The present article establishes some results on Hermite-Hadamard type inequalities of coordinated (p1,h1)-(p2,h2)-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.

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 give some correct quantum type Simpson’s inequalities via the application of q-Hölder’s inequality. The inequalities of this study are compatible with famous Simpson’s $1/8$ and $3/8$ quadrature rules for four and six panels, respectively. Several special cases from our results are discussed in detail. A counter example is presented to explain the limitation of Hölder’s inequality in the quantum framework.

Convex bodies are symmetric in nature. Between the two variables of symmetry and convexity, a correlation connection is also perceptible. Due to the interchangeable analogous properties, the application on either of them has been practicable in these modern years. The current analysis sheds insight on a general new identity involving a number of parameters for a twice partial quantum differentiable function. We find several unique quantum integral inequalities by using the new identity and a twice partial quantum differentiable function whose absolute value is coordinated convex. In addition, we present several novel and interesting error estimation-like results related to the well-known quantum Hermite–Hadamard inequality. Some examples are provided at the end to support and demonstrate the effectiveness of the new outcomes.

This study reveals the error analysis of Hermite-Hadamard inequality for coordinated convexity related to post-quantum integrals. At first, we establish a multi-parameter identity pertaining coordinated convexity via post-quantum integrals followed by new integrals to construct our main results. By utilizing this generic identity, we analyze the error estimates of classical Hermite-Hadamard inequality in the post-quantum context. Application of power mean inequality enables the refinement of bounds involved and extends it further. We make use of graphical representation with the help of concrete examples to verify the validity of presented results. In the end, to focus on usability and significance of the results, two applications through polynomial functions are produced.