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 $e^{1/t}-\psi'(t)$ 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. 
Feng Qi and Shu-Hong Wang
math.CA math.CV