Inequalities for Real Powers of Completely Monotonic Functions

Let L N denote the class of functions defined by fgL Nm y1. k f k. t . G0, ;t)0, ;k, 0FkFN. For Nª` we write fgL. Functions in L are called completely monotonic on 0, `.. We derive several inequalities involving completely monotonic functions. In particular, we prove that the implication fgL« ;a )1: f aw gL N x, is true for 0FNF7, but false for NG13.