Geometric Versions of Schwarz's Lemma for Spherically Convex Functions

TL;DR

This paper establishes sharp distortion and monotonicity theorems for spherically convex functions, extending Schwarz's lemma into a geometric context involving spherical metrics and curvature.

Contribution

It introduces geometric variants of Schwarz's lemma specifically for spherically convex functions, with new sharp bounds involving spherical length, area, and curvature.

Findings

Sharp distortion theorems for spherical length and area

Monotonicity results for spherical curvature

Extensions of Schwarz's lemma to spherical geometry

Abstract

We prove several sharp distortion and monotonicity theorems for spherically convex functions defined on the unit disk involving geometric quantities such as spherical length, spherical area and total spherical curvature. These results can be viewed as geometric variants of the classical Schwarz lemma for spherically convex functions.