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

Instead of just quoting the Schwarz Lemma, let's build the bound on $f'(0)$ from the maximum modulus principle directly, then use it to eliminate the wrong options.

Since $f(0) = 0$ and $f$ is analytic on $D$, we can write $f(z) = z\, g(z)$ for some function $g$ that is analytic on all of $D$, which comes from factoring out the zero of $f$ at the origin using the power series of $f$. Note that $g(0) = f'(0)$, since the power series of $f$ starts as $f(z) = f'(0)z + \cdots$, so dividing by $z$ and setting $z=0$ gives $g(0) = f'(0)$.

Now fix any radius $r$ with $0 < r < 1$. On the circle $|z| = r$, we have $|f(z)| < 1$ since $f$ maps into $D$, so

\[ |g(z)| = \frac{|f(z)|}{|z|} < \frac{1}{r} \quad \text{on } |z| = r \]

By the maximum modulus principle, this bound also holds inside the circle, so $|g(0)| \le 1/r$. Letting $r \to 1$ gives $|g(0)| \le 1$, that is,

\[ |f'(0)| \le 1 \]

This is the same conclusion as the Schwarz Lemma, just derived directly. Now check the four options against $|f'(0)| \le 1$:

1. (A) $\frac{5}{2}i$: magnitude $2.5$, too big, ruled out.
2. (B) $\frac{i}{10}$: magnitude $0.1$, satisfies the bound comfortably.
3. (C) $\frac{3}{2}$: magnitude $1.5$, too big, ruled out.
4. (D) $-\frac{5}{2}i$: magnitude $2.5$, too big, ruled out.

Only option (B) is small enough in magnitude to be a valid derivative value. A concrete example that actually achieves it is the linear map $f(z) = \frac{i}{10}z$, which clearly maps $D$ into $D$, fixes $0$, and has $f'(0) = \frac{i}{10}$.

Let's summarize:

• Writing $f(z) = zg(z)$ and applying the maximum modulus principle to $g$ gives $|f'(0)| \le 1$, the same bound the Schwarz Lemma provides.
• Three of the four options have magnitude bigger than $1$ and are immediately ruled out.
• Only $\frac{i}{10}$ satisfies the bound, and it is realized by a genuine example, $f(z) = \frac{i}{10}z$.

\[ \boxed{f'(0) = \dfrac{i}{10}} \]