Completely Monotonic and Related Functions

We study completely monotonic and related functions whose first N derivatives are definite on an interval. Let L N denote the class of functions defined by fgL Nm y1.kf k. t.G0, ;t)0, ;k, 0FkFN. For Nª` we write fgL; such functions are called completely monotonic on 0, `.. We prove in particular that the implication fgL N« ;a )1: f aw gL Nx, is true for 0FNF5, but false for NG6. In contrast, L. Lorch and D. J. Newman J. London Math. Soc. 2., 28, 1983, 31]45, Theorem 8. claimed this implication to be false for Ns5. Further we prove that the implication fgL« ;a )1: f aw gL Nx, is true for 0FNF6, but false for NG20.