A conjecture concerning a completely monotonic function

On the basis of a sequence of numerical computations, a conjecture is presented regarding the class of functions H.xI a/ D exp.a/􀀀.1Ca=x/x, and the open problem of determining the values of a for which the functions are completely monotonic with respect to x. The critical value of a is determined here to sufficient accuracy to show that it is not a simple symbolic quantity.