Unified treatment of Gautschi-Kershaw type inequalities for the gamma function

Gautschi, Kershaw, Lorch, Laforgia and other authors gave several inequalities for the ratio Γ(x + 1)Γ(x + s) where, as usual, Γ denotes the gamma function. In this paper we give a unified treatment of all their results and prove, among other things, new inequalities for the above ratio, which involve the psi function. Inequalities for the ratio of two gamma functions are useful, for example, to deduce Bernstein-type inequalities for ultraspherical polynomials. We give an example of this type.