Characteristic properties of Nehari functions (Q1306251)

The Nehari class \(N\) consists of all meromorphic functions \(f\) in the unit disk \(D\) such that \(|Sf(z)|\leq \frac{2}{(1-|z|^2)^2}\), where \(Sf\) denotes the Schwarzian derivative of \(f\). The class \(N_0\) consists of functions in \(N\) with the normalization \(f(0)= 0\), \(f'(0)= 1\), \(f''(0)= 0\). Important extremal functions are \(L(z)= \frac 12 \log \frac{1+z}{1-z}\) and \(n(z)= \frac{1}{\sqrt{2}} \frac{(1+z)^{\sqrt{2}}- (1-z)^{\sqrt{2}}} {(1+z)^{\sqrt{2}}+ (1-z)^{\sqrt{2}}}\). The authors find sharp upper and lower bounds in terms of the functions \(n(z)\) and \(L(z)\) for the spherical distortion \(\frac{|f'(z)|} {1+|f(z)|^2}\), where \(f\in N_0\). The authors also find a necessary and sufficient condition for a meromorphic and locally univalent function to belong to the class \(N\). This condition involves the distortion at two points in \(D\). Next, the authors consider functions in \(N\) with the normalization \(f(z)= \frac 1z+ b_0+ b_1z+\cdots\). Let \(B(z)= \{w:|z|\leq|w|\leq 1\), \(|\arg(w)- \arg(z)|\leq \pi(1-|z|)\}\). The authors prove that \(\text{diam } f(B(z))\leq K(1-|z|^2) |f'(z)|L(|z|)\) for some absolute constant. The paper concludes with two theorems on the boundary behavior of functions in \(N\). In particular, it is shown how \(f(D)\) might fail to be a bounded quasidisk.