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.

conjecturedde Branges' theorem. $\endgroup$ – abx Sep 24 '20 at 8:25