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Given $z, w \in D$ (unit disk, open), what are the functions (analytic, from unit disk to unit disk) $f$ that maximize the norm of $f(z)-f(w)$?

My attempt: We have that $$|f(z)-f(w)| \leq |f(z)|+|f(w)| \leq |z|+|w| \tag{1}$$ using the triangle inequality and Schwarz lemma. Moreover, $$|f(z)| \leq |z| \text{ and } |f(w)| \leq |w|. \tag{2}$$

Now, I found the challenge to be deciding which inequality in (1) to maximize. In the collinear case (for $z$, $w$ and $0$), by Schwarz lemma, (2) are equalities only if $f(z) = az$ for $z$ unimodular. Now, this would also make the first inequality in (1) an equality. Thus, we found the class of functions maximizing the distance.

Now I do not fully understand what to do/which inequality to maximize in the non-collinear case (for $0$, $z$ and $w$). A unimodular constant multiple would preserve the distance, which intuitively does not seem like a max to me. My intuitions are probably wrong. Any help is appreciate!

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Do you assume that $f(0)=0?$ If not, Schwarz lemma cannot imply $|f(z)|\le |z|$. –  23rd Nov 30 '12 at 3:29
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There is a fractional linear transformation $\varphi$ that preserves $D$ such that for some $x \in D$, $\varphi(z) = -\varphi(w)$. If $F: D \to D$ is analytic, consider the function $g(\zeta) = \dfrac{F(\zeta) - F(-\zeta)}{2 \zeta}$, with $g(0) = F'(0)$. Applying the maximum modulus principle to $g$, conclude that $|g(\zeta)| \le 1$. Thus $|F(\zeta) - F(-\zeta)| \le 2 |\zeta|$ (which is attained by the identity function). Taking $F = f \circ \varphi^{-1}$, we get $$|f(z) - f(w)| = |F(\varphi(z)) - F(-\varphi(z))| \le 2 |\varphi(z)| = |\varphi(z) - \varphi(w)|$$

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