Univalent functions whitch are not starlike or convex

Question

$S$ denotes the univalent function class, $S=\{f \in \mathcal{A}:f \in H_{u}(\mathrm{U})\}$. $S^{\ast}=\left\{f\in\mathcal{A}:\operatorname{Re}\frac{zf^{\prime}(z)}{f(z)}>0, \;z\in\mathrm{U}\right\}$ is the starlike functions class and $K=\left\{f\in\mathcal{A}:\operatorname{Re}\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}+1>0,\;z\in\mathrm{U}\right\}$ is the convex functions class. We know that $K \subset S^{\ast} \subset S$,and i need 2 examples for an univalent function ($S$), but not starlike ($S^{\ast}$), and a function whitch is univalent ($S$) and not convex ($K$), with proof if it's possible. (I know that the Koebe function is starlike but not convex)

Answer 1

user147263 Jun 4 '14 at 22:21 gave an answer.

Let D be domain in w-plane defined by -\pi/4< arg w <5 \pi/4 and let G=D +(-1) and f conformal mapping of the unit disk onto G such that f(0)=0.

Since G is non-convex / non-starlike (wrt 0) domain, f is not starlike.