Science and Education Publishing Turkish Journal of Analysis and Number Theory 2 5 2014 09 09 Bounds for the Ratio of Two Gamma Functions: from Gautschi's and Kershaw's Inequalities to Complete Monotonicity 152 164 EN Feng Qi Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, China TJANT2014251 10.12691/tjant-2-5-1 2014 07 23 2014 08 04 2014 09 09 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.