Hadamard And Simpson Type Inequalities For Several Classes Of Generalized Convex Functions And Their Applications

In this paper, based on the properties of some generalized convex functions and H?lder、Power-mean integral inequalities, we study Hadamard and Simpson inequalities for several classes of generalized convex functions and their applications. And we obtain some new inequalities about these generalized convex functions. When the parameters take different values, some new Hadamard or Simpson inequalities can be obtained, which extend and improve the previously known results.The chapter 1 of this article is introduction, we mainly introduce the research background of convex function, and the development situation of Hadamard and Simpson type inequality for convex function.In chapter 2, we mainly study Simpson type inequality for(a, m)- GA-convex function. Firstly, we introduce some inequalities for GA-convex function, and give the related definition and Lemmas. Then we establish some Simpson type inequalities for(a, m)- GA-convex function. Finally, these inequalities are applied to estimate error between the mean and the logarithmic mean.In chapter 3, with the help of previous definitions of extended s-convex and m-convex function, we introduce the concept of extended (s,m)-preinvex function, and estainblish Hadamard-Simpson type integral identity with parameters. Based on this integral identity, we combine the property of extended (s,m)-preinvex function and H?lder、Power-mean integral inequality, then obtain some new Hadamard-Simpson type inequalities for extended (s,m)_φ-preinvex function.Chapter 4 makes an extension and improvement on the basis of chapter 3. We propose the concept of extended (s,m)_φ-preinvex function, and estainblish Simpson type integral identity with parameters. Then we use the same mathematical methods of chapter 3, and some new Simpson type inequalities for extended (s,m)_φ-preinvex functions are obtained,these new inequalities improve the previously known results.In the last chapter, we summarize the whole work of the thesis and point out the developing directions that needs to be further explored in the future.