On the localization of the poles of the best Möbius approximations of f

We study the localization of the poles of the best Möbius approximations for locally univalent functions in the unit disk. Sharp geometric bounds for the pole function are established in terms of Pommerenke’s linear invariant orders, refining classical criteria for convexity and concavity. The behavior of poles is further analyzed for starlike mappings, convex functions of order, Janowski functions, and Robertson’s class. For polygonal mappings, we describe the regions covered by the poles and obtain exact multiplicity results. We also derive new convexity conditions based on bounds of the Schwarzian derivative.