Q: If f is analytic on |z|<1, |f|<1, f(0) = 0, and f(1/2) = 1/2, what is f? Schwarz: |f(z)| ≤ |z|; here |f(1/2)| = 1/2 = |1/2|: EQUALITY! Equality ⟹ f(z) = e^(iθ)z (rotation!). 1/2 = e^(iθ)(1/2): e^(iθ) = 1: θ = 0. f(z) = z! (Schwarz rigidity: extremal ⟹ rotation!)