Completely and logarithmically completely monotone functions
V. E. Sándor Szabó
TL;DR
The paper addresses gaps and errors in the theory of completely monotone (CM) and logarithmically completely monotone (log-CM) functions, with a focus on gamma and digamma ($\psi$) related functions. It develops streamlined proofs and generalizes several results, notably correcting a flawed argument in Chen (2007) and extending monotonicity criteria for function classes built from products and compositions. Key contributions include new CM/log-CM results for functions like $f_{\alpha,\beta}$ and ratios of Gamma functions, as well as a general framework for when fractional or transformed gamma expressions preserve CM or log-CM. The work offers sharper conditions, fixes to prior proofs, and opens questions (e.g., about optimal parameter thresholds) that strengthen and extend the monotonicity theory in analysis, with implications for special functions and their inequalities.
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.