Sharpness of certain Campbell and Pommerenke estimates (Q1280625)

Let \(\Delta\) denote the open unit disk. A set \({\mathfrak M}\) of functions \(f(z)= z+\sum^\infty_{n= 2}a_n(f)z^n\) analytic in \(\Delta\) is called a linearly invariant family if for each \(f\in{\mathfrak M}\) the following two conditions are satisfied: (1) \(f'(z)\neq 0\) in \(\Delta\) and (2) for any conformal automorphism \(\varphi(z)\) of \(\Delta\), the function \({f(\varphi(z))- f(\varphi(0))\over f'(\varphi(0))\varphi'(0)}= z+\cdots\) also belongs to \({\mathfrak M}\). This notion was introduced by C. Pommerenke in 1964. Pommerenke defined the order of a linearly invariant family \({\mathfrak M}\) as \(\sup_{f\in{\mathfrak M}}| a_2(f)|\). The universal linearly invariant family \(U_\alpha\) is the union of all linearly invariant families \({\mathfrak M}\) such that the order of \({\mathfrak M}\) is at most \(\alpha\) (for \(\alpha\geq 1\)). The authors examine several estimates for \(|\text{arg }f'(z)|\) and determine extremal functions, where \(f\) belongs to some linearly invariant family. Other results in the paper include a lower estimate for the Schwarzian derivative for functions in \(U_\alpha\) and sharp upper bounds for the order of the functions \(f_r(z)= {f(rz)\over r}\), where \(f\in U_\alpha\).