On stationary points of the conformal radius of spiral domains (Q2387108)

In this paper the following problem is investigated. Theorem 1: Let a function \(f\) of a form \[ f(z)=z+a_{n+1}z^{n+1}+a_{n+2} z^{n+2}+ \cdots \] be regular in \(E= \{z:|z|<1\}\) and let \[ \frac{zf'(z)}{f(z)} \prec \frac{1+\delta e^{-2i\gamma} z} {1-\alpha z}, \] for some \(n, \alpha,\delta,\gamma\) such that \(|\alpha|+n|\delta| /(n+1)\leq 1,\) \(|\alpha+\delta e^{-2i\gamma}|\leq n/(n+2)\). (i) Then the conformal radius \({\mathcal R}(f(E),f(z))\) of a domain \(f(E)\) with respect to a point \(f(z)\) has the unique critical point \(z=0\) if in the case of equality in (i) and \(n= 2\) we exclude extremal domains \(f(E)\) where \(f\) are defined by the relation \[ \frac {zf'(z)} {f(z)}=\frac{1+\delta e^{-2i\gamma}z^2}{1-\alpha z^2}. \] Some related problems are also investigated.