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Let $z$ and $w$ be two points in the complex unit disk, and let $f$ be a holomorphic function from the unit disk to itself (i.e. $|f| < 1$). Intuitively, it seems that the maximum value of $|f(z) - f(w)|$ over all such $f$ should occur when $f$ is fractional linear. I have tried to prove this using an argument similar to that for the Schwarz lemma, without much success. First, is this fact true? And, second, how would I prove it?

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Interesting. I don't know any results regarding the Euclidean distance between the image points, but the Schwarz-Pick lemma states that all such maps are contractions with regard to hyperbolic distance, and that the best you can do is to leave the distance fixed, in which case f is a (linear fractional) automorphism of the disc. – John Adamski May 8 '12 at 17:10
up vote 1 down vote accepted

Yes, the maximum value of $|f(z)-f(w)|$ is attained when $f$ is a Möbius map (fractional linear map that maps the unit disk onto itself) with the additional condition that $f(w)=-f(z)$. You can find such a map explicitly by solving $$\frac{z-a}{1-\bar a z}=-\frac{w-a}{1-\bar a w}$$ It's advisable to first reduce to the case $z=0$ and $w>0$, when $a$ can be taken to be real as well.

Explanation. Let $a$ be the midpoint of the hyperbolic geodesic between $z$ and $w$. The points $z$ and $w$ are contained in the hyperbolic disk $\{\zeta\colon \rho(\zeta,a)\le r\}$ where $r=\frac{1}{2}\rho(z,w)$. Here $\rho$ is the hyperbolic (Poincaré) metric, defined as $\tanh^{-1}\left| \frac{z-w}{1-\bar w z}\right|$ or the infimum of $\int_\gamma \frac{|dz|}{1-|z|^2}$ over smooth curves connecting $z$ to $w$.

By the Schwarz-Pick lemma, $f(z)$ and $f(w)$ are contained in the hyperbolic disk $\{\zeta\colon \rho(\zeta,f(a))\le r\}$. We want to estimate the Euclidean diameter of this set. Luckily, a hyperbolic disk is also a Euclidean disk, only with a different center and radius. (See, e.g., the book "Hyperbolic geometry" by James W. Anderson. Its Google Books preview might give enough information to you.) It's an instructive exercise to calculate the Euclidean radius of a hyperbolic disk in terms of $a$ and $r$ and to check that for a fixed $r$, the Euclidean radius is maximal when $a=0$.

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