  We prove some new results and unify the proofs of old ones involving complete
monotonicity of expressions involving gamma and $q$-gamma functions, $0 < q <
1$. Each of these results implies the infinite divisibility of a related
probability measure. In a few cases, we are able to get simple monotonicity
without having complete monotonicity. All of the results lead to inequalities
for these functions. Many of these were motivated by the bounds in a 1959 paper
by Walter Gautschi. We show that some of the bounds can be extended to complex
arguments.
