eng Science and Education Publishing Turkish Journal of Analysis and Number Theory 2014-09-09 2 5 152 164 10.12691/tjant-2-5-1 TJANT2014251 article Feng Qi qifeng618@gmail.com, qifeng618@hotmail.com, qifeng618@qq.com 1 Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, China In the expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, the author looks back and analyses 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. http://pubs.sciepub.com/tjant/2/5/1/tjant-2-5-1.pdf bound ratio of two gamma functions completely monotonic function logarithmically completely monotonic function divided difference gamma function q-gamma function psi function polygamma function inequality