Let \(f(z) \in H({\mathbb D})\), \(\text{Re}(f(z)) >0\) and \(f(0)= a>0\). Show that \[ \abs{ \frac{f(z)-a}{f(z)+a}} \leq |z|, \; \; \; |f'(0)| \leq 2a .\]
Show that the above is still true if \(\text{Re}(f(z)) >0\) is replaced with \(\text{Re}(f(z)) \geq 0\).
Solution
Goal: (1) For \(f \in H(\DD)\) with \(\Re f > 0\) and \(f(0) = a > 0\), prove \(\abs{\frac{f(z) - a}{f(z) + a}} \leq \abs z\) and \(\abs{f'(0)} \leq 2a\); (2) same conclusion when \(\Re f \geq 0\).
1. The Cayley transform \(T(w) := \frac{w - a}{w + a}\) maps the right half-plane \(\theset{\Re w > 0}\) conformally onto \(\DD\), with \(T(a) = 0\).
\(T\) is a M"obius map; \(w = a\) (the point in the right half-plane with \(\abs{T(w)} = 0\)) maps to \(0\), the boundary \(\Re w = 0\) maps to the unit circle, and the interior \(\Re w > 0\) maps to the interior \(\abs{T(w)} < 1\) (check at \(w = \infty\)-type points or a test point like \(w = 1\): \(T(1) = \frac{1-a}{1+a} \in (-1,1)\)).
2. Define \(g(z) := T(f(z)) = \frac{f(z) - a}{f(z) + a}\); then \(g: \DD \to \DD\) is analytic and \(g(0) = 0\).
2.1. \(g\) is analytic on \(\DD\).
2.2. \(\abs{g(z)} < 1\) on \(\DD\).
\(T\) maps the right half-plane into \(\DD\) (<1>1) and \(\Re f > 0\).
2.3. \(g(0) = T(a) = 0\).
\(f(0) = a\) and 1.
3. By Schwarz’s lemma, \(\abs{g(z)} \leq \abs z\) for all \(z \in \DD\).
Schwarz lemma applied to \(g\) (<1>2).
4. This proves the first inequality: \(\abs{\frac{f(z) - a}{f(z) + a}} \leq \abs z\).
\(g(z) = \frac{f(z) - a}{f(z) + a}\) by definition.
5. \(\abs{f'(0)} \leq 2a\).
5.1. \(g'(0) = T'(f(0)) \cdot f'(0) = T'(a) f'(0)\).
Chain rule.
5.2. \(T'(a) = \frac{2a}{(2a)^2} = \frac{1}{2a}\).
\(T'(w) = \frac{(w+a) - (w-a)}{(w+a)^2} = \frac{2a}{(w+a)^2}\); evaluate at \(w = a\).
5.3. Schwarz’s lemma also gives \(\abs{g'(0)} \leq 1\).
Schwarz lemma, derivative form.
5.4. \(\abs{f'(0)} = 2a \abs{g'(0)} \leq 2a\).
<2>1–<2>3.
6. Part (2): the same conclusions hold when \(\Re f \geq 0\).
6.1. \(g = T \circ f\) still maps \(\DD\) into \(\overline{\DD}\) and is analytic with \(g(0) = 0\).
\(T\) maps the closed right half-plane \(\Re w \geq 0\) into the closed unit disk; \(f(z) + a \neq 0\) still holds since \(\Re(f(z)+a) \geq a > 0\).
6.2. Schwarz’s lemma applies to \(g: \DD \to \overline{\DD}\) with \(g(0) = 0\).
Schwarz’s lemma requires only \(\abs g \leq 1\), not strict.
6.3. Hence \(\abs{g(z)} \leq \abs z\) and \(\abs{g'(0)} \leq 1\), giving the same two inequalities.
Same argument as <1>3–<1>5.
7. Q.E.D.
4 and 5 prove (1); 6 proves (2).