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SOLVED: Let f(z) be an analytic function on the unit disc such that |f(z)| = 1 for all |z| < 1. Prove that f(z) is constant using: (1) the Open Mapping Theorem, and (2) the Cauchy-Riemann equations and the identity |f(x iy)|^2 = u^2(x,y) v^2(x,y).
SOLVED: Let f(z) be an analytic function on the unit disc such that |f(z)| = 1 for all |z| < 1. Prove that f(z) is constant using: (1) the Open Mapping Theorem, and (2) the Cauchy-Riemann equations and the identity |f(x iy)|^2 = u^2(x,y) v^2(x,y).
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