Logarithmically complete monotonicity properties and characterizations of the gamma function

Abstract

n this paper, the logarithmically complete monotonic properties of the functions $ \prod_{i=1}^{n}\frac{\Gamma(x-a_i)}{\Gamma(x-b_i)} $ ,$ \Gamma(x)^{\alpha}\Gamma\Big(x-\sum_{i=1}^n a_i\Big)/\prod_{i=1}^{n}\Gamma(x-a_i) $, and $ x^r(e/x)^x \Gamma(x) $ are obtained. Some characterizations of the gamma function are deduced.