On some k-fractional integral inequalities of~Hermite--Hadamard type for twice differentiable generalized beta (r, g)-preinvex functions

Abstract

Abstract In the present paper, a new class of generalized beta ( r , g ) {(r,g)} -preinvex functions is introduced and some new integral inequalities for the left-hand side of Gauss–Jacobi type quadrature formula involving generalized beta ( r , g ) {(r,g)} -preinvex functions are given. Moreover, some generalizations of Hermite–Hadamard type inequalities for generalized beta ( r , g ) {(r,g)} -preinvex functions that are twice differentiable via k-fractional integrals are established. These general inequalities give us some new estimates for Hermite–Hadamard type k-fractional integral inequalities and also extend some results appeared in the literature; see [A. Kashuri and R. Liko, Ostrowski type fractional integral inequalities for generalized ( s , m , φ ) (s,m,\varphi) -preinvex functions, Aust. J. Math. Anal. Appl. 13 2016, 1, Article ID 16]. At the end, some applications to special means are given.