Approximations for certain hyperbolic functions by partial sums of their Taylor series and completely monotonic functions related to gamma function

In this paper, we establish some lower and upper bounds for certain hyperbolic functions in terms of partial sums of their Taylor series. These allow us to present two completely monotonic functions involving gamma function. As consequences, sharp Burnside type approximations for gamma function, sharp Detemple–Wang type approximations for psi function, and sharp bounds for polygamma functions are deduced, which generalize and improve some known results.