| Fractal is a branch subject of the nonlinear field.It can give a good inter-pretation for some of the phenomena which can not be described in Euclidean space.So it is widely used in geophysics,image compression,information dissemination and many other fields.Local fractional calculus is a kind of cal-culus which is established on the fractal sets.It not only can effectively solve the continuous but non-differentiable problems in fractal geometry,but also can be used to describe the dynamic behavior of fractal space.The convex function is an important function in the field of mathematical analysis,and it is widely used to judge the extreme value of functions,describe the image of the functions,prove the inequalities and so on.Therefore,it is necessary to introduce the convex function and study its related properties on the fractal sets.In this paper,we will introduce the generalized s-convex func-tions and study related properties on the fractal sets.And on this basis,we will establish some inequalities related to the generalized s-convex function too.The structure arrangement of this paper is as follows:In Chapter one,the research background of this paper,the concept of real linear fractal sets and related operation properties,the definitions and related properties of local fractional calculus are introduced.In Chapter two,we introduce two kinds of generalized s-convex functions on the fractal sets,and study the related properties.And we also present some applications for the generalized s-convex functions.In Chapter three,some inequalities related to the generalized s-convex functions on the fractal sets are established.Firstly,we establish the generalized Hermite-Hadamard’ s inequality related to the generalized s-convex function-s on the fractal sets.Then,we study the generalized Hermite-Hadamard’ s inequality for the local differentiable functions by using the properties of the generalized s-convex functions and the generalized Holder’ s inequality. |
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