Mappings with rotation symmetry (Q1589401)

Let \(S_p(M)\), \(p\in\mathbb{N}\), \(M> 1\), be the class of functions \(f(z)= z+\cdots\), holomorphic in the unit disc \(E_1\), where \(E_r= \{z:|z|< r\}\), which univalently map \(E_1\) to domains from the disc \(E_M\) and have \(p\)-multiple rotation symmetry with respect to the point \(0\). It is known, that the set \(\widetilde S_p(M)\) of functions of the form \(W= M\zeta(z,\ln M)\), where \(\zeta(z,\tau)\) \((0\leq \tau\leq\ln M)\) is a solution of the Löwner equation [see \textit{I. A. Aleksandrov}, Parametric continuations in the theory of schlicht functions, Nauka, Moscow (1976; Zbl 0462.30008), p. 27] \({d\zeta\over d\tau}= -\zeta{\mu^p(\tau)+ \zeta^p\over \mu^p(\tau)- \zeta^p}\), \(\zeta(z, 0)= z\in E_1\), \(\mu(\tau)\) -- a continuous function with the module equal to \(1\), is a subclass dense in \(S_p(M)\) (in the topology of uniform convergence on compact subsets of \(E_1\)). In this paper the authors give a form of the control function \(\mu\), such that the Löwner equation leads to a formula for \(\zeta(z, \tau)\), which is more general than the hitherto known one.