Nehari's theorem on univalence

Remarks

Locally, $f = u_1/u_2$ for two independent solutions of the linear equation $u'' + \frac{1}{2} S(f)\, u = 0$. If $f(a) = f(b)$ with $a \ne b$, then some solution of this equation vanishes at both $a$ and $b$. A disc automorphism moves $a$ and $b$ onto a diameter. The bound on $S(f)$ then allows a comparison of Sturm type with the equation $u'' + (1 - x^2)^{-2} u = 0$ on $(-1, 1)$. That equation has the solution $\sqrt{1 - x^2}$ without zeros, which leads to a contradiction. The constant $2$ cannot be replaced by any larger constant. In the other direction, the Kraus–Nehari theorem says that every univalent $f$ on $\mathbb{D}$ satisfies $\abs{S(f)(z)} \le 6/(1 - \abs{z}^2)^2$, with equality at $z = 0$ for the Koebe function. Example: $f(z) = e^{cz}$ has $S(f) = -c^2/2$, so for $\abs{c} \le 2$ the bound holds, since $(1 - \abs{z}^2)^{-2} \ge 1$; such $f$ is univalent on $\mathbb{D}$, as two points of $\mathbb{D}$ cannot differ by $2\pi i k / c$ with $k \ne 0$.

Going deeper

For a student

The weight $(1 - \abs{z}^2)^{-2}$ is the square of the density of the hyperbolic metric, so the condition is invariant under automorphisms of the disc. Together with the necessary bound $6$, it shows that the Schwarzian derivatives of univalent functions all have bounded hyperbolic size, and that all small enough ones come from univalent functions. This point of view leads to the Bers embedding of the universal Teichmüller space. It also leads to the theorem of Ahlfors and Weill that a slightly stronger bound gives a quasiconformal extension of $f$ to the plane.