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Fall 2010 - Problem 9

Grunsky coefficients

Let

f\p{z} = \sum_{n=0}^\infty a_nz^n

be a holomorphic function in $\D$ . Show that if

\sum_{n=2}^\infty n\abs{a_n} \leq \abs{a_1}

with $a_1 \neq 0$ then $f$ is injective.

Solution.

Consider the function

g\p{z, w} = \frac{g\p{z} - g\p{w}}{z - w}

for $z, w \in \D$ . Since $f$ is analytic, we have the following expansion for $g$ :

\begin{aligned} g\p{z, w} &= \frac{1}{z - w} \p{\sum_{n=0}^\infty a_nz^n - \sum_{n=0}^\infty a_nw^n} \\ &= a_1 + \sum_{n=2}^\infty \p{a_n \sum_{k=0}^{n-1} z^{n-1-k}w^k}. \end{aligned}

Given $z \neq w$ in $\D$ , let $r = \max\,\set{\abs{z}, \abs{w}}$ so that $\abs{z}, \abs{w} \leq r < 1$ . Applying this along with the triangle inequality yields

\begin{aligned} \abs{g\p{z, w}} \geq \abs{a_1} - r\sum_{n=2}^\infty n\abs{a_n} \geq \abs{a_1}\p{1 - r} > 0, \end{aligned}

since $a_1 \neq 0$ . Thus, $g\p{z, w} \neq 0$ , and so $f\p{z} \neq f\p{w}$ for any distinct $z, w \in \D$ .