Research Of Local Fractional Midpoint-type Integral Inequalities And Their Applications

In this thesis,by using the local fractional theory,we generalize the existing generalized convex functions and use some classical integral inequalities to further establish the midpoint-type integral inequalities of the first and second order differentiable functions on fractal space.Also,some fractal results are obtained from the application perspective.In Chapter 1,the research background,purpose and significance of the topic are introduced,including the research status of convex functions and local fractional calculus both at home and aboard,as well as the relevant results of the H(?)lder integral inequalities and the power mean integral inequalities.In Chapter 2,firstly,an improved version of the power-mean integral inequality on fractal sets is established.Then,the concept of the generalized(s,P)-convex functions is proposed and its related properties are analyzed.The Hermite-Hadamard’s integral inequalities related to the generalized(s,P)-convex functions are also established.In addition,according to the midpoint-type integral identities on fractal sets,we derive some midpoint-type integral inequalities for functions whose first-order derivatives in absolute value belong to the generalized(s,P)-convex functions.As applications,three outcomes regarding special means,numerical integration and probability density functions are acquired in the end.In Chapter 3,in order to study the midpoint-type integral inequalities involving second-order differentiable functions on fractal sets,firstly,the generalized left and right sides integral operators on fractal sets are proposed,and the Hermite-Hadamard type integral inequalities related to them are established.Using the introduced integral operators,we construct a midpoint-type integral identity on fractal sets involving second-order differentiable functions.And based on them,five fractal integral inequalities are acquired.Finally,some fractal results for special means,moments of random variables,and wave equations are obtained.In Chapter 4,the main work of the paper is summarized,and some directions for further research are proposed.