Complete Monotonicity of Functions Connected with the Exponential Function and Derivatives

Abstract

Some complete monotonicity results that the functions $\pm 1/\left(e^{\pm t}-1\right)$ are logarithmically completely monotonic, and that differences between consecutive derivatives of these two functions are completely monotonic, and that the ratios between consecutive derivatives of these two functions are decreasing on $\left(0, \infty \right)$ are discovered. As applications of these newly discovered results, some complete monotonicity results concerning the polylogarithm are found. Finally a conjecture on the complete monotonicity of the above-mentioned ratios is posed.