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ABSTRACT. The principal result here is that each positive zero of the function

A function f(x), such as e-X or (x-a)-~, y~0, IS said to be completely monotonic over (a, b), where -00~a < b~+ 00, if(1)If, in addition, f(x) is continuous at x = a, then it is called completely monotonic over [a, b), with similar definitions for (a, b] and [a, b]. A function is absolutely monotonic if all its derivatives are non-negative. A detailed study of these concepts can be found, for example, in [11, Chapter IV].Our chief concern here is for the standard case in which a = 0, b = + 00.

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