# The Pseudo-Hyperbolic Metric and Lindelöf's Inequality (cont.)

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**Theorem (Lindelöf's Inequality).** *Every holomorphic function $f:\Delta\to\Delta$ satisfies *
$$\frac{|f(0)|-|z|}{1-|f(0)||z|}\leq |f(z)|\leq \frac{|f(0)|+|z|}{1+|f(0)||z|}\quad\text{for all $z\in\Delta$}.$$

*do*know." (Fun fact: In 1938, Lars Ahlfors proved* that the Schwarz Lemma is actually a statement about curvature of Riemannian metrics! For more on this, see, for instance, ch. 2 of Krantz's

*Geometric Function Theory*.)

Now to prove the theorem, we first need to derive a useful inequality concerning the metric $d$.

**Claim.** *For all $z,a\in \Delta$, the metric $d$ satisfies*
$$d(|z|,|a|)\leq d(z,a)\leq d(|z|,-|a|).$$

*Proof.* We need only prove the first inequality since the second was verified during the proof of the proposition in our previous post. To this end, note that
$$d(|z|,|a|)=\bigg| \frac{|z|-|a|}{1-|z||a|} \bigg| = \frac{| |z|-|a| |}{1-|z||a|} $$

We are now ready to derive Lindelöf's inequality and do so by verifying the rightmost inequality first. Again let $z\in\Delta$. By our claim above we know that $$d(|f(z)|,|f(0)|)\; \leq \;d(f(z),f(0))\; \leq \; d(|z|,0)$$ which by definition of $d$ implies \begin{align}\label{blue} \bigg| \frac{|f(z)|-|f(0)|}{1-|f(z)||f(0)|} \bigg| \leq |z| \end{align} and so $||f(z)|-|f(0)||\leq |z|(1-|f(z)||f(0)|)$. Thus \begin{align*} |f(z)|&\leq ||f(z)|-|f(0|| + |f(0)| \\ &\leq |z| (1-|f(z)||f(0)|) + |f(0)|. \end{align*} Rearranging terms we find that $|f(z)|( 1 + |z||f(0)|)\leq |f(0)| + |z|$ and so $$|f(z)|\leq \frac{|f(0)|+|z|}{1+|f(0)||z|}$$ as desired. For the lower bound on $|f(z)|$ observe that (\ref{blue}) implies \begin{align*} |f(0)|&\leq ||f(0)|-|f(z)||+|f(z)|\\ &\leq |z|(1-|f(z)||f(0)|)+|f(z)| \end{align*} and hence, again with some rearranging, $|f(0)|+|z||f(z)||f(0)|\leq|f(z)|+|z|$ and thus $$\frac{|f(0)|-|z|}{1-|f(0)||z|}\leq |f(z)|.$$

QED!

* L. Ahlfors, An extension of Schwarz’s lemma, *Trans. Amer. Math.* *Soc.* 43 (1938), 359–364.