Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions

In the paper, the authors verify the complete monotonicity of the difference $e^{1\mathrm{/}t}-{\psi }^{\mathrm{\prime }}(t)$ on $(0,\mathrm{\infty })$, compute the completely monotonic degree and establish integral representations of the remainder of the Laurent series expansion of $e^{1\mathrm{/}z}$, and derive an inequality which gives a lower bound for the first order modified Bessel function of the first kind.