Distinct ellipsoids are not biholomorphic in $\mathbb{C}^2$

Question

I am stuck on the following problem:

For each real $p\geq 1$ consider the the set $$D_p=\{(z,w)\in\mathbb{C}^2:|z|^{2p}+|w|^2<1\}.$$ Let $p\neq q$, then, there does not exist any biholomorphism between $D_p$ and $D_q$.

I can't even think how to start the proof. Any help will be appreciated.

Edit: I found the question from the following article: http://www.jstor.org/stable/2322989