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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 author (1) presents the decreasing monotonicity of a ratio constituted via three derivatives of a function involving trigamma function; (2) discovers necessary and sufficient conditions for a function constituted via three derivatives of a function involving trigamma function to be completely monotonic. These results conform previous guesses posed by the author.

In the paper, the authors establish some Hermite-Hadamard type inequalities for the product of two (α, m)-convex functions.

Abstract. In the paper, the authors discuss the Schur m -power convexity on (0,∞) × (0,∞) for ratios of some famous means, such as the arithmetic, geometric, harmonic, root-square means, and the like, and obtain some inequalities related to ratios of means. Mathematics subject classification (2010): Primary 26B25; Secondary 26E60, 26D20.

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