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Is anything non-trivial known about univalent functions with non-negative coefficients?

Let $U$ be the unit disc, and $f$ a univalent (=injective) holomorphic function, $f(0)=0$, $f'(0)>0$. There is a one-to-one correspondence between such functions and simply connected regions $D$ in the plane, containing $0$. The function is real iff $D$ is symmetric with respect to the real line.

  1. What can be said about $D$ if it is known that coefficients are non-negative? (Except the trivial fact that the boundary point of the largest modulus is on the positive ray).

  2. What can be said about non-negative coefficients of a power series if it is known that it represents a univalent function?

  3. Is it possible to parametrize this class?

I am especially interested in the case when the positive ray is contained in $D$, that is $f(1)=\infty$.

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