Abstract :
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.
