Maps Preserving the Sum-to-Difference Ratio
Abstract
For a field F, what are all functions f:F→F that satisfy the functional equation f((x+y)/(x−y))=(f(x)+f(y))/(f(x)−f(y)) for all x≠y in F? We solve this problem for the fields Q,R, and a class of its subfields that includes the real constructible numbers, the real algebraic numbers, and all quadratic number fields. We also solve it over the complex numbers, and on any subfield of R, if f is continuous over the reals. The proofs involve a mix of algebra in all fields, analysis over the real line, and some topology in the complex plane.
Fields of Study & Subjects
Meromorphic and Entire Functions
Functional Equations Stability Results
Advanced Topology and Set Theory
Publication details
- Journal / source
- American Mathematical Monthly
- Publisher
- Informa UK Limited
- Pages
- 1–14
- Publication date
- 17 August 2026
- Language
- EN