Consider the class $S$ of univalent functions on the unit disk $D$ normalized so that $f(0)=0$ and $f'(0)=1$. Each function in $S$ satisfy the Bieberbach conjecture, that is the $n$-th coefficient in the power series expansion is less or equal than $n$. I am looking at the convex hull of class $S$. Any function there satisfies the Bieberbach conjecture by an easy computation.
Are there functions which do not belong to the convex hull of $S$ but satisfy the Bieberbach conjecture?
More generally, how do you prove that a function is not in the convex hull of $S$ without using the Bieberbach conjecture? There are characterizations of convex hulls of subfamilies of $S$, for example starlike maps, in terms of integral representations.