How do show that $w=z-\frac12 z^2$ maps $|z|<1$ onto the interior of a cardioid? How do i show that $w=z-\frac12 z^2$ maps $|z|<1$ onto the interior of a cardioid?
 A: This is an incomplete answer, but it may be sufficient for your purposes. I can show that the boundary of the unit disk (i.e. the unit circle $|z|=1$) maps to a cardioid curve, but proving that every point of the interior of the unit disk maps to the interior of the cardioid (and vice versa) became a very difficult problem.
Let $z = re^{\theta i}$. Then
$\newcommand{\half}{\frac{1}{2}}$
\begin{align}
w &= re^{\theta i} - \half r^2 e^{2\theta i} \\
&= r \cos \theta + i r \sin \theta - \half r^2 \cos 2\theta - \half i r^2 \sin 2\theta \ \ \ \ \textrm{(by Euler's Formula)}
\end{align}
By the Double Angle Identities, we know
\begin{align}
\cos 2\theta &= 2 \cos^2 \theta - 1 \\
\sin 2\theta &= 2 \sin\theta \cos \theta
\end{align}
Hence
\begin{align}
w &= r \cos \theta - r^2 \left(\cos^2 \theta - \half \right) + i \left(r \sin \theta - r^2 \sin \theta \cos \theta \right) \\
&= \left(r \cos \theta - r^2 \cos^2 \theta + r^2/2 \right) + i \left(r \sin \theta - r^2 \sin \theta \cos \theta \right) \\
&= (1 - r \cos \theta) \, r \cos \theta + r^2/2 + i (1 - r \cos \theta) \, r \sin \theta \\
&= (r \cos \theta + i \, r \sin \theta)(1-r \cos \theta) + r^2/2 \\
&= r e^{\theta i} (1-r \cos \theta) + r^2/2
\end{align}
If we now set $r = 1$ (i.e. we restrict the domain to the unit circle) we obtain
$$ e^{\theta i}(1-\cos \theta) + 1/2 $$
which is the polar form for an equation of a cardioid (see Cardioid - Wikipedia), but shifted half a unit to the right. Hence, the boundary of the unit disk maps to a cardioid curve, but again, showing the interior of the unit disk maps to the interior of the cardioid appears to be very difficult.
One technique I tried was to take any point $z$ in the open unit disk and show that
$$|w(z)| < \left|e^{\theta' i}(1-\cos \theta') + 1/2 \right|$$
where $\theta'$ represents the angle of the image $w(z)$. But attempting to prove this inequality was excessively difficult and ultimately may have been unfruitful.
Likewise, I imagine proving the converse that every point in the interior of the cardioid traces to a point in the open unit disk is a similarly difficult problem.
