 【摘要】Some new Hermite-Hadamard type's integral equations and inequalities are established. The results in  and  which refined the upper bound of distance between the middle and left of the typical Hermite-Hadamard's integral inequality are generalized.

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Chin. Quart. J. of Math. 2018, 33 (3): 278—286 Generalizations of Hermite-Hadamard Type Inequalities Involving S-convex Functions LIAN Tie-yan1 , TANG Wei2 (1.College of Light Industry Science and Engineering, Shaanxi University of Science & Technology, Xi’an 710021, P.R. China; 2.College of Electrical &Information Engineering, Shaanxi University of Science & Technology, Xi’an 710021, P.R. China) Abstract: Some new Hermite-Hadamard type’s integral equations and inequalities are es- tablished. The results in  and  which refined the upper bound of distance between the middle and left of the typical Hermite-Hadamard’s integral inequality are generalized. Key words: Hermite-Hadamard’s integral inequality; s-convex function; Hölder’s integral inequality 2000 MR Subject Classification: 26D15; 26A51 CLC number: O177.1 Document code: A Article ID: 1002–0462 (2018) 03–0278–09 §1. Introduction Let R be the set of real numbers, I ⊂ R, Io be the interior of I. The following is the definition of convex functions which is well known knowledge in mathematical literature : f : I → R is said to be convex on an interval I if the inequality f(λx + (1− λ)y) ≤ λf(x) + (1− λ)f(y) (1.1) holds for all x, y ∈ I and λ ∈ [0, 1]. For convex function, many equalities or inequalities have been established but the Hermite- Hadamard’s integral inequality is one of the most important, which is stated as follow : Received date: 2016-10-31 Foundation item: Supported by the key scientific and technolo