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

Abstract

We study the localization of the poles of the best Mobius 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 alpha, 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.