Completely and logarithmically completely monotone functions

TL;DR

This paper simplifies and extends the theory of completely and logarithmically completely monotone functions, correcting errors, generalizing key results, and exploring their properties, especially in relation to the Gamma and psi functions.

Contribution

It provides simplified proofs, corrects previous errors, generalizes existing theorems, and introduces new results in the theory of monotone functions related to special functions.

Findings

Simplified proofs of key theorems

Corrected previous erroneous results

New insights into monotonicity of Gamma-related functions

Abstract

We simplify the proof of some widely used theoretical theorems, extending their applicability, while correcting some erroneous results. We also generalize key results and present new results that contribute to the development of the theory. Furthermore, we use the results obtained to investigate the monotonicity properties of some specific functions related to the Gamma function. Finally, we formulate an open problem related to the psi function.