At first, we recall the q-operators in the context of q-calculus and by examining these operators we will introduce new definitions of the partial q-operators. Then, we investigate some new refinements inequalities of Hermite–Hadamard ($H-H$ H − H ) type on the coordinated convex functions involving the new defined partial q-operators. From our main results, we establish several specific inequalities and we point out the existing results which had already been obtained in the literature.

In this paper, we establish the error bounds of different versions of mid-point type inequalities. At first, we prove two identities for fractional integrals involving the extended generalized Mittag-Leffler function and generalized exponential fractional integrals, and then we establish the corresponding error bound inequalities. Furthermore, we find a generalized inequality for error bound inequalities using a generalized identity. Also, we find some inequalities which formulate all error bound inequalities for various versions of Hadamard inequality. Finally, we present some examples of the central moment of a random variable.

In this paper, we establish a generic identity for (p, q)-differentiable functions involving multi-parameter. Using this identity as an auxiliary result, we derive some new post-quantum fundamental integral inequalities, including Simpson-type and Hermite-Hadamard-type pertaining n-polynomial convex functions. Also, by considering the boundedness and the Lipschitz condition of (p, q)-differentiable functions further results are given. Finally, in order to show the significance of our results, we present two illustrative examples and some applications to special means for different positive real numbers.

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The present article establishes some results on Hermite-Hadamard type inequalities of coordinated (p 1 , h 1 )-(p 2 , h 2 )-convex functions by using Katugampola fractional integrals. The results are in generalized form and deduced for various special cases like coordinated pq-convex functions, coordinated (p, s)-convex functions, coordinated s-convex functions and classical case of coordinated convex functions.