On the linear connectivity of the regular part of Gakhov set

The study of extrema of the inner mapping (conformal) radii of the plane domains has been initiated by G. P´olya and G. Szeg¨o in connection with the isoperimetrical inequalities and by F.D. Gakhov in concern with construction of the correction classes for the exterior boundary value problems. Both traditions have been unified in the works of L.A. Aksent’ev and his successors. These works also were seminal for the systematic accumulation of the uniqueness criteria for the critical points of the conformal radii. The process of such an accumulation has led to the introduction of the Gakhov set. Let be the class of functions holomorphic in the unit disk D = { ∈ ∈ C : | |