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Analytic functions with hyperbolic range and Bohr's inequality

Yusuf Abu Muhanna, Issam Louhichi

Abstract

We use properties of the hyperbolic metric and properties of the modular function to show that the Bohr's radius for covering maps onto hyperbolic domains is greater or equal to exponential minus pi. This includes almost all known classes of analytic functions.

Analytic functions with hyperbolic range and Bohr's inequality

Abstract

We use properties of the hyperbolic metric and properties of the modular function to show that the Bohr's radius for covering maps onto hyperbolic domains is greater or equal to exponential minus pi. This includes almost all known classes of analytic functions.
Paper Structure (9 sections, 12 theorems, 41 equations)

This paper contains 9 sections, 12 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Bohr's operator
  3. Hyperbolic metric.
  4. The Modular function and consequences
  5. Main results
  6. Harmonic maps
  7. No conflict of interest
  8. Compliance with Ethical Standards
  9. Data availability statement

Key Result

Theorem 1.1

(Uniformization Theorem [5]) If $D$ is hyperbolic, then there is a universal cover $F$ (conformal) from $U$ onto $D.$ This cover is unique with the normalization $F(0)=a$ and $F^{\prime }(0)>0,$ for some $a\in D.$

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 1.5
  • Lemma 1.6
  • Lemma 1.7
  • Corollary 1.8
  • Theorem 2.1
  • proof
  • ...and 5 more