In the paper, the authors find necessary and sufficient conditions for a difference between the exponential function αeβ/t, α, β > 0, and the trigamma function ψ
(t) to be completely monotonic on (0,∞).
While proving the complete onotonicity, the authors discover some properties related to the first order modified Bessel function of the first kind I1, including inequalities, monotonicity, unimodality, and convexity.