Let $D_1\subset D_2$ be simply connected domains in the complex plane. Let $\lambda_1$ and $\lambda_2$ be the corresponding hyperbolic (Poincare) metrics. It seems intuitive to me that $\lambda_2$ is greater than or equal to $\lambda_1$, but I cannot prove it. Can someone offer a quick proof or a reference?
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1$\begingroup$ What is your order relation on metrics? If you say one metric is less than another if and only if one domain is contained in another, then I have a proof. $\endgroup$– Ryan BudneyCommented Mar 18, 2015 at 20:34
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1$\begingroup$ This is true essentially by the Schwarz lemma. en.wikipedia.org/wiki/Schwarz_lemma $\endgroup$– Ian AgolCommented Mar 18, 2015 at 20:36
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1$\begingroup$ @DanGallo could you answer Ryan's request? it's maybe standard language for you and and Sam Nead, but not everybody. $\endgroup$– YCorCommented Mar 18, 2015 at 21:14
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$\begingroup$ YCor- I am not familiar with order relations on metrics. $\endgroup$– Dan GalloCommented Mar 18, 2015 at 21:33
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1$\begingroup$ @DanGallo: thank you. If you say "greater than", it means that you have an order relation in mind, which you finally answered (I didn't guess the order because "metric" can refer both to the Riemannian 2-tensor and to the distance it determines) $\endgroup$– YCorCommented Mar 18, 2015 at 21:56
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As Ian says, this is answered by the Schwarz lemma. The precise statement you want is due to Ahlfors - here is the Wikipedia page for the Ahlfors-Schwarz-Pick lemma.
