TITLE: Necessary and sufficient conditions for a holomorphic function defined in the unit disk to be univalent?
QUESTION [5 upvotes]: de Branges has proved de Branges's theorem (the famous Bieberbach conjecture) that if a holomorphic function $f(z) = z+\sum_{n=2}^{\infty} a_nz^n$ in the unit disk $D = \{z\in \mathbb{C},|z| \leq 1\}$ is univalent, then we have $|a_n| \leq n,\forall n\geq 2$.
Conversely, let's consider a holomorphic function $g(z) = z+\sum_{n=2}^{\infty} b_nz^n$ which is defined in $D$ and satifies $|b_n| \leq n$, then what are the general sufficient conditions(I've known some special conditions on this problems, such as Nehari's univalence criterion and other criterions, unfortunately, they are not in full generality) to ensure $g(z)$ is univalent. Any clues and facts are welcomed, best regards !

Updated question: necessary and sufficient conditions for a holomorphic function defined in the unit disk to be univalent (as far as I known, several conditions have be proposed, but all of them seem to be not practical), simple forms and only depend on function g(z) or its derivatives, integrals, their combinations, and so on. For example something like Milin's inequality. Unfortunately, I've tried several variants of this inequality (together with some additional conditons), but fails.

REPLY [3 votes]: It is highly unlikely that anything "reasonable" can be said in terms of the coefficients.
Already the allowable region for $(a_2,a_3)$ (so they are the first non-trivial coefficients of $z+a_2z^2+a_3z^3+...$ univalent) is quite complicated as for example the sharpness (ie for each $0 \le \alpha \le 1$ there is an univalent functions for which the equality holds) of the Fekete-Szego inequalities $|a_3-\alpha a_2^2| \le 1+2e^{\frac{-2\alpha}{1-\alpha}}, 0 \le \alpha \le 1$ shows - here $\alpha =0$ corresponds to the highly non-trivial third coefficient bound $|a_3| \le 3$, while $\alpha=1$ corresponds to the easy $|a_3-a_2^2| \le 1$
See also Schaeffer Spencer book on the coefficient regions for Schlicht functions https://www.ams.org/books/coll/035/