Extended Class of Symmetric Quantum Operators and Inequality-Preserving Unified ANN Framework

Symmetric quantum calculus offers a dynamic framework for investigating the various classes of functions. However, the symmetric quantum operators become inconclusive at certain points. To overcome the limitations of existing calculi, operators over finite intervals have been extensively explored. To develop a more general and applicable setup, we introduce the symmetric quantum derivative and integral operators governed by an arbitrary point. Furthermore, we discuss structural properties of the newly developed operators and special cases to relate to the existing literature. Then, by applying the concepts of general symmetric quantum operators, convexity, Lipschitzian property, and Korkine’s identity, we derive a new set of inequalities, including Hermite–Hadamard, Ostrowski, Hólder, Minkowski, and Gruss-type inequalities, respectively. The proposed inequalities are useful to derive the bounds of generalized symmetric quantum integrals. Furthermore, the newly developed operators can be applied to study the impulsive difference equations and their dynamics. Additionally, a feed-forward ANN model is established to approximate the analytic expressions involved in inequalities and to observe the consistency of integral bounds. The results of the ANN analysis suggest a significant agreement between analytical solutions and approximations. Lastly, we focus on an applicable analysis of our derived results. The generic nature of operators will lead to new developments in quantum calculus. The hybrid approach evolved in this study will bring new applicable insights to the mathematical analysis.