Cauchy's functional equation and extensions: Goldie's equation and inequality, the Go{\l}\k{a}b-Schinzel equation and Beurling's equation

TL;DR

This paper explores fundamental functional equations like Cauchy's, Goldie's, Beurling's, and Golab-Schinzel, analyzing their roles in regular variation and extending classical results with new insights.

Contribution

It provides a comprehensive study of these equations and inequalities, highlighting their interrelations and applications in regular variation theory.

Findings

Unified framework for classical and Beurling regular variation

New insights into the Golab-Schinzel equation

Extensions of Goldie's equation and inequality

Abstract

The Cauchy functional equation is not only the most important single functional equation, it is also central to regular variation. Classical Karamata regular variation involves a functional equation and inequality due to Goldie; we study this, and its counterpart in Beurling regular variation, together with the related Go{\l}\k{a}b-Schinzel equation.