Logarithmically completely monotonic functions related to the q-gamma function and its applications

Our main goal in this paper is to introduce new classes of logarithmically completely monotonic functions involving q-gamma function. As applications, new classes of Bernstein functions related to the q-gamma function and dilogarithm are established with its integral representation. Moreover, various new sharp bounds for the q-digamma and q-trigamma functions are derived. In particular, sharp bounds for the q-analogue of harmonic numbers are established as a consequence. The results obtained in this work are new. The limiting case q -> 1, in the results obtained in this paper leads to the results for a class of Bernstein functions and logarithmically completely monotonic function involving Euler's gamma function and dilogarithm, which are also new in the literature.

Automated summary

The main goal of this paper is to introduce new classes of logarithmically completely monotonic functions involving q-gamma function. Applications of these classes are made in establishing new classes of Bernstein functions related to the q-gamma function and dilogarithm, as well as deriving various new sharp bounds for the q-digamma and q-trigamma functions. The results obtained in this work are new and the limiting case q -> 1 leads to new results for a class of Bernstein functions and logarithmically completely monotonic function involving Euler's gamma function and dilogarithm, which are also new in the literature.