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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 andestablish integral representations of the remainder of the Laurent seriesexpansion of $e^{1/z}$, and derive an inequality which gives a lower bound forthe first order modified Bessel function of the first kind.

global journal of mathematical analysis

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