How to show this mapping is quasiconformal. And the integrability of the gradient. A complex map $f$ on the unit disk defined as  $f(re^{i\theta})=r^ke^{i\theta}$ where $k>1$. I hope to know how to show this map is quasiconformal and what is the largest $p$ such that the gradient of the map is $L^p$ on the disk. Are there any interesting properties of this map? In general, how do we approach 'strange' maps like this?

For the first part of your questions, I think that you can look at this maps as
$$
f(z)=\frac{z}{|z|}|z|^k\,,
$$
and they are classical examples of very importat class of mappings caled \emph{radial stretchings} (a complete discussion of the basics this class of maps can be found in section 2.6 of Astala, Iwaniec and Martin "Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane"). For the class of maps that you are considering, the Beltrami coefficient $\mu(z)$ is given by
$$
\mu(z)=\frac{k-1}{k+1}\frac{z}{\overline{z}}
$$
and the norm of the gradient can be explicitly obtained too:
$$
|Df(z)|=k|z|^{k-1}.
$$
I am sure that from the expressions above you will be able to solve your problems about quasiconformality and integrability of $Df(z)$.
The second part of your questions (about the importance of this maps) is more deep, and they appear as a kind of extremal examples when we are studying Hölder continuity of of q.c. mappings. I am not an expert in the field but I am sure that you can find more information in the overcited book of Astala, Iwaniec and Martin.
