Estimates for the hyperbolic and quasihyperbolic metrics in hyperbolic regions

TL;DR

This paper derives sharp bounds for derivatives of universal covering maps in hyperbolic regions, linking these bounds to hyperbolic density and providing conditions for their applicability in complex analysis.

Contribution

It introduces new sharp bounds for derivatives of covering maps in hyperbolic regions and establishes necessary and sufficient conditions for these bounds in arbitrary domains.

Findings

Sharp bounds for |f'(z)|/dist(f(z),∂f(D)) in simply connected domains

Necessary and sufficient conditions for bounds in arbitrary domains

Connection between bounds for derivatives and the Schwarzian derivative

Abstract

In this paper we consider ordinary derivative of universal covering mappings $f$ of hyperbolic regions $D$ in the complex plane. We obtain sharp bounds for the ratio $|f'(z)|/{\rm dist}(f(z),\partial f(D))$ in terms of the hyperbolic density in simply connection domains. In arbitrary domains, we find a necessary and sufficient condition for an upper bound for the quantity $|f'(z)|/{\rm dist}(f(z),\partial f(D))$ to hold in terms of the hyperbolic density. As an application of the above results, it is observed that the bounds for the quantity of the above type are closely connected with similar bounds for $|f''(z)/f'(z)|$.