- Let \(f:H\to D\) be a holomorphic map from the upper half plane
\(H = \left\{ z \in \mathbb { C } : \operatorname { I m } z > 0 \right\}\) to the unit disk \(D = \{ z \in \mathbb { C } : | z | < 1 \}\)
Suppose that \(f ( i ) = 1 / 2\) . Determine the maximal possible value of \(| f ^ { \prime } ( i ) |\)
Solution
The maximum is \(\boxed{3/8}\).
1. Normalize both the source and the target to a disk map fixing zero.
The maps \[ \phi(z)=\frac{z-i}{z+i},\qquad \psi(w)=i\frac{1+w}{1-w} \] are inverse holomorphic bijections \(H\to D\) and \(D\to H\). Indeed, direct substitution verifies the inverse identities, and for \(z\in H\), \(|z+i|^2-|z-i|^2=4\operatorname{Im}z>0\). For \(w\in D\), the inverse satisfies \(\operatorname{Im}\psi(w)=(1-|w|^2)/|1-w|^2>0\). They send \(i\) to zero and zero to \(i\), respectively.
Likewise \[ T(v)=\frac{v-1/2}{1-v/2},\qquad T^{-1}(u)=\frac{u+1/2}{1+u/2} \] are inverse disk automorphisms, since \[ 1-|T(v)|^2=\frac{(3/4)(1-|v|^2)}{|1-v/2|^2}>0 \] and the analogous identity holds for the inverse. Therefore \(F=T\circ f\circ\psi\) maps \(D\) holomorphically into \(D\) and satisfies \(F(0)=0\).
2. Schwarz’s lemma gives the numerical bound.
Direct differentiation yields \(\psi'(0)=2i\) and \(T'(1/2)=4/3\). The chain rule gives \[ F'(0)=\frac43f'(i)\,2i. \] Schwarz’s lemma implies \(|F'(0)|\leq1\) [@SS03]. Hence \[ |f'(i)|\leq\frac38. \]
3. The bound is attained, and the extremizers are explicit.
For any \(\lambda\in\mathbb C\) with \(|\lambda|=1\), define \[ f_\lambda(z)=T^{-1}(\lambda\phi(z)) =\frac{\lambda\phi(z)+1/2}{1+\lambda\phi(z)/2}. \] This is a holomorphic map \(H\to D\) with \(f_\lambda(i)=1/2\). Its normalized map is \(F(w)=\lambda w\), so the derivative identity in step 2 gives \(|f_\lambda'(i)|=3/8\). For example \(\lambda=1\) gives \(f_1(z)=(3z-i)/(3z+i)\).
Conversely, equality in the derivative bound forces \(|F'(0)|=1\), so the equality case of Schwarz’s lemma gives \(F(w)=\lambda w\) for some \(|\lambda|=1\) [@SS03]. Undoing the two normalizations gives exactly \(f_\lambda\). This proves attainability and the asserted maximum.