Let Ω⊆ℂ be open and connected and contain U={|z|≤1/2}. Let J consist of analytic f on Ω with sup_(z,w∈U)|f(z)−f(w)|=1. P:some f∈J has |f′(0)|≥2. Q:every f∈J has |f⁽³⁾(0)|≤48. Select all correct statements.
Set g(z)=f(z)−f(0), so |g|≤1 on U. Cauchy gives |g′(0)|≤2 and |g⁽³⁾(0)|≤3!/(1/2)³=48. Equality in the first bound would force g(z)=2e^(iθ)z on the disk, whose image diameter is 2, contradicting the stipulated diameter 1. Therefore |f′(0)|<2, P is false and Q true. **C,D**.
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