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Let $f$ be a schlicht function. (That is, a univalent function on the unit disk so that $f(0)=0$ and $f'(0)=1$.)

Let us further suppose that $f$ and $f'$ extends to a continuous function on the boundary.

I'm wondering if there was a distortion theorem for such functions, similar to Koebe distortion theorem. I'm actually interested in the logarithmic derivative.

Let $h(z) = z \frac{f'(z)}{f(z)}$. Then, for any schlicht function

\begin{equation} \left(\frac{(1-|z|)}{1+|z|}\right) \le |h(z)| \le \left(\frac{(1+|z|)}{1-|z|}\right) \end{equation}

However, $h(z)$ should be bounded in this case, so I feel like there should be a better estimate.

EDIT: Just to clarify. What I want to be able to do is to somehow characterize the family of functions that $f$ that satisfy $\sup |h-1| < k < 1$.

EDIT2: I was originally thinking that I could get away with restricting the function to be $C^p$ for some $p>0$, but then someone pointed out the following example:

Let $f(z) = z/(1-z)^2$. (The Koebe function.) Then let $f_r(z) =\frac{1}{r}f(rz)$. Then, $$h_r(z) = z \frac{f_r'(z)}{f_r(z)} = \frac{r+z}{r-z}.$$

For $r>1$, $f_r$ has a analytic extension to $|z|<r$, and $h_r$ gets arbitrarily close to $h(z) = \frac{1+z}{1-z}$.

So it seems like adding smoothness to the extension doesn't do anything...

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