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

Hugo Arbelaez, Martin Chuaqui, Rodrigo Hernandez, Willy Sierra

TL;DR

This work extends the polarity relation between best Möbius approximations and geometric convexity/concavity for locally univalent maps in the unit disk. By articulating precise links between the pole function $P_f$, Pommerenke’s orders $\alpha_f$ and $\mu_f$, and the Schwarzian derivative, the authors derive sharp pole-location bounds for broad subclasses (including convex of order $\alpha$, Janowski and Robertson classes, and starlike mappings) and provide exact multiplicity/region results for polygonal mappings. The results yield new convexity criteria via Schwarzian bounds and illuminate how pole geometry encodes strong geometric properties of conformal maps, with potential implications for Teichmüller theory and related complex-analytic frameworks. Overall, the paper deepens the understanding of how the BMAs’ poles govern convexity, concavity, and boundary behavior through explicit, order-dependent inequalities.

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.

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

TL;DR

This work extends the polarity relation between best Möbius approximations and geometric convexity/concavity for locally univalent maps in the unit disk. By articulating precise links between the pole function , Pommerenke’s orders and , and the Schwarzian derivative, the authors derive sharp pole-location bounds for broad subclasses (including convex of order , Janowski and Robertson classes, and starlike mappings) and provide exact multiplicity/region results for polygonal mappings. The results yield new convexity criteria via Schwarzian bounds and illuminate how pole geometry encodes strong geometric properties of conformal maps, with potential implications for Teichmüller theory and related complex-analytic frameworks. Overall, the paper deepens the understanding of how the BMAs’ poles govern convexity, concavity, and boundary behavior through explicit, order-dependent inequalities.

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.
Paper Structure (6 sections, 21 theorems, 83 equations)

This paper contains 6 sections, 21 theorems, 83 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Best Möbius Approximation (BMA)
  3. Properties of $P_f$ function
  4. Location of $P_f$ in terms of $\mu_f$ and $\alpha_f$
  5. Special Subclasses
  6. $P_f$ and Schwarzian derivative

Key Result

Theorem 1

Let $f$ be locally injective in a domain $D$, and let $\zeta \in D$. Then the pole $P_f(\zeta)$ of the BMA of $f$ at $\zeta$ satisfies the following:

Theorems & Definitions (48)

  • Theorem 1
  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Theorem 3.1
  • proof
  • Example 3.2
  • Proposition 3.3
  • ...and 38 more