  In this expository and survey paper, along one of main lines of bounding the
ratio of two gamma functions, we look back and analyse some inequalities, the
complete monotonicity of several functions involving ratios of two gamma or
$q$-gamma functions, the logarithmically complete monotonicity of a function
involving the ratio of two gamma functions, some new bounds for the ratio of
two gamma functions and divided differences of polygamma functions, and related
monotonicity results.
