In the article, sufficient and necessary conditions that a class of functions involving ratio of Euler's gamma functions and originating from Wendel's and Gautschi's inequalities are logarithmically completely monotonic are presented. From this, Wendel's, Gautschi's, Kershaw's, Laforgia's, Bustoz-Ismail's, Merkle's and Elezovic-Giordano-Pecaric's inequalities are refined, extended and sharpened, and a double inequality on the divided differences of the psi and polygamma functions is deduced straightforwardly.