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 this paper, we introduce the Orlicz space corresponding to the Young function and, by virtue of the equivalent theorem between the modified K-functional and modulus of smoothness, establish the direct, inverse, and equivalent theorems for linear combination of the Jacobi weighted Baskakov-Kantorovich operators in the Orlicz spaces.
The main aim of this present note is to establish three new Hermite-Hadamard type integral inequalities for <i>r</i>-convex functions. The three new Hermite-Hadamard type integral inequalities for <i>r</i>-convex functions improve the result of original one by Hölder’s integral inequality, Stolarsky mean and convexity of function
In this paper, the authors introduce the Orlicz spaces corresponding to the Young function and, by virtue of the equivalent theorem between the modified K-functional and modulus of smoothness, establish the direct, inverse, and equivalent theorems for linear combinations of modified summation operators of integral type in the Orlicz spaces.
Utilizing some properties of multivariate Baskakov-Kantorovich operators and using K-functional and a decomposition technique, the authors find two equivalent theorems between the K-functional and modulus of smoothness, and obtain a direct theorem in the Orlicz spaces.
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