Is there an example of a schlicht function $f(z)=z+a\_2z^2+a\_3z^3+\cdots$, which is analytic and injective on the open unit disk $\mathbb{D}$, such that $-1/a\_2$ belongs to the range $f(\mathbb{D})$? Or is $-1/a\_2$ necessarily an omitted value of $f$? 



Can $-1/a_2$ belong to the range of a schlicht function $z+a_2z^2+\cdots$?   Or is $-1/a_2$ necessarily an omitted value?
It is easy to design a function like $z/(1-az)$ with small $a$ that maps the circle to a nice domain whose closure does not contain $-1/{a\_2}$. Now take this domain and grow a blob that contains $-1/{a\_2}$ deep in its interior but is connected to our initial domain by a very thin stem. Then the conformal mapping to this new domain has almost the same coefficients as the original one (almost being controlled by the width of the stem, not the size of the blob), so after renormalizing you get a counterexample.


