Using this definition of $D_{\alpha}f(z)$, if $D(z)=\frac{\max_{\alpha}|D_{\alpha}f(z)|}{\min_{\alpha}|D_{\alpha}f(z)|}$, find $D(z)$

Question

I don't know what to do with this problem:

Let $D_{\alpha}f(z)=\lim_{r\to 0}\frac{f(z+re^{i\alpha})-f(z)}{re^{i\alpha}}$ and $D(z)=\frac{\max_{\alpha}|D_{\alpha}f(z)|}{\min_{\alpha}|D_{\alpha}f(z)|}$. Find $D(z)$ if $f(z)=\bar{z}$.

So my try was this:

To find $D(z)$, we need to find $D_{\alpha}f(z)$:

$D_{\alpha}f(z)=\lim_{r\to 0}\frac{f(z+re^{i\alpha})-f(z)}{re^{i\alpha}}=\lim_{r\to 0}\frac{\overline{z+re^{i\alpha}}-\bar{z}}{re^{i\alpha}}$

Which is the correct way to compute this limit?

Thanks for your time.

Answer 1

Hint:

1. For $u,v\in\mathbb C$ there holds $\overline{u+v}=\overline u+\overline v$ and $\overline{uv}=\overline{u}\cdot\overline{v}$.

2. For $\alpha\in\mathbb R$ there holds $e^{i\alpha}=\cos(\alpha)+i\sin(\alpha)$. So, you can conclude $\overline{e^{i\alpha}}=e^{-i\alpha}$