Real-variables characterization of generalized Stieltjes functions

We obtain a characterization of generalized Stieltjes functions of any order λ>0λ>0 in terms of inequalities for their derivatives on (0,∞)(0,∞). When λ=1λ=1, this provides a new and simple proof of a characterization of Stieltjes functions first obtained by Widder in 1938.