entirely to the question of the boundary behaviour of $f$ (unless I am missing something).

For example, consider for simplicity the case where $\Omega=\mathbb{D}$ is the unit disc, and that $\partial\Omega'$ is sufficiently nice to ensure that $f'$ extends continuously to the boundary (see Chapter 3 of Pommerenke's book "Boundary behaviour of conformal maps"). Then it would seem that your requirement comes down precisely to the requirement that the harmonic measure of any piece of $\partial \Omega'$, when viewed from the point $z_0 := f^{-1}(0)$, is at most as large as the Euclidean length of this piece.

There are many techniques for estimating harmonic measure (I recommend the book by Garnett & Marshall), but it is hard to know what you are looking for without further details.