  In the paper, the authors verify the complete monotonicity of the difference
$e^{1/t}-\psi'(t)$ on $(0,\infty)$, compute the completely monotonic degree and
establish integral representations of the remainder of the Laurent series
expansion of $e^{1/z}$, and derive an inequality which gives a lower bound for
the first order modified Bessel function of the first kind.
