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 main objective of this paper is to introduce a new class of convexity called left-right–bi-convex fuzzy interval-valued functions. We study this class from the perspective of fractional Hermite–Hadamard inequalities, involving a new fractional integral called the left-right–AB fractional integral. We discuss several special cases that demonstrate that our results are quite unifying. We provide non-trivial numerical examples regarding special means for positive real numbers in order to check the validity of our outcomes.

The intent of the current study is to explore convex stochastic processes within a broader context. We introduce the concept of unified stochastic processes to analyze both convex and non-convex stochastic processes simultaneously. We employ weighted quasi-mean, non-negative mapping $\gamma$, and center-radius ordering relations to establish a class of extended $cr$-interval-valued convex stochastic processes. This class yields a combination of innovative convex and non-convex stochastic processes. We characterize our class by illustrating its relationships with other classes as well as certain key attributes and sufficient conditions for this class of processes. Additionally, leveraging Riemann–Liouville stochastic fractional operators and our proposed class, we prove parametric fractional variants of Jensen’s inequality, Hermite–Hadamard’s inequality, Fejer’s inequality, and product Hermite–Hadamard’s like inequality. We establish an interesting relation between means by means of Hermite–Hadamard’s inequality. We utilize the numerical and graphical approaches to showcase the significance and effectiveness of primary findings. Also, the proposed results are powerful tools to evaluate the bounds for stochastic Riemann–Liouville fractional operators in different scenarios for a larger space of processes.

The theory of integral inequality has gained considerable attention due to its influential impact on several fields of mathematics and applied sciences. Over the years, numerous refinements, generalizations, and extensions of convexity have been explored to achieve more precise variants of already established results. The principal idea of this article is to establish some interval-valued integral inequalities of the Hermite–Hadamard type in the fractional domain. First, we propose the idea of generalized interval-valued convexity with respect to the continuous monotonic functions ⋎, bifunction $\zeta$, and based on the containment ordering relation, which is termed as $(⋎,\mathfrak{h})$ pre-invex functions. This class is innovative due to its generic characteristics. We generate numerous known and new classes of convexity by considering various values for ⋎ and $\mathfrak{h}$. Moreover, we use the notion of $(⋎,\mathfrak{h})$-pre-invexity and Atangana–Baleanu (AB) fractional operators to develop some fresh fractional variants of the Hermite–Hadamard (HH), Pachpatte, and Hermite–Hadamard–Fejer (HHF) types of inequalities. The outcomes obtained here are the most unified forms of existing results. We provide several specific cases, as well as a numerical and graphical study, to show the significance of the major results.