Question

Let \( D = \{ z \in \mathbb{C} \mid |z| < 1 \} \) denote the unit disc in the complex plane \( \mathbb{C} \).
Let \( f: D \to D \) be an analytic function which satisfies \( f(0) = 0 \). Then which one of the following is a possible value of \( f'(0) \)?

• A. \( \dfrac{5}{2} i \)
• B. \( \dfrac{i}{10} \)
• C. \( \dfrac{3}{2} \)
• D. \( -\dfrac{5}{2} i \)

Hint:

Apply the Schwarz Lemma to bound the magnitude of f'(0) for an analytic self-map of the disc that fixes 0.

Answer 1

The Correct Option is B

Step 1: Identify the tool needed.
We are told \( D = \{z \in \mathbb{C} : |z| < 1\} \) is the open unit disc, and \( f : D \to D \) is analytic with \( f(0) = 0 \). This is exactly the setup for the Schwarz Lemma.

Step 2: State the Schwarz Lemma.
The Schwarz Lemma says that if \( f : D \to D \) is analytic and \( f(0) = 0 \), then for every \( z \in D \),
\[ |f(z)| \leq |z| \]
and in particular, letting \( z \to 0 \), the derivative at the origin satisfies
\[ |f'(0)| \leq 1 \]

Step 3: Apply this bound to each option.
Since \( f'(0) \) must satisfy \( |f'(0)| \leq 1 \), any option whose magnitude exceeds \( 1 \) cannot be a possible value of \( f'(0) \). Check the magnitude of each option:
\[ \left|\frac{5}{2}i\right| = 2.5, \quad \left|\frac{i}{10}\right| = 0.1, \quad \left|\frac{3}{2}\right| = 1.5, \quad \left|-\frac{5}{2}i\right| = 2.5 \]
Options (A), (C), and (D) all have magnitude greater than \( 1 \), so none of them can be \( f'(0) \) for an analytic self-map of the disc fixing \( 0 \).
Option (B), \( \dfrac{i}{10} \), has magnitude \( 0.1 \), which is well within the allowed bound of \( 1 \).

Step 4: Confirm it is actually achievable.
Take the simple linear map \( f(z) = \dfrac{i}{10} z \). This sends \( D \) into \( D \) since \( |f(z)| = \dfrac{1}{10}|z| < |z| < 1 \) for \( z \in D \), it is analytic since it is a polynomial, and \( f(0) = 0 \) with \( f'(0) = \dfrac{i}{10} \). So this value is genuinely attainable, not just consistent with the bound.

Final Answer:
Only \( \dfrac{i}{10} \) satisfies \( |f'(0)| \leq 1 \) from the Schwarz Lemma, and it is realized by \( f(z) = \frac{i}{10}z \). \[ \boxed{f'(0) = \dfrac{i}{10}} \]