Quantum calculus provides a significant generalization of classical concepts and overcomes the limitations of classical calculus in tackling non-differentiable functions. Implementing the q-concepts to obtain fresh variants of classical outcomes is a very intriguing aspect of research in mathematical analysis. The objective of this article is to establish novel Milne-type integral inequalities through the utilization of the Mercer inequality for q-differentiable convex mappings. In order to accomplish this task, we begin by demonstrating a new quantum identity of the Milne type linked to left and right q derivatives. This serves as a supporting result for our primary findings. Our approach involves using the q-equality, well-known inequalities, and convex mappings to obtain new error bounds of the Milne–Mercer type. We also provide some special cases, numerical examples, and graphical analysis to evaluate the efficacy of our results. To the best of our knowledge, this is the first article to focus on quantum Milne–Mercer-type inequalities and we hope that our methods and findings inspire readers to conduct further investigation into this problem.

In recent years, the theory of convexity has influenced every field of mathematics due to its unique characteristics. Numerous generalizations, extensions, and refinements of convexity have been introduced, and one of them is set-valued convexity. Interval-valued convex mappings are a special type of set-valued maps. These have a close relationship with symmetry analysis. One of the important aspects of the relationship between convex and symmetric analysis is the ability to work on one field and apply its principles to another. In this paper, we introduce a novel class of interval-valued (I.V.) functions called $\mathcal{CR}$-$\gamma$-convex functions based on a non-negative mapping $\gamma$ and center-radius ordering relation. Due to its generic property, a set of new and known forms of convexity can be obtained. First, we derive new generalized discrete and integral forms of Jensen’s inequalities using $\mathcal{CR}$-$\gamma$-convex I.V. functions. We employ this definition and Riemann-Liouville fractional operators to develop new fractional versions of Hermite-Hadamard’s, Hermite-Hadamard-Fejer, and Pachpatte’s type integral inequalities. We examine various key properties of this class of functions by considering them as special cases. Finally, we support our findings with interesting examples and graphical representations.

The aim of this research is to explore fractional integral inequalities that involve interval-valued preinvex functions. Initially, a new set of fractional operators is introduced that uses the extended generalized Mittag-Leffler function $E_{\mu ,\alpha ,l}^{\gamma ,\delta ,k,c}(\tau ;p)$ as a kernel in the interval domain. Additionally, a new form of Atangana–Baleanu operator is defined using the same kernel, which unifies multiple existing integral operators. By varying the parameters in $E_{\mu ,\alpha ,l}^{\gamma ,\delta ,k,c}(\tau ;p)$, several new fractional operators are obtained. This study then utilizes the generalized AB integral operators and the preinvex interval-valued property of functions to establish new Hermite–Hadamard, Pachapatte, and Hermite–Hadamard–Fejer inequalities. The results are supported by numerical examples, graphical illustrations, and special cases.

In this paper, we propose two new hybrid methods for solving nonlinear equations, utilizing the advantages of classical methods (bisection, trisection, and modified false position), i.e., bisection-modified false position (Bi-MFP) and trisection-modified false position (Tri-MFP). We implemented the proposed algorithms for several benchmark problems. We discuss the graphical analysis of these problems with respect to the number of iterations and the average CPU time.