Complete monotonicity of a difference between the exponential and trigamma functions

In the paper, by directly verifying an inequality which gives a lower bound for the first order modified Bessel function of the first kind, the authors supply a new proof for the complete monotonicity of a difference between the exponential function $e^{1\mathrm{/}t}$ and the trigamma function ${\psi }^{\mathrm{\prime }}(t)$ on $(0,\mathrm{\infty })$.