 Solution

ID: past-exam-of-the-complex-analysis-conformal-maps-and-geometry-course-of-the-university-of-oxford/2023/1/b/i/solution

ϕ \in S

satisfies

\frac{∣ z ∣}{( 1 + ∣ z ∣ ) ^{2}} \leq ∣ ϕ (z) ∣ \leq \frac{∣ z ∣}{( 1 - ∣ z ∣ ) ^{2}} (z \in D) .

Both bounds are sharp. Rotations of the Koebe function,

ϕ_{θ} (z) = z / (1 - e^{i θ} z)^{2}

, attain them along suitable radial directions.

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