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

TL;DR

This paper investigates the localization of poles in best Möbius approximations of certain univalent functions, providing geometric bounds, analyzing specific function classes, and deriving new convexity criteria.

Contribution

It introduces sharp bounds for pole locations, analyzes their behavior across various function classes, and establishes new convexity conditions based on Schwarzian derivative bounds.

Findings

Sharp geometric bounds for pole locations

Regions covered by poles for polygonal mappings

New convexity conditions from Schwarzian derivative bounds

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.