Research On Fractional Simpson-type Integral Inequalities And Their Applications

In this paper,we study fractional Simpson-type integral inequalities by using the related properties of some convex functions,H(?)lder integral inequality,power-mean integral inequality and their improved integral inequalities,and apply these results to special means of real numbers,error estimate for Simpson-like quadrature formula,and continuous random variables,and so on.In Chapter 1,the concepts of Riemann-Liouville fractional integrals,local fractional calculus,quantum integrals and the related results of H(?)lder integral inequality are introduced.And the research status of convex functions and Simpson-type integral inequalities at home and abroad is introduced.In Chapter 2,we establish a Simpson-like type identity and several Simpson-like type inequalities for Riemann-Liouville fractional integrals.As applications,we apply the obtained results to special means of real numbers,an error estimate for Simpson-like quadrature formula,and q-digamma function,respectively.In Chapter 3,we investigate some inequalities for generalized s-convex functions on fractal sets(0 1)?R<? ?,which are related to Simpson-like inequalities.For this purpose,an improved version of H(?)lder integral inequality and a Simpson-like identity on fractal sets are established,based on which we give some estimation-type results involving Simpson-like inequalities for the first-order differentiable mappings.As applications with respect to local fractional integrals,we propose certain inequalities for generalized probability density mappings and ?-type special means.In Chapter 4,we study the parameterized inequalities of Hadamard-Simpson type for quantum integrals.By employing a multi-parameter quantum integral identity,we establish novel inequalities for a class of q-differentiable mappings,which are related to s-(?,m)-convex mappings.Moreover,we acquire estimation-type results by considering the boundedness and the Lipschitz condition.As applications,we present two illustrative examples and several quantum integral inequalities for the special means.In Chapter 5,we summarize the main contents of this paper and provide some further research directions.