Properties and integral inequalities of Hadamard- Simpson type for the generalized $(s,m)$-preinvex functions

Volume 9, Issue 5, pp 3112--3126 Publication Date: May 28, 2016
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Authors

Ting-Song Du - College of Science, China Three Gorges University, 443002, Yichang, P. R. China. - Hubei Province Key Laboratory of System Science in Metallurgical Process, Wuhan University of Science and Technology, 430081, Wuhan, P. R. China. Jia-Gen Liao - College of Science, China Three Gorges University, 443002, Yichang, P. R. China. Yu-Jiao Li - College of Science, China Three Gorges University, 443002, Yichang, P. R. China.

Abstract

The authors introduce the concepts of m-invex set, generalized $(s,m)$-preinvex function, and explicitly $(s,m)$-preinvex function, provide some properties for the newly introduced functions, and establish new Hadamard-Simpson type integral inequalities for a function of which the power of the absolute of the first derivative is generalized $(s,m)$-preinvex function. By taking different values of the parameters, Hadamardtype and Simpson-type integral inequalities can be deduced. Furthermore, inequalities obtained in special case present a refinement and improvement of previously known results.

Keywords

• Integral inequalities of Hadamard-Simpson type
• Hölder's inequality
• $(s،m)$-preinvex function.

•  26D15
•  26A51
•  26B12

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