- Let H denote the upper half plane \(\{ z \in \mathbb { C } : \operatorname { I m } z > 0 \}\) . Suppose that \(f : H \to H\) is holomorphic, and \(f ( 3 + 1 7 i ) = 3 + 1 7 i\) What is the maximum possible value of \(f ^ { \prime } ( 3 { + } 1 7 i )\) . Give a reason for your answer (and try not to do any lengthy computations).
A complex derivative need not be real, so a maximum requires a real-valued objective. The set of possible derivatives, determined in the solution, gives both the maximum modulus and the largest possible real derivative, without assuming that all derivatives are real.
Solution
Let \(a=3+17i\) and \(D=\{w\in\mathbb C:|w|<1\}\). The possible values are exactly \[ \boxed{\{f'(a):f:H\to H\text{ holomorphic},\ f(a)=a\} =\{\lambda\in\mathbb C:|\lambda|\leq1\}.} \] In particular \(\max|f'(a)|=1\), and the largest possible real value of \(f'(a)\) is also one, attained by \(f(z)=z\).
1. Conjugation to the disk bounds the derivative.
The fractional map \[ \phi(z)=\frac{z-a}{z-\overline a},\qquad \phi^{-1}(w)=\frac{a-\overline a w}{1-w} \] is a biholomorphism \(H\to D\), with \(\phi(a)=0\). Indeed, \[ 1-|\phi(z)|^2 =\frac{4\operatorname{Im}a\operatorname{Im}z}{|z-\overline a|^2}>0, \qquad \operatorname{Im}\phi^{-1}(w) =\operatorname{Im}a\frac{1-|w|^2}{|1-w|^2}>0, \] and substitution verifies that the formulas are inverse. For \(F=\phi\circ f\circ\phi^{-1}\), one has \(F:D\to D\) holomorphic and \(F(0)=0\). The chain rule and \((\phi^{-1})'(0)=1/\phi'(a)\) give \(F'(0)=f'(a)\). Schwarz’s lemma therefore gives \(|f'(a)|\leq1\) [@SS03].
2. Every derivative in that disk is attained.
For any \(|\lambda|\leq1\), the map \(w\mapsto\lambda w\) takes \(D\) into \(D\), including when \(\lambda=0\). Thus \[ f_\lambda(z)=\phi^{-1}\bigl(\lambda\phi(z)\bigr) \] is a holomorphic self-map of \(H\), fixes \(a\) and has derivative \(f_\lambda'(a)=\lambda\) by the same chain-rule cancellation. This proves equality of the stated sets. For example \(\lambda=i\) gives a nonreal derivative, while \(\lambda=1\) gives the identity, which attains both \(\max|f'(a)|=1\) and the largest real value \(f'(a)=1\).