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 $e^{1/t}$ and the trigamma function $\psi'(t)$ on $(0,\infty)$.
Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions
Integral representations and complete monotonicity related to the remainder of Burnside's formula for the gamma function
Complete monotonicity of a family of functions involving the tri- and tetra-gamma functions
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