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In the paper, by 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 (1) verify decreasing monotonicity of a ratio between three derivatives of a function involving trigamma function; (2) find 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 the first author and generalize corresponding known conclusions.