More convex functions by Artin`s method

TL;DR

This paper generalizes Artin's method to find log-convex solutions of functional equations, enhancing understanding of functions through their representers and contributing to the theory of convex functions.

Contribution

It introduces a generalized approach inspired by Artin's trick to identify log-convex solutions to functional equations, expanding the analytical toolkit.

Findings

Established a method to find log-convex solutions to functional equations.

Connected the solutions to their representers for better understanding.

Extended the concept of convexity and log-convexity in functional analysis.

Abstract

First we recall the notion of conxity and log-convexity for real-valued. Then we generalize the trick used by Artin in his famous paper on the Gamma function to find log-convex solutions to the functional equations f(x+1)=g(x)f(x). This gives rise to understand a function by its representer.