Discussion
Let \(U = \{z \in \mathbb{C} : \text{Im}(z) \gt 0\}\) and \(D = \{z \in \mathbb{C} : |z| \lt 1\}\), where Im(z) denotes the imaginary part of z. Let S be the set of all bijective analytic functions \(f: U \to D\) such that \(f(i) = 0\). Then, the value of \(\sup_{f \in S} |f(4i)|\) is
Answer: D
The supremum is \(\dfrac{3}{5}\).
Every bijective analytic map of the upper half-plane onto the unit disc sending i to 0 has the form
\[ f(z) = e^{i\theta}\,\frac{z - i}{z + i} \]
for some real \(\theta\),
because any two such maps differ by an automorphism of the disc fixing 0, which is a rotation. So for every f in S,
\[ |f(4i)| = \left|\frac{4i - i}{4i + i}\right| = \frac{3}{5} \]
The value is the same for all of them, so the supremum is 3/5. Zero would need \(f(4i) = f(i)\), impossible for an injective map, and 1/4 and 1/2 do not come from the Cayley form.
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