Integral Inequalities Based On N-Polynomial Convex Functions And Their Applications

In this thesis,based on the n-polynomial convex function,some classical inequalities are used to study some related properties and integral inequalities of generalized n-polynomial convexity in fractal space,and some classification results are obtained from the point of view of application.In addition,we generalize the n-polynomial convex functions and sub-b-convex functions on the real number set,and establish a new class of convex functions,called sub-bn-polynomial convex functions,and discuss the related properties and optimization conditions of the sub-b-n-polynomial convex functions.In Chapter 1,the research background and research significance of this topic,the research status at home and abroad,the theory of local fractional calculus,and some important inequalities related to convexity are introduced.In Chapter 2,a new generalized convex function is defined on the fractal space,which is called generalized n-polynomial convex function.Firstly,the properties of generalized npolynomial convex function are studied,and two Hermite-Hadamard integral inequalities of this function are established in the framework of fractal space.Secondly,according to the identity with parameters,for the function whose absolute value of the first derivative is generalized n-polynomial convex,the correlative integral inequalities are given.As applications,three inequalities on special mean value,numerical integration and probability density function are obtained on the basis of local fractional calculus.In Chapter 3,a generalized integral identity about twice differentiable mapping is proposed.On this basis,for the function whose absolute value of the second derivative is generalized npolynomial convex,some new integral inequalities in fractal space are derived.Finally,a series of fractal results related to special mean value,midpoint formula,moments of random variable and the wave equation on Cantor set are obtained from the perspective of application.In Chapter 4,the concepts of sub-b-n-polynomial convex sets and sub-b-n-polynomial convex functions are given in real space,and some properties of sub-b-n-polynomial convex functions in general cases and differentiable cases are discussed.For the optimization of subb-n-polynomial convexity,sufficient conditions of optimality under unconstrained and inequality constraints are given respectively.In Chapter 5,we summarize the main contents of this paper and provide some further research directions.