Bounds for the ratio of two gamma functions---From Wendel's and related inequalities to logarithmically completely monotonic functions

Abstract

In the survey paper, along one of several main lines of bounding the ratio of two gamma functions, the authors retrospect and analyse Wendel's double inequality, Kazarinoff's refinement of Wallis' formula, Watson's monotonicity, Gautschi's double inequality, Kershaw's first double inequality, and the (logarithmically) complete monotonicity results of functions involving ratios of two gamma or $q$-gamma functions obtained by Bustoz, Ismail, Lorch, Muldoon, and other mathematicians.