Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function

Abstract

In this study, using convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein’s theorem for completely monotonic functions, and other analytic techniques, the authors verify decreasing property of a ratio between three derivatives of a function involving trigamma function and find the necessary and sufficient conditions for a function defined by three derivatives of a function involving trigamma function to be completely monotonic. These results confirm previous guesses posed by Qi and generalize the corresponding known conclusions.

Keywords

• decreasing monotonicity
• complete monotonicity
• completely monotonic function
• trigamma function
• derivative
• ratio
• convolution theorem
• inequality
• monotonicity rule
• laplace transform
• bernstein’s theorem
• exponential function guess
• 33b15
• 26a48
• 26d07
• 33b10
• 44a10
• 44a35