  We consider two operations on the Mittag-Leffler function which cancel the
exponential term in the expansion at infinity, and generate a completely
monotonic function. The first one is the action of a certain
differential-difference operator, and leads to a characterization via some
necktie domain. The second one is the subtraction of the exponential term
itself multiplied by an incomplete Gamma function. These results extend
previous works by various authors.
