New estimates of Gauss-Jacobi and trapezium type inequalities for strongly $(h_{1},h_{2})$-preinvex mappings via general fractional integrals

Abstract

In this paper, authors discover two interesting identities regarding Gauss--Jacobi and trapezium type integral inequalities. By using the first lemma as an auxiliary result, some new bounds with respect to Gauss--Jacobi type integral inequalities for a new class of functions called strongly $(h_{1},h_{2})$--preinvex of order $sigma>0$ with modulus $mu>0$ via general fractional integrals are established. Also, using the second lemma, some new estimates with respect to trapezium type integral inequalities for strongly $(h_{1},h_{2})$--preinvex functions of order $sigma>0$ with modulus $mu>0$ via general fractional integrals are obtained. It is pointed out that some new special cases can be deduced from main results. Some applications to special means for different real numbers and new approximation error estimates for the trapezoidal are provided as well. These results give us the generalizations of some previous known results. The ideas and techniques of this paper may stimulate further research in the fascinating field of inequalities.