Erratum to: Geometry Behind Chordal Loewner Chains

1 Erratum to: Complex Anal. Oper. Theory (2010) 4:541–587 DOI 10.1007/s11785-010-0057-6

The authors apologize for several mistakes and misprints found after the paper “Geometry behind chordal Loewner chains” was published. The following corrections are necessary in Lemmas 2.1, 2.2 and 3.6.

I. Lemma 2.1 should be as follows:

Lemma 2.1

Let $\varphi$ be a holomorphic univalent self-mapping of the unit disk $\mathbb D$ with ${\varphi (0)=0}$. Then:

• (1) for all $z\in \mathbb D$,

  $$\begin{aligned} |\varphi (z)-z|\le |1-\varphi ^{\prime }(0)|+\frac{|z|^2}{\sqrt{1-|z|^2}}\sqrt{1-|\varphi ^{\prime }(0)|^2}; \end{aligned}$$

• (2) if $\varphi ^{\prime }(0)>0$, then for all $z\in \mathbb D$,

  $$\begin{aligned} |\varphi (z)-z|\le C_0\sqrt{1-\varphi ^{\prime }(0)}, \end{aligned}$$

  where $C_0:=1+ \dfrac{\sqrt{2}\,r^2}{\sqrt{1-r^2}}$ and $r:=|z|$.

Proof

Write $\varphi (z)=\sum _{n=1}^{\infty } a_n z^n$. Let us recall that

$$\begin{aligned} \sum _{n=1}^{\infty } n|a_n|^2=m(\varphi (\mathbb D ))\le m(\mathbb D )=1, \end{aligned}$$

where $m(\cdot )$ stands for the normalized Lebesgue measure in the unit disc. Let $z\in \mathbb D$ and $r:=|z|$. Then

$$\begin{aligned} |\varphi (z)-z|&\le \left| \sum _{n=2}^{\infty } a_nz^n\right|+|1-\varphi ^{\prime }(0)| \le \sqrt{\sum _{n=2}^{\infty } r^{2n}}~\sqrt{\sum _{n=2}^{\infty } |a_n|^2}+|1-\varphi ^{\prime }(0)|\\&\!= \dfrac{r^2}{\sqrt{1-r^2}}\sqrt{\sum _{n=2}^{\infty } |a_n|^2} \!+\!|1-\varphi ^{\prime }(0)| \!\le \! \dfrac{r^2}{\sqrt{1-r^2}}\sqrt{\sum _{n=2}^{\infty } n|a_n|^2} \!+\!|1-\varphi ^{\prime }(0)|\\&\le \dfrac{r^2}{\sqrt{1-r^2}}\sqrt{1-|\varphi ^{\prime }(0)|^2} +|1-\varphi ^{\prime }(0)|. \end{aligned}$$

This proves (1). Assertion (2) is now an immediate consequence of (1). $\square$

II. The last line in the proof of Lemma 2.2 has to be changed as follows:

$$\begin{aligned} |f(z)-g(z)|\le g^{\prime }(0)\frac{2 C_0}{(1-r)^3} \sqrt{1-\varphi ^{\prime }(0)}=\frac{2 C_0}{(1-r)^3}\sqrt{g^{\prime }(0)(g^{\prime }(0)-f^{\prime }(0))}. \end{aligned}$$

III. The proof of Lemma 3.6 contains an error. Arguing in a more direct way we get a better estimate under weaker conditions $|z|>b>0$ and $|\arg z-\pi /2|\le \pi /3$:

$$\begin{aligned}&\frac{1+r_{\mathbb H }(z,ib)}{1-r_{\mathbb H }(z,ib)}=\frac{|z+ib|+|z-ib|}{|z+ib|-|z-ib|}=\frac{\big (|z+ib|+|z-ib|\big )^2}{|z+ib|^2-|z-ib|^2}\\&\quad =\frac{\big (|z+ib|+|z-ib|\big )^2}{4b\,{\mathsf Im}\,z}\le \frac{\big (2|z|+2b\big )^2}{4b\cdot \frac{1}{2}|z|}\le \frac{2}{b}\,|z|. \end{aligned}$$

IV. Misprints:

• 1. Condition (B.2) in Theorem 4.8 should contain “ $\Omega _t$” instead of “ $\Omega _t(\mathbb D )$”.

• 2. The estimate of $|V(z,x)/z|$ in the proof of Proposition 5.9, see page 572, should contain “$|{\mathsf Re}\,z|$” instead of “${\mathsf Re}\,z$”.

• 3. The Eq. (7.5) on p. 585 should contain the formula $u(t):=H^{-1}(\lambda (t))\ne 1$ rather than $u(t):=H(\lambda (t))\ne 1$.