[Fall 1998, Problem #5] Suppose \(f : \mathbb { D } \to \mathbb { D }\) is analytic. Prove that for any \(a \in \mathbb { D }\)
\[ { \frac { | f ^ { \prime } ( a ) | } { 1 - | f ( a ) | ^ { 2 } } } \leq { \frac { 1 } { 1 - | a | ^ { 2 } } } . \]
Solution
For \(c\in\DD\), define the disk automorphism \[ \phi_c(z)=\frac{z-c}{1-\bar c z}. \]
1. The map \(\phi_c\) is an automorphism of \(\DD\) satisfying\[ \phi_c(c)=0 \]and\[ \phi_c'(c)=\frac1{1-\abs{c}^2}. \]
The standard disk automorphism formula shows that \(\phi_c\) maps \(\DD\) biholomorphically onto itself and sends \(c\) to \(0\). Differentiating gives \[ \phi_c'(z) = \frac{1-\abs{c}^2}{(1-\bar c z)^2}. \] Evaluating at \(z=c\) yields \[ \phi_c'(c) = \frac{1-\abs{c}^2}{(1-\abs{c}^2)^2} = \frac1{1-\abs{c}^2}. \]
2. Fix \(a\in\DD\), put\[ b=f(a), \]and define\[ F=\phi_b\circ f\circ\phi_a^{-1}. \]Then \(F:\DD\to\DD\) is analytic and \(F(0)=0\).
By step 1, both \(\phi_a^{-1}\) and \(\phi_b\) are disk automorphisms. Since \(f\) is an analytic self-map of \(\DD\), their composition \(F\) is also an analytic self-map of \(\DD\). Moreover, \[ F(0) = \phi_b\bigl(f(\phi_a^{-1}(0))\bigr) = \phi_b(f(a)) = \phi_b(b) = 0. \]
3. One has\[ \abs{F'(0)}\leq1. \]
This is Schwarz’s lemma applied to the function in step 2.
4. The derivative of \(F\) at the origin is\[ F'(0) = \frac{1-\abs{a}^2} {1-\abs{f(a)}^2} f'(a). \]
By the chain rule, \[ F'(0) = \phi_b'(b)\, f'(a)\, (\phi_a^{-1})'(0). \] Step 1 gives \[ \phi_b'(b) = \frac1{1-\abs{b}^2}. \] Since \(\phi_a(a)=0\), the inverse function theorem gives \[ (\phi_a^{-1})'(0) = \frac1{\phi_a'(a)} = 1-\abs{a}^2. \] Substituting \(b=f(a)\) gives the displayed formula.
5. For every \(a\in\DD\),\[ \boxed{ \frac{\abs{f'(a)}}{1-\abs{f(a)}^2} \leq \frac1{1-\abs{a}^2}. } \]
Taking absolute values in step 4 and using step 3 gives \[ \frac{1-\abs{a}^2} {1-\abs{f(a)}^2} \abs{f'(a)} \leq1. \] Both denominators are positive because \(a,f(a)\in\DD\). Dividing by \(1-\abs{a}^2\) yields the required inequality.
6. Q.E.D.
Step 5 is the claimed Schwarz–Pick derivative inequality.