On a generalized conjecture by Alzer and Matkowski

TL;DR

This paper explores a generalization of a conjecture related to the bilinearity of the Cauchy exponential difference, extending previous results from real functions to more general real or complex mappings on linear spaces.

Contribution

It extends the original conjecture by Alzer and Matkowski to broader classes of functions, including complex mappings, and discusses related generalizations.

Findings

The original conjecture was affirmatively resolved for real functions.

The paper generalizes the conjecture to complex mappings.

It provides new insights into the bilinearity property of the Cauchy exponential difference.

Abstract

We study a recent conjecture proposed by Horst Alzer and Janusz Matkowski concerning a bilinearity property of the Cauchy exponential difference for real-to-real functions. The original conjecture was affirmatively resolved by Tomasz Ma\l{}olepszy. We deal with generalizations for real or complex mappings acting on a linear space.