| Convexity is a classical concept,which plays an important basic and research tool in the fields of mathematics,economics,management and engineering technology.The convex function is closely related to inequality.In particular,the integral inequality of convexity has always played a vital role in many fields,such as mathematics,physics,chemistry and so on.With the rapid development of science and technology and the increasing complexity of research problems,the ability to recognize natural world has been continuously enhanced,and the fractional calculus theory has been developed rapidly.In 2011,the concept of Katugampola fractional integral was put forward,which extended the classical RiemannLiouville fractional integral and Hadamard fractional integral,and its properties and applications have attracted the research of many scholars.In this paper,based on Katugampola fractional integral,Hermite-Hadamard type inequality and Simpson type inequality of convex function and quasi-convex function are discussed respectively as following:(1)Hermite-Hadamard type inequalities of convex function are established for Katugampola fractional integral on the intervals from any point to the end point and from the end point to any point,respectively,and the corresponding identities are established.Some Hermite-Hadamard inequalities of differentiable convex functions are obtained for Katugampola fractional integral by using identities,convexity and differentiability of functions,and some classical inequalities.(2)Based on convex functions,Simpson type identities of differentiable functions is established for Katugampola fractional integral on the interval from any point to endpoint and from endpoint to any point,respectively.Simpson type inequalities are obtained for Katugampola fractional integral by using the established identities,the convexity and differentiability of functions.(3)Based on quasi-convex function and from another point of view,Katugampola fractional integral relation is established on the interval from the midpoint to the endpoint.The error estimation on the left side of Hermite-Hadamard type inequality is carried out.Some new Hermite-Hadamard type inequalities for Katugampola fractional integral are obtained. |
