This paper is motivated by the recent progress on the Hermite-Hadamard inequality for convex functions defined on the bounded closed interval, obtained by Z. Pavić [Z. Pavić, Improvements of the Hermite-Hadamard inequality, J. Inequal. Appl. 2015 (2015), Article ID 222]. As a generalization, we obtained a new refinement of the Hermite-Hadamard inequality for co-ordinated convex functions defined on the rectangle.
In the present paper, a new class of generalized beta (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)-preinvex functions are given. Moreover, some generalizations of Hermite-Hadamard type inequalities for generalized beta (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)-preinvex functions, Aust. J. Math. Anal. Appl. 13 (2016), no. 1, Article ID 16]. At the end, some applications to special means are given.
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