Is the analytic map from $B(0,1)$ to $B(0,1)$ such that $f(0)=1/2$ and $f'(0)=3/4$ unique?
2026-03-25 08:07:00.1774426020
Analytic Map from $B(0,1)$ to $B(0,1)$
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When in doubt, normalize the self-maps of $B(0,1)$ so that $0$ goes to $0$.
Namely, let $$g(z) = \frac{f(z)-1/2}{1-\frac12 f(z)}$$ which is $f$ composed with a Möbius transformation $\phi(z) = \frac{z-1/2}{1-z/2}$. Using the chain rule, calculate $$g'(0) = \phi'(1/2)f'(0) = \frac{4}{3}\frac{3}{4} =1$$ The (equality case of) Schwarz lemma does the rest.