Inequalities and monotonicity properties for the gamma function

We present a survey of some recent results on the ratio of two gamma functions and prove the following inequalities for the product:12Γ(1+x1+x2)⩽Γ(1+x1)Γ(1+x2)<Γ(1+x1+x2)for x1,x2>0 and x1·x2=1. Finally, we study the possible extension to k variables of the lower bound above.