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 f epsilon L(N) reversible arrow (-1)(k)f((k))(t) greater than or equal to 0, For All(t) > 0, For All(k), 0 less than or equal to k less than or equal to N. For N --> infinity we write f epsilon L(N); such functions are called completely monotonic on (0, infinity). We prove in particular that the implication f epsilon L(N) --> [For All alpha > 1: f(alpha) epsilon L(N)], is true for 0 less than or equal to N less than or equal to 5, but false for N greater than or equal to 6. In contrast, L. Lorch and D. J. Newman (J. London Math. Sec. (2), 28, 1983, 31-45, Theorem 8) claimed this implication to be false for N = 5. Further we prove that the implication f epsilon L --> [For All alpha > 1: f(alpha) epsilon L(N)], is true for 0 less than or equal to N less than or equal to 6, but false for N greater than or equal to 20. (C) 1996 Academic Press, Inc.