Study On Hermite-Hadamard Type Inequalities Based On(p,q)-Integration

Quantum calculus,also known as q-calculus,is a kind of calculus based on the idea of finite difference re-scaling and without limits.It is widely used in many fields of mathematics and physics,such as number theory,orthogonal polynomials,combinations,fundamental hyper-geometric functions,mechanics,quantum and relativity.q-calculus was first proposed by Euler in the 18th century,and Jackson gave the concept of q-definite integral in 1910,which began to research systematically on q-calculus.With the development and progress of science and technology,people probe into mathematics,and the theory of quantum calculus has also developed rapidly.Tariboon and Ntouyas introduced qa-derivative and qa-integral on finite real intervals and gave the related properties in 2014;Tun? and G?v gave(p,q)a-derivative and(p,q)a-integral in 2016;Bermudo et al.defined qb-derivative and qb-integral and Ali et al.defined(p,q)b-derivative and(p,q)b-integral in 2020.Hereafter,the research on q-derivative,q-integral,(p,q)-derivative and(p,q)-integral has been launched.In this paper,based on(p,q)-integral and(p,q)-derivative,we study the Hermite-Hadamard type(p,q)a-integral inequalities and(p,q)b-integral inequalities for convex function,h-convex function and modified h-convex function,specifically as follows:(1)Using(p,q)a-derivative,(p,q)a-integral and Jensen’s inequality,Hermite-Hadamard type(p,q)a-integral inequality for continuous convex function is generalized and refined,and some new(p,q)a-integral inequalities are obtained.(2)The(p,q)b-derivative and(p,q)b-integral are studied and some properties are obtained.Hermite-Hadamard type(p,q)b-integral inequality for continuous convex function are established.(3)Hermite-Hadamard(p,q)b-integral inequalities of h-convex function and modified h-convex function are established.