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

Yin Hong-Ping, Han Ling-Xiong, Qi Feng

Demonstratio Mathematica

2024

Vol. 57, nr 1

art. no. 20240041

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.

Increasing property and logarithmic convexity of functions involving Dirichlet lambda function

Qi Feng, Lim Dongkyu

Demonstratio Mathematica

2023

Vol. 56, nr 1

art. no. 20220243

In this article, with the help of an integral representation of the Dirichlet lambda function, by means of a monotonicity rule for the ratio of two integrals with a parameter, and by virtue of complete monotonicity and another property of an elementary function involving the exponential function, the authors find increasing property and logarithmic convexity of two functions containing the gamma function and the Dirichlet lambda function.