arXiv subjects

T. K. Carne

Publications and source records attributed to T. K. Carne.

2 recordsLinked to original sources

arXiv2007

Spherical and hyperbolic lengths of images of arcs

Let $f$ be an analytic function on the unit disc which is in the Dirichlet class, so the Euclidean area of the image, counting multiplicity, is finite. The Euclidean length of a radial arc of hyperbolic length $ρ$ is then $o(ρ^1/2)$. In this note we consider the corresponding results when $f$ maps into the unit disc with the hyperbolic metric or the Riemann sphere with the spherical metric. Similar but not identical results hold.

math.CV↗

arXiv2007

Euclidean and hyperbolic lenghs of images of arcs

Let $f$ be a function that is analytic in the unit disc. We give new estimates, and new proofs of existing estimates, of the Euclidean length of the image under $f$ of a radial segment in the unit disc. Our methods are based on the hyperbolic geometry of plane domains, and we address some new questions that follow naturally from this approach.

math.CV↗