New Inequalities for Completely Monotonic Functions

摘要

Let L(sup N) denote the class of functions defined by f (a member of) L(sup N) <--> (-1)(sup k)f(sup (k)(t) = or > 0, for all t > 0, for all k (a member of) (0, 1, ..., N). For N --> infinity we write f (a member of) L. Functions in L are called completely monotonic on (0, infinity). We prove that the implication f (a member of) L --> (left bracket) for all alpha > 1: f (sup alpha) (a member of) L(sup N) (right bracket) is true for N=8, leaving the cases N=9, 10, 11, and 12 to be open problems. We derive several new inequalities involving completely monotonic functions. (Copyright (c) 1995 by Faculty of Technical Mathematics and Informatics, Delft, The Netherlands.)