A Schur's theorem via a monotonicity and the expansion module

TL;DR

This paper extends Schur's classical theorem to convex plane and spherical curves using a new monotonicity approach, providing broader geometric inequalities and insights.

Contribution

It introduces a novel monotonicity method that generalizes Schur's theorem to convex plane and spherical curves, extending classical results.

Findings

Extended Schur's theorem to convex plane curves

Proved a spherical curve version of Schur's theorem

Established a new monotonicity principle in geometric inequalities

Abstract

In this paper we present a monotonicity which extends a classical theorem of A. Schur comparing the chord length of a convex plane curve with a space curve of smaller curvature. We also prove a Schur's Theorem for spherical curves, which extends the Cauchy's Arm Lemma.